research article distortion of quasiregular mappings and...

Research ArticleDistortion of Quasiregular Mappings and Equivalent Norms onLipschitz-Type Spaces

Miodrag MateljeviT

Faculty of Mathematics University of Belgrade Studentski Trg 16 11000 Belgrade Serbia

Correspondence should be addressed to Miodrag Mateljevic miodragmatfbgacrs

Received 24 January 2014 Accepted 15 May 2014 Published 21 October 2014

Academic Editor Ljubomir B Ciric

Copyright copy 2014 Miodrag Mateljevic This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We prove a quasiconformal analogue of Koebersquos theorem related to the average Jacobian and use a normal family argument here toprove a quasiregular analogue of this result in certain domains in 119899-dimensional space As an applicationwe establish that Lipschitz-type properties are inherited by a quasiregular function from its modulo We also prove some results of Hardy-Littlewood type forLipschitz-type spaces in several dimensions give the characterization of Lipschitz-type spaces for quasiregular mappings by theaverage Jacobian and give a short review of the subject

1 Introduction

The Koebe distortion theorem gives a series of bounds for aunivalent function and its derivative A result of Hardy andLittlewood relates Holder continuity of analytic functions inthe unit disk with a bound on the derivative (we refer to thisresult shortly as HL-result)

Astala and Gehring [1] observed that for certain dis-tortion property of quasiconformal mappings the function119886119891 defined in Section 2 plays analogous role as |1198911015840| when119899 = 2 and 119891 is conformal and they establish quasiconformalversion of the well-known result due to Koebe cited here asLemma 4 and Hardy-Littlewood cited here as Lemma 5

In Section 2 we give a short proof of Lemma 4 usinga version with the average Jacobian 119869

119891instead of 119886119891 and

we also characterize bi-Lipschitz mappings with respect toquasihyperbolic metrics by Jacobian and the average Jaco-bian see Theorems 8 9 and 10 and Proposition 13 Gehringand Martio [2] extended HL-result to the class of uniformdomains and characterized the domains119863 with the propertythat functions which satisfy a local Lipschitz condition in 119863for some 120572 always satisfy the corresponding global conditionthere

The main result of the Nolder paper [3] generalizes aquasiconformal version of a theorem due to Astala and

Gehring [4 Theorems 19 and 317] (stated here as Lemma 5)to a quasiregular version (Lemma 33) involving a somewhatlarger class of moduli of continuity than 119905120572 0 lt 120572 lt 1

In the paper [5] several properties of a domain whichsatisfies the Hardy-Littlewood property with the inner lengthmetric are given and also some results on the Holdercontinuity are obtained

The fact that Lipschitz-type properties are sometimesinherited by an analytic function from its modulus wasfirst detected in [6] Later this property was considered fordifferent classes of functions and we will call shortly resultsof this type Dyk-type results Theorem 22 yields a simpleapproach to Dyk-type result (the part (ii1) see also [7])and estimate of the average Jacobian for quasiconformalmappings in space The characterization of Lipschitz-typespaces for quasiconformal mappings by the average Jacobianis established in Theorem 23 in space case and Theorem 24yields Dyk-type result for quasiregular mappings in planarcase

In Section 4 we establish quasiregular versions of thewell-known result due to Koebe Theorem 39 here and usethis result to obtain an extension of Dyakonovrsquos theoremfor quasiregular mappings in space (without Dyakonovrsquoshypothesis that it is a quasiregular local homeomorphism)Theorem 40 The characterization of Lipschitz-type spaces

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 895074 20 pageshttpdxdoiorg1011552014895074

2 Abstract and Applied Analysis

for quasiregular mappings by the average Jacobian is alsoestablished inTheorem 40

By R119899 = 119909 = (1199091 119909119899) 1199091 119909119899 isin R denotethe real vector space of dimension 119899 For a domain 119863 in R119899

with nonempty boundary we define the distance function119889 = 119889(119863) = dist(119863) by 119889(119909) = 119889(119909 120597119863) = dist(119863)(119909) =inf|119909minus119910| 119910 isin 120597119863 and if 119891maps119863 onto1198631015840 sub R119899 in somesettings it is convenient to use short notation 119889lowast = 119889119891(119909) for119889(119891(119909) 120597119863

1015840) It is clear that 119889(119909) = dist(119909 119863119888) where 119863119888 is

the complement of119863 in R119899Let 119866 be an open set in R119899 A mapping 119891 119866 rarr R119898

is differentiable at 119909 isin 119866 if there is a linear mapping 1198911015840(119909) R119899 rarr R119899 called the derivative of 119891 at 119909 such that

119891 (119909 + ℎ) minus 119891 (119909) = 1198911015840(119909) ℎ + |ℎ| 120576 (119909 ℎ) (1)

where 120576(119909 ℎ) rarr 0 as ℎ rarr 0 For a vector-valued function119891 119866 rarr R119899 where 119866 sub R119899 is a domain we define100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816= max|ℎ|=1

100381610038161003816100381610038161198911015840(119909) ℎ

10038161003816100381610038161003816 119897 (119891

1015840(119909)) = min

|ℎ|=1

100381610038161003816100381610038161198911015840(119909) ℎ

10038161003816100381610038161003816

(2)

when 119891 is differentiable at 119909 isin 119866In Section 3 we review some results from [7 8] For

example in [7] under some conditions concerning amajorant120596 we showed the following

Let 119891 isin 1198621(119863R119899) and let 120596 be a continuous majorant

such that 120596lowast(119905) = 120596(119905)119905 is nonincreasing for 119905 gt 0Assume 119891 satisfies the following property (which we call

Hardy-Littlewood (119888 120596)-property)

(HL (119888 120596)) 119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 119888120596 (119889 (119909))

119909 isin 119863 where 119889 (119909) = dist (119909119863119888) (3)

Then

(locΛ) 119891 isin locΛ 120596 (119866) (4)

If in addition 119891 is harmonic in 119863 or more generally119891 isin 119874119862

2(119863) then (HL(119888 120596)) is equivalent to (locΛ) If 119863

is a Λ 120596-extension domain then (HL(119888 120596)) is equivalent to119891 isin Λ 120596(119863)

In Section 3 we also consider Lipschitz-type spaces ofpluriharmonic mappings and extend some results from [9]

In Appendices A and B we discuss briefly distortion ofharmonic qc maps background of the subject and basicproperty of qr mappings respectively For more details onrelated qr mappings we refer the interested reader to [10]

2 Quasiconformal Analogue of KoebersquosTheorem and Applications

Throughout the paper we denote byΩ 119866 and119863 open subsetof R119899 119899 ge 1

Let119861119899(119909 119903) = 119911 isin R119899 |119911minus119909| lt 119903 119878119899minus1(119909 119903) = 120597119861119899(119909 119903)(abbreviated 119878(119909 119903)) and let B119899 119878119899minus1 stand for the unit balland the unit sphere in R119899 respectively Sometimes we writeD and T instead ofB2 and 120597D respectively For a domain119866 sub

R119899 let 120588 119866 rarr [0infin) be a continuous function We say that120588 is a weight function or a metric density if for every locallyrectifiable curve 120574 in 119866 the integral

119897120588 (120574) = int120574120588 (119909) 119889119904 (5)

exists In this case we call 119897120588(120574) the 120588-length of 120574 A metricdensity defines a metric 119889120588 119866 times 119866 rarr [0infin) as follows For119886 119887 isin 119866 let

119889120588 (119886 119887) = inf120574119897120588 (120574) (6)

where the infimum is taken over all locally rectifiable curvesin 119866 joining 119886 and 119887

For the modern mapping theory which also considersdimensions 119899 ge 3 we do not have a Riemann mappingtheorem and therefore it is natural to look for counterparts ofthe hyperbolicmetric So-called hyperbolic typemetrics havebeen the subject of many recent papers Perhaps the mostimportant metrics of these metrics are the quasihyperbolicmetric120581119866 and the distance ratiometric 119895119866 of a domain119866 sub R119899

(see [11 12]) The quasihyperbolic metric 120581 = 120581119866 of 119866 is aparticular case of the metric 119889120588 when 120588(119909) = 1119889(119909 120597119866) (see[11 12])

Given a subset 119864 of C119899 or R119899 a function 119891 119864 rarr C (ormore generally a mapping 119891 from 119864 into C119898 or R119898) is saidto belong to the Lipschitz space Λ 120596(119864) if there is a constant119871 = 119871(119891) = 119871(119891 119864) = 119871(119891 120596 119864) which we call Lipschitzconstant such that

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le 119871120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (7)

for all 119909 119910 isin 119864 The norm 119891Λ120596(119864) is defined as the smallest

119871 in (7)There has been much work on Lipschitz-type properties

of quasiconformal mappings This topic was treated amongother places in [1ndash5 7 13ndash22]

As in most of those papers we will currently restrictourselves to the simplest majorants 120596120572(119905) = 119905

120572(0 lt 120572 le 1)

The classes Λ 120596(119864) with 120596 = 120596120572 will be denoted by Λ120572 (or byLip(120572 119864) = Lip(120572 119871 119864)) 119871(119891) and 120572 are called respectivelyLipschitz constant and exponent (of 119891 on 119864) We say that adomain 119866 sub 119877

119899 is uniform if there are constants 119886 and 119887 suchthat each pair of points 1199091 1199092 isin 119866 can be joined by rectifiablearc 120574 in 119866 for which

119897 (120574) le 11988610038161003816100381610038161199091 minus 1199092

1003816100381610038161003816

min 119897 (120574119895) le 119887119889 (119909 120597119866)(8)

for each 119909 isin 120574 here 119897(120574) denotes the length of 120574 and 1205741 1205742the components of 120574 119909 We define 119862(119866) = max119886 119887The smallest 119862(119866) for which the previous inequalities holdis called the uniformity constant of 119866 and we denote it by119888lowast= 119888

lowast(119866) Following [2 17] we say that a function119891 belongs

to the local Lipschitz space locΛ 120596(119866) if (7) holds with a fixed119862 gt 0 whenever 119909 isin 119866 and |119910 minus 119909| lt (12)119889(119909 120597119866) We saythat 119866 is a Λ 120596-extension domain if Λ 120596(119866) = locΛ 120596(119866) In

Abstract and Applied Analysis 3

particular if 120596 = 120596120572 we say that 119866 is a Λ120572-extension domainthis class includes the uniform domains mentioned above

Suppose that Γ is a curve family inR119899We denote byF(Γ)

the family whose elements are nonnegative Borel-measurablefunctions 120588 which satisfy the condition int

120574120588(119909)|119889119909| ge 1 for

every locally rectifiable curve 120574 isin Γ where 119889119904 = |119889119909| denotesthe arc length element For 119901 ge 1 with the notation

119860119901 (120588) = intR119899

1003816100381610038161003816120588 (119909)

1003816100381610038161003816

119901119889119881 (119909) (9)

where 119889119881(119909) = 119889119909 denotes the Euclidean volume element11988911990911198891199092 sdot sdot sdot 119889119909119899 we define the 119901-modulus of Γ by

119872119901 (Γ) = inf 119860119901 (120588) 120588 isin F (Γ) (10)

Wewill denote119872119899(Γ) simply by119872(Γ) and call it themod-ulus of Γ

Suppose that 119891 Ω rarr Ωlowast is a homeomorphism

Consider a path family Γ inΩ and its image family Γlowast = 119891∘120574 120574 isin Γ We introduce the quantities

119870119868 (119891) = sup119872(Γ

lowast)

119872 (Γ)

119870119874 (119891) = sup 119872(Γ)

119872 (Γlowast)

(11)

where the suprema are taken over all path families Γ inΩ suchthat119872(Γ) and119872(Γ

lowast) are not simultaneously 0 orinfin

Definition 1 Suppose that 119891 Ω rarr Ωlowast is a homeomor-

phism we call 119870119868(119891) the inner dilatation and 119870119874(119891) theouter dilatation of 119891 The maximal dilatation of 119891 is 119870(119891) =max119870119874(119891) 119870119868(119891) If 119870(119891) le 119870 lt infin we say 119891 is 119870-quasiconformal (abbreviated qc)

Suppose that 119891 Ω rarr Ωlowast is a homeomorphism and

119909 isin Ω 119909 = infin and 119891(119909) = infinFor each 119903 gt 0 such that 119878(119909 119903) sub Ω we set 119871(119909 119891 119903) =

max|119910minus119909|=119903|119891(119910) minus 119891(119909)| 119897(119909 119891 119903) = min|119910minus119909|=119903|119891(119910) minus 119891(119909)|

Definition 2 The linear dilatation of 119891 at 119909 is

119867(119909 119891) = lim sup119903rarr0

119871 (119909 119891 119903)

119897 (119909 119891 119903)

(12)

Theorem 3 (the metric definition of quasiconformality) Ahomeomorphism 119891 Ω rarr Ω

lowast is qc if and only if 119867(119909 119891) isbounded on Ω

Let Ω be a domain in 119877119899 and let 119891 Ω rarr 119877119899 be con-

tinuous We say that 119891 is quasiregular (abbreviated qr) if

(1) 119891 belongs to Sobolev space1198821198991loc(Ω)

(2) there exists 119870 1 le 119870 lt infin such that100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119869119891 (119909) ae (13)

The smallest119870 in (13) is called the outer dilatation119870119874(119891)A qr mapping is a qc if and only if it is a homeomorphismFirst we need Gehringrsquos result on the distortion property ofqc (see [23 page 383] [24 page 63])

GehringrsquosTheorem For every119870 ge 1 and 119899 ge 2 there exists afunction 120579119899119870 (0 1) rarr R with the following properties

(1) 120579119899119870 is increasing(2) lim119903rarr0120579

119899119870(119903) = 0

(3) lim119903rarr1120579119899119870(119903) = infin

(4) Let Ω and Ω1015840 be proper subdomains of R119899 and let 119891 Ω rarr Ω

1015840 be a 119870-qc If 119909 and 119910 are points in Ω suchthat 0 lt |119910 minus 119909| lt 119889(119909 120597Ω) then

1003816100381610038161003816119891 (119910) minus 119891 (119909)

1003816100381610038161003816

119889 (119891 (119909) 120597Ω1015840)

le 120579119899119870 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816

119889 (119909 120597Ω)

) (14)

Introduce the quantity mentioned in the introduction

119886119891 (119909) = 119886119891119866 (119909) = exp( 1

11989910038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

log 119869119891 (119911) 119889119911) 119909 isin 119866

(15)

associated with a quasiconformal mapping 119891 119866 rarr 119891(119866) sub

R119899 here 119869119891 is the Jacobian of 119891 while B119909 = B119909119866 stands forthe ball 119861(119909 119889(119909 120597119866)) and |B119909| for its volume

Lemma 4 (see [4]) Suppose that119866 and1198661015840 are domains in 119877119899If 119891 119866 rarr 119866

1015840 is 119870-quasiconformal then

1

119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

le 119886119891119866 (119909) le 119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

119909 isin 119866

(16)

where 119888 is a constant which depends only on 119870 and 119899

Set 1205720 = 120572119899119870 = 119870

1(1minus119899)

Lemma 5 (see [4]) Suppose that119863 is a uniform domain in 119877119899and that 120572 and 119898 are constants with 0 lt 120572 le 1 and 119898 ge 0 If119891 is 119870-quasiconformal in119863 with 119891(119863) sub 119877119899 and if

119886119891119863 (119909) le 119898119889(119909 120597119863)120572minus1 for 119909 isin 119863 (17)

then 119891 has a continuous extension to 119863 infin and1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119863))

120572 (18)

for 1199091 1199092 isin 119863 infin where the constant 119862 = 119888(119863) dependsonly on 119870 119899 119886 119898 and the uniformity constant 119888lowast = 119888

lowast(119863)

for 119863 In the case 120572 le 1205720 (18) can be replaced by the strongerconclusion that1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(19)

Example 6 The mapping 119891(119909) = |119909|119886minus1

119909 119886 = 1198701(1minus119899) is 119870-

qc with 119886119891 bounded in the unit ball Hence 119891 satisfies thehypothesis of Lemma 5 with 120572 = 1 Since |119891(119909) minus 119891(0)| le

|119909 minus 0|119886= |119909|

119886 we see that when 119870 gt 1 that is 119886 lt 1 = 120572the conclusion (18) in Lemma 5 cannot be replaced by thestronger assertion |119891(1199091) minus 119891(1199092)| le 119888119898|1199091 minus 1199092|


LetΩ isin R119899 andR+ = [0infin) and 119891 119892 Ω rarr R+ If thereis a positive constant 119888 such that 119891(119909) le 119888119892(119909) 119909 isin Ω wewrite 119891 ⪯ 119892 onΩ If there is a positive constant 119888 such that

1

119888

119892 (119909) le 119891 (119909) le 119888119892 (119909) 119909 isin Ω (20)

we write 119891 asymp 119892 (or 119891 ≍ 119892) onΩLet119866 sub R2 be a domain and let119891 119866 rarr R2119891 = (1198911 1198912)

be a harmonic mapping This means that 119891 is a map from 119866

into R2 and both 1198911 and 1198912 are harmonic functions that issolutions of the two-dimensional Laplace equation

Δ119906 = 0 (21)

The above definition of a harmonic mapping extends in anatural way to the case of vector-valued mappings 119891 119866 rarr

R119899 119891 = (1198911 119891119899) defined on a domain 119866 sub R119899 119899 ge 2Let ℎ be a harmonic univalent orientation preserving

mapping on a domain 119863 1198631015840 = ℎ(119863) and 119889ℎ(119911) = 119889(ℎ(119911)

1205971198631015840) If ℎ = 119891 + 119892 has the form where 119891 and 119892 are analytic

we define 120582ℎ(119911) = 119863minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892

1015840(119911)|

21 Quasihyperbolic Metrics and the Average Jacobian Forharmonic qc mappings we refer the interested reader to [25ndash28] and references cited therein

Proposition 7 Suppose 119863 and 1198631015840 are proper domains in R2If ℎ 119863 rarr 119863

1015840 is 119870-qc and harmonic then it is bi-Lipschitzwith respect to quasihyperbolic metrics on 119863 and 1198631015840

Results of this type have been known to the participantsof Belgrade Complex Analysis seminar see for example[29 30] and Section 22 (Proposition 13 Remark 14 andCorollary 16) This version has been proved by Manojlovic[31] as an application of Lemma 4 In [8] we refine herapproach

Proof Let ℎ = 119891 + 119892 be local representation on 119861119911Since ℎ is119870-qc then

(1 minus 1198962)

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2le 119869ℎ le 119870

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2 (22)

on 119861119911 and since log |1198911015840(120577)| is harmonic

log 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816=

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578 (23)

Hence

log 119886ℎ119863 (119911) le1

2

log119870 +

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578

= logradic119870 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816

(24)

Hence 119886ℎ119863(119911) le radic119870|1198911015840(119911)| andradic1 minus 119896|1198911015840(119911)| le 119886ℎ119863(119911)

Using Astala-Gehring result we get

Λ (ℎ 119911) ≍

119889 (ℎ119911 1205971198631015840)

119889 (119911 120597119863)

≍ 120582 (ℎ 119911) (25)

This pointwise result combined with integration alongcurves easily gives

1198961198631015840 (ℎ (1199111) ℎ (1199112)) ≍ 119896119863 (1199111 1199112) 1199111 1199112 isin 119863 (26)

Note that we do not use that log(1119869ℎ) is a subharmonicfunction

The following follows from the proof of Proposition 7

(I) Λ(ℎ 119911) asymp 119886ℎ119863 asymp 120582(ℎ 119911) and 2radic119869119891 asymp 119869119891 see below for

the definition of average jacobian 119869119891

When underlining a symbol (we also use other latex-symbols)wewant to emphasize that there is a specialmeaningof it for example we denote by 119888 a constant and by 119888 119888 119888 somespecific constants

Our next result concerns the quantity

119864119891119866 (119909) =1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (27)


R119899 here 119869119891 is the Jacobian of 119891 while 119861119909 = 119861119909119866 stands forthe ball 119861(119909 119889(119909 120597119866)2) and |119861119909| for its volume

Define

119869119891= 119869

119891119866=119899

radic119864119891119866 (28)

Using the distortion property of qc (see [24 page 63])we give short proof of a quasiconformal analogue of Koebersquostheorem (related to Astala and Gehringrsquos results from [4]cited as Lemma 4 here)

Theorem 8 Suppose that 119866 and 1198661015840 are domains in R119899 If 119891

119866 rarr 1198661015840 is 119870-quasiconformal then

1

119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

le 119869119891119866

(119909) le 119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

119909 isin 119866

(29)


Proof By the distortion property of qc (see [23 page 383][24 page 63]) there are the constants119862lowast and 119888lowast which dependon 119899 and119870 only such that

119861 (119891 (119909) 119888lowast119889lowast) sub 119891 (119861119909) sub 119861 (119891 (119909) 119862lowast119889lowast) 119909 isin 119866

(30)

where 119889lowast(119909) = 119889(119891(119909)) = 119889(119891(119909) 1205971198661015840) and 119889(119909) = 119889(119909

120597119866) Hence

1003816100381610038161003816119861 (119891 (119909) 119888lowast119889lowast)

1003816100381610038161003816le int

119861119909

119869119891 (119911) 119889119911 le1003816100381610038161003816119861 (119891 (119909) 119862lowast119889lowast)

1003816100381610038161003816 (31)

and therefore we proveTheorem 8

We only outline proofs in the rest of this subsectionSuppose that Ω and Ω1015840 are domains in R119899 different from

R119899


Theorem 9 Suppose 119891 Ω rarr Ω1015840 is a 1198621 and the following

hold

(i1) 119891 is c-Lipschitz with respect to quasihyperbolic metricson Ω and Ω1015840 then

(I1) 119889|1198911015840(1199090)| le 119888119889lowast(1199090) for every 1199090 isin Ω

(i2) If 119891 is a qc 119888-quasihyperbolic-isometry then

(I2) 119891 is a 1198882-qc mapping(I3) |1198911015840| asymp 119889lowast119889

Proof Since Ω and Ω1015840 are different from R119899 there are quas-ihyperbolic metrics onΩ andΩ1015840 Then for a fixed 1199090 isin Ω wehave

119896 (119909 1199090) = 119889(1199090)minus1 1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816(1 + 119900 (1)) 119909 997888rarr 1199090

119896 (119891 (119909) 119891 (1199090)) =

1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816

119889lowast (119891 (1199090))(1 + 119900 (1))

when 119909 997888rarr 1199090

(32)

Hence (i1) implies (I1) If 119891 is a 119888-quasihyperbolic-isom-etry then 119889|1198911015840(1199090)| le 119888119889lowast and 119889119897(1198911015840(1199090)) ge 119889lowast119888 Hence(I2) and (I3) follow

Theorem 10 Suppose 119891 Ω rarr Ω1015840 is a 1198621 qc homeomor-

phism The following conditions are equivalent

(a1) 119891 is bi-Lipschitz with respect to quasihyperbolic metricson Ω and Ω1015840

(b1) 119899radic119869119891 asymp 119889lowast119889

(c1) 119899radic119869119891 asymp 119886119891

(d1) 119899radic119869119891 asymp 119869119891

where 119889(119909) = 119889(119909 120597Ω) and 119889lowast(119909) = 119889(119891(119909) 120597Ω1015840)

Proof It follows fromTheorem 9 that (a) is equivalent to (b)(see eg [32 33])

Theorem 8 states that 119869119891asymp 119889lowast119889 By Lemma 4 119886119891 asymp 119889lowast119889

and therefore (b) is equivalent to (c) The rest of the proof isstraightforward

Lemma 11 If119891 isin 11986211 is a119870-quasiconformalmapping defined

in a domain Ω sub R119899 (119899 ge 3) then

119869119891 (119909) gt 0 119909 isin Ω (33)

provided that 119870 lt 2119899minus1 The constant 2119899minus1 is sharp

If 119866 sub Ω then it is bi-Lipschitz with respect to Euclideanand quasihyperbolic metrics on 119866 and 1198661015840 = 119891(119866)

It is a natural question whether is there an analogy ofTheorem 10 if we drop the 1198621 hypothesis

Suppose 119891 Ω rarr Ω1015840 is onto qc mapping We can

consider the following conditions

(a2) 119891 is bi-Lipschitz with respect to quasihyperbolicmetrics onΩ andΩ1015840

(b2) 119899radic119869119891 asymp 119889lowast119889 ae inΩ

(c2) 119899radic119869119891 asymp 119886119891 ae inΩ

(d2) 119899radic119869119891 asymp 119869119891 ae inΩ

It seems that the above conditions are equivalent but wedid not check details

If Ω is a planar domain and 119891 harmonic qc then weproved that (d2) holds (see Proposition 18)

22 Quasi-Isometry in Planar Case For a function ℎ we usenotations 120597ℎ = (12)(ℎ

1015840119909 minus 119894ℎ

1015840119910) and 120597ℎ = (12)(ℎ

1015840119909 + 119894ℎ

1015840119910)

we also use notations 119863ℎ and 119863ℎ instead of 120597ℎ and 120597ℎrespectively when it seems convenient Now we give anotherproof of Proposition 7 using the following

Proposition 12 (see [29]) Let ℎ be an euclidean harmonicorientation preserving univalentmapping of the unit discD intoC such that ℎ(D) contains a disc 119861119877 = 119861(119886 119877) and ℎ(0) = 119886Then

|120597ℎ (0)| ge

119877

4

(34)

If in addition ℎ is 119870-qc then

120582ℎ (0) ge1 minus 119896

4

119877 (35)

For more details in connection with material consid-ered in this subsection see also Appendix A in particularProposition A7

Let ℎ = 119891 + 119892 be a harmonic univalent orientationpreserving mapping on the unit disk D Ω = ℎ(D) 119889ℎ(119911) =119889(ℎ(119911) 120597Ω) 120582ℎ(119911) = 119863

minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892

1015840(119911)| By the harmonic analogue of the

KoebeTheorem then

1

16

119863minus(0) le 119889ℎ (0) (36)

and therefore

(1 minus |119911|2)

1

16

119863minus(119911) le 119889ℎ (119911) (37)

If in addition ℎ is119870-qc then

(1 minus |119911|2)119863

+(119911) le 16119870119889ℎ (119911) (38)

Using Proposition 12 we also prove

(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911) (39)

If we summarize the above considerations we haveproved the part (a3) of the following proposition


Proposition 13 (e-qch hyperbolic distance version) (a3)Let ℎ = 119891 + 119892 be a harmonic univalent orientation preserving119870-qc mapping on the unit disk D Ω = ℎ(D) Then for 119911 isin D

(1 minus |119911|2)Λ ℎ (119911) le 16119870119889ℎ (119911)

(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911)

(40)

(b3) If ℎ is a harmonic univalent orientation preserving119870-qc mapping of domain119863 onto1198631015840 then

119889 (119911) Λ ℎ (119911) le 16119870119889ℎ (119911)

119889 (119911) 120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911)

(41)

and1 minus 119896

4

120581119863 (119911 1199111015840) le 1205811198631015840 (ℎ119911 ℎ119911

1015840)

le 16119870120581119863 (119911 1199111015840)

(42)

where 119911 1199111015840 isin 119863

Remark 14 In particular we have Proposition 7 but here theproof of Proposition 13 is very simple and it it is not based onLemma 4

Proof of (b3) Applying (a3) to the disk119861119911 119911 isin 119863 we get (41)It is clear that (41) implies (42)

For a planar hyperbolic domain inC we denote by120588 = 120588119863and 119889hyp = 119889hyp119863 the hyperbolic density and metric of 119863respectively

We say that a domain 119863 sub C is strongly hyperbolic ifit is hyperbolic and diameters of boundary components areuniformly bounded from below by a positive constant

Example 15 The Poincare metric of the punctured disk D1015840 =0 lt |119911| lt 1 is obtained by mapping its universal coveringan infinitely-sheeted disk on the half plane Πminus = Re119908 lt 0

by means of 119908 = ln 119911 (ie 119911 = 119890119908) The metric is

|119889119908|

|Re119908|=

|119889119911|

|119911| ln (1 |119911|) (43)

Since a boundary component is the point 0 the punctureddisk is not a strongly hyperbolic domain Note also that 120588119889and 1 ln(1119889) tend to 0 if 119911 tends to 0 Here 119889 = 119889(119911) =

dist(119911 120597D1015840) and 120588 is the hyperbolic density of D1015840 Thereforeone has the following

(A1) for 119911 isin D1015840 120581D1015840(119911 1199111015840) = ln(1119889(119911))119889hypD1015840(119911 119911

1015840) when

1199111015840rarr 119911

There is no constant 119888 such that

(A2) 120581D1015840(119911 1199111015840) le 119888120581Πminus(119908 119908

1015840) for every 1199081199081015840 isin Πminus where

119911 = 119890119908 and 1199111015840 = 119890119908

1015840

Since 119889hypΠminus asymp 120581Πminus if (A2) holds we conclude that 120581D1015840 asymp119889hypD1015840 which is a contradiction by (A1)

Let119863 be a planar hyperbolic domain Then if119863 is simplyconnected

1

2119889

le 120588 le

2

119889

(44)

For general domain as 119911 rarr 120597119863

1 + 119900 (1)

119889 ln (1119889)le 120588 le

2

119889

(45)

Hence 119889hyp(119911 1199111015840) le 2120581119863(119911 119911

1015840) 119911 1199111015840 isin 119863

If Ω is a strongly hyperbolic domain then there is ahyperbolic density 120588 on the domain Ω such that 120588 asymp 119889

minus1where 119889(119908) = 119889(119908 120597Ω) see for example [34] Thus 119889hyp119863 asymp

120581119863 Hence we find the following

Corollary 16 Every 119890-harmonic quasiconformal mapping ofthe unit disc (more generally of a strongly hyperbolic domain)is a quasi-isometry with respect to hyperbolic distances

Remark 17 Let 119863 be a hyperbolic domain let ℎ be e-har-monic quasiconformalmapping of119863 ontoΩ and let120601 D rarr

119863 be a covering and ℎlowast = ℎ ∘ 120601

(a4) Suppose that 119863 is simply connected Thus 120601 and ℎlowastare one-to-one Then

(1 minus |119911|2) 120582ℎlowast (120577) ge

1 minus 119896

4

119889ℎ (119911) (46)

where 119911 = 120601(120577)Hence since 120582ℎlowast(120577) = 119897ℎ(119911)|120601

1015840(120577)| and 120588hyp119863(119911)|120601

1015840(120577)| =

120588D(120577)

120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911) 120588119863 (119911) (47)

Using Hallrsquos sharp result one can also improve theconstant in the second inequality in Propositions 13 and 12(ie the constant 14 can be replaced by 1205910 see below formore details)

(1 minus |119911|2) 120582ℎ (119911) ge (1 minus 119896) 1205910119889ℎ (119911) (48)

where 1205910 = 3radic32120587

(b4) Suppose that 119863 is not a simply connected Then ℎlowast isnot one-to-one and we cannot apply the procedure asin (a4)

(c4) It seems natural to consider whether there is ananalogue in higher dimensions of Proposition 13

Proposition 18 For every 119890-harmonic quasiconformal map-ping 119891 of the unit disc (more generally of a hyperbolic domain119863) the following holds

(e2) radic119869119891 asymp 119889lowast119889

In particular it is a quasi-isometry with respect to quasihyper-bolic distances


Proof For 119911 isin 119863 by the distortion property1003816100381610038161003816119891 (1199111) minus 119891 (119911)

1003816100381610038161003816le 119888119889 (119891 (119911)) for every 1199111 isin 119861119911 (49)

Hence by Schwarz lemma for harmonic maps 119889|1198911015840(119911)| le1198881119889(119891(119911)) Proposition 12 yields (e) and an application ofProposition 13 gives the proof

Recall 119861119909 stands for the ball 119861(119909 119889(119909 120597119866)2) and |119861119909| forits volume If 119881 is a subset ofR119899 and 119906 119881 rarr R119898 we define

osc119881119906 = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

1003816100381610038161003816 119909 119910 isin 119881

120596119906 (119903 119909) = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

10038161003816100381610038161003816100381610038161003816119909 minus 119910

1003816100381610038161003816= 119903

(50)

Suppose that 119866 sub R119899 and 119861119909 = 119861(119909 119889(119909)2) Let119874119862

1(119866) = 119874119862

1(119866 1198881) denote the class of 119891 isin 119862

1(119866) such

that

119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 1198881osc119861

119909

119891 (51)

for every 119909 isin 119866

Proposition 19 Suppose 119891 Ω rarr Ω1015840 is a 119862

1 Then119891 isin 119874119862

1(Ω 119888) if and only if 119891 is 119888-Lipschitz with respect to

quasihyperbolic metrics on 119861 and 119891(119861) for every ball 119861 sub Ω

23 Dyk-Type Results The characterization of Lipschitz-typespaces for quasiconformal mappings in space and planarquasiregular mappings by the average Jacobian are the mainresults in this subsection In particular using the distortionproperty of qc mappings we give a short proof of a quasicon-formal version of a Dyakonov theorem which states

Suppose 119866 is a Λ120572-extension domain in R119899 and 119891 isa 119870-quasiconformal mapping of 119866 onto 119891(119866) sub R119899Then 119891 isin Λ

120572(119866) if and only if |119891| isin Λ120572(119866)

This isTheorem B3 inAppendix B below It is convenientto refer to this result asTheoremDy see alsoTheorems 23ndash24and Proposition A Appendix B

First we give some definitions and auxiliary resultsRecall Dyakonov [15] used the quantity

119864119891119866 (119909) 1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (52)


119877119899 here 119869119891 is the Jacobian of 119891 while 119861119909 = 119861119909119866 stands for

the ball 119861(119909 119889(119909 120597119866)2) and |119861119909| for its volumeDefine 1205720 = 120572

119899119870 = 119870

1(1minus119899) 119862lowast = 120579119899119870(12) 120579

119899119870(119888lowast) = 12

and

119869119891119866

=119899

radic119864119891119866 (53)

For a ball 119861 = 119861(119909 119903) sub R119899 and a mapping 119891 119861 rarr R119899we define

119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (54)

and 119869119886V119891 (119861 119909) = 119899

radic119864119891(119861 119909) we also use the notation 119869119891(119861 119909)and 119869119891(119909 119903) instead of 119869119886V119891 (119861 119909)

For 119909 119910 isin R119899 we define the Euclidean inner product ⟨sdot sdot⟩by

⟨119909 119910⟩ = 11990911199101 + sdot sdot sdot + 119909119899119910119899 (55)

By E119899 we denote R119899 with the Euclidean inner product andcall it Euclidean space 119899-space (space of dimension 119899) In thispaper for simplicity we will use also notationR119899 forE119899Thenthe Euclidean length of 119909 isin R119899 is defined by

|119909| = ⟨119909 119909⟩12

= (100381610038161003816100381611990911003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199091198991003816100381610038161003816

2)

12 (56)

The minimal analytic assumptions necessary for a viabletheory appear in the following definition

Let Ω be a domain in R119899 and let 119891 Ω rarr R119899 becontinuous We say that 119891 has finite distortion if


(2) the Jacobian determinant of119891 is locally integrable anddoes not change sign inΩ

(3) there is a measurable function119870119874 = 119870119874(119909) ge 1 finiteae such that

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119874 (119909)

10038161003816100381610038161003816119869119891 (119909)

10038161003816100381610038161003816ae (57)

The assumptions (1) (2) and (3) do not imply that 119891 isin

1198821198991loc(Ω) unless of course119870119874 is a bounded functionIf 119870119874 is a bounded function then 119891 is qr In this setting

the smallest 119870 in (57) is called the outer dilatation119870119874(119891)If 119891 is qr also

119869119891 (119909) le 1198701015840119897(119891

1015840(119909))

119899ae (58)

for some 1198701015840 1 le 1198701015840lt infin where 119897(1198911015840(119909)) = inf|1198911015840(119909)ℎ|

|ℎ| = 1 The smallest 1198701015840 in (58) is called the inner dilatation119870119868(119891) and 119870(119891) = max(119870119874(119891) 119870119868(119891)) is called the maximaldilatation of 119891 If 119870(119891) le 119870 119891 is called 119870-quasiregular

In a highly significant series of papers published in 1966ndash1969 Reshetnyak proved the fundamental properties of qrmappings and in particular the main theorem concerningtopological properties of qr mappings every nonconstant qrmap is discrete and open cf [11 35] and references cited there

Lemma 20 (see Morreyrsquos Lemma Lemma 671 [10 page170]) Let 119891 be a function of the Sobolev class 1198821119901

(BE119899) inthe ball B = 2119861 = 119861(1199090 2119877) 119861 = 119861(1199090 119877) 1 le 119901 lt infin suchthat

119863119901 (119891 119861) = (1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119901)

1119901

le 119872119903120574minus1

(59)

where 0 lt 120574 le 1 holds for every ball 119861 = 119861(119886 119903) sub B Then 119891is Holder continuous in119861with exponent 120574 and one has |119891(119909)minus119891(119910)| le 119862|119909 minus 119910|

120574 for all 119909 119910 isin 119861 where 119888 = 4119872120574minus12minus120574 Here

and in some places we omit to write the volume element 119889119909

We need a quasiregular version of this Lemma


Lemma 21 Let 119891 B rarr R119899 be a 119870-quasiregular mappingsuch that

119869119891 (119886 119903) le 119872119903120574minus1

(60)


120574 for all 119909 119910 isin 119861 where 119862 = 41198721198701119899120574minus12minus120574

Proof By hypothesis119891 satisfies (57) and therefore119863119899(119891 119861) le1198701119899119869119891(119886 119903) An application of Lemma 20 to 119901 = 119899 yields

proof

(a0) By 119861(119866) denote a family of ball 119861 = 119861(119909) = 119861(119909 119903)

such that B = 119861(119909 2119903) sub 119866 For 119861 isin 119861(119866) define1198891015840= 119889

1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)

Define 120596loc(119891 119903) = 120596loc119866(119891 119903) = sup osc119861119891 and 1199030(119866) =sup 119903(119861) where supremum is taken over all balls 119861 = 119861(119886) =

119861(119886 119903) such that 1198611 = 119861(119886 2119903) sub 119866 and 119903(119861) denotes radiusof 119861

Theorem 22 Let 0 lt 120572 le 1 119870 ge 1 Suppose that (a5) 119866 isa domain in R119899 and 119891 119866 rarr R119899 is 119870-quasiconformal and1198661015840= 119891(119866) Then one has the following

(i1) For every 119909 isin 119866 there exists two points 1199091 1199092 isin 119861119909

such that |1199102|minus|1199101| ge 119877(119909) where119910119896 = 119891(119909119896) 119896 = 1 2and 119877(119909) = 119888lowast119889lowast(119909)

(ii1) 120596loc119866(119891 119903) le 21198880120596loc119866(|119891| 119903) 0 le 119903 le 1199030(119866) where1198880 = 2119862lowast119888lowast

(iii1) If in addition one supposes that (b5) |119891| isin locΛ120572(119866)then for all balls 119861 = 119861(119909) = 119861(119909 119903) such that 1198611 =119861(119909 2119903) sub 119866 1198891 ⪯ 119903

120572 and in particular 119877(119909) =

119888lowast119889lowast(119909) le 119871119889(119909)120572 119909 isin 119866

(iv1) There is a constant 119888 such that

119869119891 (119909 119903) le 119888 119903120572minus1 (61)

for every 1199090 isin 119866 and 119861 = 119861(119909 119903) sub 1198611199090

(v1) If in addition one supposes that (c5) 119866 is a Λ

120572-extension domain in R119899 then 119891 isin Lip(120572 1198712 119866)

Proof By the distortion property (30) we will prove the fol-lowing

(vi1) For a ball 119861 = 119861(119909) = 119861(119909 119903) such that 1198611 = 119861(119909

2119903) sub 119866 there exist two points 1199091 1199092 isin 119861(119909) such that|1199102| minus |1199101| ge 119877(119909) where 119910119896 = 119891(119909119896) 119896 = 1 2

Let 119897 be line throughout 0 and 119891(119909) which intersects the119878(119891(119909) (119909)) at points 1199101 and 1199102 and 119909119896 = 119891

minus1(119910119896) 119896 = 1 2

By the left side of (30) 1199091 1199092 isin 119861(119909) We consider two cases

(a) if 0 notin 119861(119891(119909) 119877(119909)) and |1199102| ge |1199101| then |1199102| minus |1199101| =2119877(119909) and |1199102| minus |1199101| = 2119877(119909)

(b) if 0 isin 119861(119891(119909) 119877(119909)) then for example 0 isin [1199101 119891(119909)]and if we choose 1199091 = 119909 we find |1199102| minus |119891(119909)| = 119877(119909)and this yields (vi1)

Let 119909 isin 119866 An application of (vi1) to 119861 = 119861119909 yields (i1)If |119909 minus 1199091015840| le 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| and therefore we have

the following

(vi11015840) |119891(119909) minus 119891(1199091015840)| le 119862lowast1198891015840le 1198880(|1199102| minus |1199101|) where 1198880 =

119862lowast119888lowast

If 1199111 1199112 isin 119861(119909 119903) then |119891(1199111) minus 119891(1199112)| le |119891(1199111) minus 119891(119909)|+|119891(1199112) minus 119891(119909)| Hence using (vi11015840) we find (ii1) Proof of(iii1) If the hypothesis (b) holds withmultiplicative constant119871 and |119909 minus 119909

1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572

and therefore |119891(119909) minus 119891(1199091015840)| le 119862lowast119889

1015840le 1198881119903

120572 where 1198881 =

(119862lowast119888lowast)2120572119871 Hence 1198891 ⪯ 119903

120572 and therefore in particular119877(119909) = 119888lowast119889lowast(119909) le 119871119889(119909)

120572 119909 isin 119866It is clear by the hypothesis (b) that there is a fixed con-

stant 1198711 such that (vii1) |119891| belongs to Lip(120572 1198711 119861) for everyball 119861 = 119861(119909 119903) such that 1198611 = 119861(119909 2119903) sub 119866 and by (ii1) wehave the following

(viii1) 119871(119891 119861) le 1198712 for a fixed constant 1198712 and 119891 isin locLip(120572 1198712 119866)

Hence since by the hypothesis (c5) 119866 is a 119871120572-extensiondomain we get (v1)

Proof of (iv1) Let 119861 = 119861(119909) = 119861(119909 119903) be a ball such thatB = 119861(119909 2119903) sub 119866 Then 119864119892(119861 119909) asymp 119889

1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)

on 119866Note that one can also combine (iii1) and Lemma 33 (for

details see proof of Theorem 40 below) to obtain (v1)

Note that as an immediate corollary of (ii1) we get asimple proof of Dyakonov results for quasiconformal map-pings (without appeal to Lemma 33 or Lemma 21 which is aversion of Morreyrsquos Lemma)

We enclose this section by proving Theorems 23 and 24mentioned in the introduction in particular these resultsgive further extensions of Theorem-Dy

Theorem 23 Let 0 lt 120572 le 1 119870 ge 1 and 119899 ge 2 Suppose 119866is a Λ120572-extension domain in R119899 and 119891 is a 119870-quasiconformalmapping of 119866 onto 119891(119866) sub R119899 The following are equivalent

(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)

(iii2) there is a constant 1198883 such that

119869119891 (119909 119903) le 1198883119903120572minus1 (63)


If in addition119866 is a uniform domain and if 120572 le 1205720 = 119870

1(1minus119899) then(i2) and (ii2) are equivalent to

(iv2) |119891| isin Λ120572(119866 120597119866)


Proof Suppose (ii2) holds so that |119891| is 119871-Lipschitz in 119866Then (iv1) shows that (iii2) holds By Lemma 21 (iii2)implies (i2)

We outline less direct proof that (ii2) implies (ii1)One can show first that (ii2) implies (18) (or more gen-

erally (iv1) see Theorem 22) Using 119860(119911) = 119886 + 119877119911 weconclude that 119888lowast(119861(119886 119877)) = 119888

lowast(B) and 119888(119861) = 119888 is a fixed

constant (which depends only on119870 119899 1198881) for all balls 119861 sub 119866Lemma 5 tells us that (18) holds with a fixed constant for allballs 119861 sub 119866 and all pairs of points 1199091 1199092 isin 119861 Further wepick two points 119909 119910 isin 119866 with |119909 minus 119910| lt 119889(119909 120597119866) and apply(18) with 119863 = 119861(119909 |119909 minus 119910|) letting 1199091 = 119909 and 1199092 = 119910 Theresulting inequality

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)

shows that 119891 isin locΛ120572(119866) and since 119866 is a Λ120572-extensiondomain we conclude that 119891 isin Λ

120572(119866)

The implication (ii4)rArr (i4) is thus establishedThe con-verse is clear For the proof that (iii4) implies (i4) for 120572 le 1205720see [15]


119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)

and 119869119891(119861 119909) = 119899radic119864119891(119861 119909)We use the factorization of planar quasiregular mappings

to prove the following

Theorem 24 Let 0 lt 120572 le 1 119870 ge 1 Suppose 119863 is a Λ120572-extension domain in C and 119891 is a 119870-quasiregular mapping of119863 onto 119891(119863) sub C The following are equivalent

(i3) 119891 isin Λ120572(119863)

(ii3) |119891| isin Λ120572(119863)(iii3) there is a constant 119888 such that

119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)


Proof Let 119863 and 119882 be domains in 119862 and 119891 119863 rarr 119882 qrmapping Then there is a domain 119866 and analytic function 120601on 119866 such that 119891 = 120601 ∘ 119892 where 119892 is quasiconformal see [36page 247]

Our proof will rely on distortion property of quasiconfor-mal mappings By the triangle inequality (i4) implies (ii4)Now we prove that (ii4) implies (iii4)

Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110

As in analytic case there is 1205771 isin 119892(119861) suchthat ||120601(1205771)| minus |120601(1205770)|| ge 1198881198891(1205770)|120601

1015840(1205770)| If 1199111 = 119892

minus1(1205771) then

||120601(1205771)|minus|120601(1205770)|| = ||119891(1199111)|minus|119891(1199110)|| Hence if |119891| is120572-Holderthen 1198891(120577)|120601

1015840(120577)| le 1198881119903

120572 for 120577 isin 1198660 Hence since

119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)

and therefore using 119864119892(119861 119911) asymp 11988921119903

2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)

Thus we have (iii3) with 119888 = 1198883Proof of the implication (iii4) rArr (i4)An application of Lemma 21 (see also a version of Astala-

Gehring lemma) shows that 119891 is 120572-Holder on 1198611199110

with aLipschitz (multiplicative) constant which depends only on 120572and it gives the result

3 Lipschitz-Type Spaces of Harmonicand Pluriharmonic Mappings

31 Higher Dimensional Version of Schwarz Lemma Beforegiving a proof of the higher dimensional version of theSchwarz lemma we first establish notation

Suppose that ℎ 119861119899(119886 119903) rarr R119898 is a continuous vector-valued function harmonic on 119861119899(119886 119903) and let

119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)

Let ℎ = (ℎ1 ℎ2 ℎ119898) A modification of the estimate in[37 Equation (231)] gives

119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)

We next extend this result to the case of vector-valuedfunctions See also [38] and [39 Theorem 616]

Lemma 25 Suppose that ℎ 119861119899(119886 119903) rarr R119898 is a continuousmapping harmonic in 119861119899(119886 119903) Then

119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)

Proof Without loss of generality we may suppose that 119886 = 0

and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)

where120596119899 is the volume of the unit ball inR119899 Hence as in [8]for 1 le 119895 le 119899 we have

120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)

Let 120578 isin 119878119899minus1 be a unit vector and |120585| = 119903 For given 120585

it is convenient to write 119870120585(119909) = 119870(119909 120585) and consider 119870120585 asfunction of 119909

Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)


Since |(120585 120578)| le |120585||120578| = 119903 we see that

100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)

This last inequality yields

10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)

where 119889120590(119910) is the surface element on the sphere and theproof is complete

Let C119899 = 119911 = (1199111 119911119899) 1199111 119911119899 isin C denote thecomplex vector space of dimension 119899 For 119886 = (1198861 119886119899) isin

C119899 we define the Euclidean inner product ⟨sdot sdot⟩ by

⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)

where 119886119896 (119896 isin 1 119899) denotes the complex conjugate of119886119896 Then the Euclidean length of 119911 is defined by

|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)

Denote a ball in C119899 with center 1199111015840 and radius 119903 gt 0 by

B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)

In particular B119899 denotes the unit ball B119899(0 1) and S119899minus1 thesphere 119911 isin C119899 |119911| = 1 Set D = B1 the open unit disk inC and let 119879 = S0 be the unit circle in C

A continuous complex-valued function 119891 defined in adomain Ω sub C119899 is said to be pluriharmonic if for fixed119911 isin Ω and 120579 isin S119899minus1 the function 119891(119911 + 120579120577) is harmonic in120577 isin C |120579120577 minus 119911| lt 119889Ω(119911) where 119889Ω(119911) denotes the distancefrom 119911 to the boundary 120597Ω of Ω It is easy to verify that thereal part of any holomorphic function is pluriharmonic cf[40]

Let 120596 [0 +infin) rarr [0 +infin) with 120596(0) = 0 be a con-tinuous function We say that 120596 is amajorant if

(1) 120596(119905) is increasing(2) 120596(119905)119905 is nonincreasing for 119905 gt 0

If in addition there is a constant 119862 gt 0 depending onlyon 120596 such that

int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)

for some 1205750 thenwe say that120596 is a regularmajorant Amajor-ant is called fast (resp slow) if condition (81) (resp (82)) isfulfilled

Given a majorant 120596 we define Λ 120596(Ω) (resp Λ 120596(120597Ω)) tobe the Lipschitz-type space consisting of all complex-valued

functions 119891 for which there exists a constant 119862 such that forall 119911 and 119908 isin Ω (resp 119911 and 119908 isin 120597Ω)

1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)

Using Lemma 25 one can prove the following

Proposition 26 Let 120596 be a regular majorant and let119891 be har-monic mapping in a simply connected Λ 120596-extension domain119863 sub 119877

119899Then ℎ isin Λ 120596(119863) if and only if |nabla119891| le 119862(120596(1minus|119911|)(1minus|119911|))

It is easy to verify that the real part of any holomorphicfunction is pluriharmonic It is interesting that the converseis true in simply connected domains

Lemma 27 (i) Let 119906 be pluriharmonic in 1198610 = 119861(119886 119903) Thenthere is an analytic function 119891 in 119861 such that 119891 = 119906 + 119894V

(ii) Let Ω be simply connected and 119906 be pluriharmonic inΩ Then there is analytic function 119891 inΩ such that 119891 = 119906 + 119894V

Proof (i) Let 1198610 = 119861(119886 119903) sub Ω 119875119896 = minus119906119910119896

and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =

119906 + 119894V which is analytic on 1198610(ii) If 119911 isin Ω there is a chain 119862 = (1198610 1198611 119861119899) in Ω

such that 119911 is center of 119861119899 and by the lemma there is analyticchain (119861119896 119891119896) 119896 = 0 119899 We define 119891(119911) = 119891119899(119911) As in inthe proof of monodromy theorem in one complex variableone can show that this definition does not depend of chains119862 and that 119891 = 119906 + 119894V in Ω

The following three theorems in [9] are a generalizationof the corresponding one in [15]

Theorem 28 Let 120596 be a fast majorant and let 119891 = ℎ + 119892 bea pluriharmonic mapping in a simply connected Λ 120596-extensiondomain Ω Then the following are equivalent

(1) 119891 isin Λ 120596(Ω)(2) ℎ isin Λ 120596(Ω) and 119892 isin Λ 120596(Ω)(3) |ℎ| isin Λ 120596(Ω) and |119892| isin Λ 120596(Ω)(4) |ℎ| isin Λ 120596(Ω 120597Ω) and |119892| isin Λ 120596(Ω 120597Ω)

where Λ 120596(Ω 120597Ω) denotes the class of all continuous functions119891 onΩ cup 120597Ω which satisfy (83) with some positive constant 119862whenever 119911 isin Ω and 119908 isin 120597Ω

Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899

Theorem 29 Let Ω sub C119899 be a domain and 119891 analytic in ΩThen

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)

If |119891| isin Λ 120596(Ω) then 119889Ω(119911)|nabla119891(119911)| le 4119862120596(119889Ω(119911))


If 119891 = ℎ + 119892 is a pluriharmonic mapping where 119892 and ℎare analytic inΩ then |1198921015840| le |1198911015840| and |ℎ1015840| le |1198911015840| In particular|119863119894119892| |119863119894ℎ| le |119891

1015840| le |119863119892| + |119863ℎ|

Proof Using a version of Koebe theorem for analytic func-tions (we can also use Bloch theorem) we outline a proofLet 119911 isin Ω 119886 = (1198861 119886119899) isin C119899 |119886| = 1 119865(120582) = 119865(120582 119886) =

119891(119911 + 120582119886) 120582 isin 119861(0 119889Ω(119911)2) and 119906 = |119865|By the version of Koebe theorem for analytic functions

for every line 119871 which contains 119908 = 119891(119911) there are points1199081 1199082 isin 119871 such that 119889Ω(119911)|119865

1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2

and 119908 isin [1199081 1199082] Hence 119889Ω(119911)|1198651015840(0)| le 4120596119906(119889Ω(119911)) and

119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891

1015840(119911) 119886) we find |nabla119891(119911)| = |1198911015840(119911)| where

119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)

Finally if |119891| isin Λ 120596(Ω) we have 119889Ω(119911)|nabla119891(119911)| le

4119862120596(119889Ω(119911))

For 119861119899 the following result is proved in [9]

Theorem30 Let120596 be a regularmajorant and letΩ be a simplyconnected Λ 120596-extension domain A function 119891 pluriharmonicin belongs to Λ 120596(Ω) if and only if for each 119894 isin 1 2 119899119889Ω(119911)|119863119894119891(119911)| le 120596(119889Ω(119911)) and 119889Ω(119911) |119863119894119891(119911)| le 120596(119889Ω(119911))

for some constant C depending only on 119891 120596 Ω and 119899

Weonly outline a proof let119891 = ℎ+119892 Note that119863119894119891 = 119863119894ℎ

and119863119894119891 = 119863119894119892We can also use Proposition 26

32 Lipschitz-Type Spaces Let 119891 119866 rarr 1198661015840 be a 1198622 function

and 119861119909 = 119861(119909 119889(119909)2) We denote by 1198741198622(119866) the class offunctions which satisfy the following condition

sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)

for every 119909 isin 119866It was observed in [8] that 1198741198622(119866) sub 119874119862

1(119866) In [7] we

proved the following results

Theorem 31 Suppose that

(a1) 119863 is aΛ120572-extension domain inR119899 0 lt 120572 le 1 and 119891 iscontinuous on119863which is a119870-quasiconformalmappingof119863 onto 119866 = 119891(119863) sub R119899

(a2) 120597119863 is connected(a3) 119891 is Holder on 120597119863 with exponent 120572

(a4) 119891 isin 1198741198622(119863) Then 119891 is Holder on119863 with exponent 120572

The proof in [7] is based on Lemmas 3 and 8 in thepaper of Martio and Nakki [18] In the setting of Lemma 8119889|119891

1015840(119910)| le 119889

120572 In the setting of Lemma 3 using the fact that120597119863 is connected we get similar estimate for 119889 small enough

Theorem 32 Suppose that 119863 is a domain in R119899 0 lt 120572 le 1and 119891 is harmonic (more generally 1198741198622(119863)) in 119863 Then onehas the following

(i1) 119891 isin Lip(120596 119888 119863) implies(ii1) |119891

1015840(119909)| le 119888119889(119909)

120572minus1 119909 isin 119863(ii1) implies(iii1) 119891 isin loc Lip(120572 1198711 119863)

4 Theorems of Koebe and Bloch Type forQuasiregular Mappings

We assume throughout that 119866 sub R119899 is an open connected setwhose boundary 120597119866 is nonempty Also 119861(119909 119877) is the openball centered at 119909 isin 119866 with radius 119877 If 119861 sub R119899 is a ball then120590119861 120590 gt 0 denotes the ball with the same center as 119861 and withradius equal to 120590 times that of 119861

The spherical (chordal) distance between two points 119886119887 isin R

119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)

where 119901 R119899rarr 119878(119890119899+12 12) is stereographic projection

defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)

Explicitly if 119886 = infin = 119887

119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)

When 119891 119866 rarr R119899 is differentiable we denote itsJacobi matrix by 119863119891 or 1198911015840 and the norm of the Jacobi matrixas a linear transformation by |1198911015840| When 119863119891 exists ae wedenote the local Dirichlet integral of 119891 at 119909 isin 119866 by 119863119891(119909) =

119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is

no chance of confusion we will omit the index 119866 If B = B119909then119863119891119866(119909) = 119863119891B(119909) and ifB is the unit ball wewrite119863(119891)instead of119863119891B(0)

When the measure is omitted from an integral as hereintegration with respect to 119899-dimensional Lebesgue measureis assumed

A continuous increasing function 120596(119905) [0infin) rarr

[0infin) is amajorant if120596(0) = 0 and if120596(1199051+1199052) le 120596(1199051)+120596(1199052)for all 1199051 1199052 ge 0

The main result of the paper [3] generalizes Lemma 5 toa quasiregular version involving a somewhat larger class ofmoduli of continuity than 119905120572 0 lt 120572 lt 1

Lemma 33 (see [3]) Suppose that 119866 is Λ 120596-extension domainin R119899 If 119891 is K-quasiregular in 119866 with 119891(119866) sub R119899 and if

119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)


for 1199091 1199092 isin 119863 infin where the constant 1198622 depends only on119870 119899 120596 1198621 and 119866

If 119866 is uniform the constant 1198622 = 119888(119866) depends only on119870 119899 1198621 and the uniformity constant 119888lowast = 119888lowast(119866) for 119866

Conversely if there exists a constant1198622 such that (92) holdsfor all 1199091 1199092 isin 119866 then (91) holds for all 119909 isin 119866 with 1198621

depending only on 1198622 119870 119899 120596 and 119866

Now suppose that 120596 = 120596120572 and 120572 le 1205720Also in [3] Nolder using suitable modification of a theo-

rem of Nakki and Palka [41] shows that (92) can be replacedby the stronger conclusion that

1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)

Remark 34 Simple examples show that the term 119889(1199091 120597119866)

cannot in general be omitted For example 119891(119909) = 119909|119909|119886minus1

with 119886 = 1198701(1minus119899) is 119870-quasiconformal in 119861 = 119861(0 1) 119863119891(119909)

is bounded over 119909 isin 119861 yet 119891 isin Lip119886(119861) see Example 6If 119899 = 2 then |1198911015840(119911)| asymp 120588

119886minus1 where 119911 = 120588119890119894120579 and if 119861119903 =

119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1

Note that we will show below that if |119891| isin 119871120572(119866) the

conclusion (92) in Lemma 33 can be replaced by the strongerassertion |119891(1199091) minus 119891(1199092)| le 119888119898|1199091 minus 1199092|

120572

Lemma 35 (see [3]) If 119891 = (1198911 1198912 119891119899) is K-quasiregularin 119866 with 119891(119866) sub 119877

119899 and if 119861 is a ball with 120590119861 sub 119866 120590 gt 1then there exists a constant 119862 depending only on 119899 such that

(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)

for all 119886 isin R and all 119895 = 1 2 119899 Here and in some placeswe omit to write the volume and the surface element

Lemma 36 (see[42] second version of Koebe theorem foranalytic functions) Let 119861 = 119861(119886 119903) let 119891 be holomorphicfunction on 119861 119863 = 119891(119861) 119891(119886) = 119887 and let the unboundedcomponent 119863infin of 119863 be not empty and let 119889infin = 119889infin(119887) =

dist(119887 119863infin) Then (a1) 119903|1198911015840(119886)| le 4119889infin (b1) if in addition119863 is simply connected then119863 contains the disk119861(119887 120588) of radius120588 where 120588 = 120588119891(119886) = 119903|1198911015840(119886)|4

The following result can be considered as a version of thislemma for quasiregular mappings in space

Theorem 37 Suppose that 119891 is a 119870-quasiregular mapping onthe unit ball B 119891(0) = 0 and |119863(119891)| ge 1

Then there exists an absolute constant 120572 such that for every119909 isin S there exists a point 119910 on the half-line Λ 119909 = Λ(0 119909) =

120588119909 120588 ge 0 which belongs to 119891(B) such that |119910| ge 2120572If 119891 is a 119870-quasiconformal mapping then there exists an

absolute constant 1205881 such that 119891(B) contains B(0 1205881)

For the proof of the theorem we need also the followingresult Theorem 1881 [10]

Lemma 38 For each 119870 ge 1 there is 119898 = 119898(119899119870) with thefollowing property Let 120598 gt 0 let 119866 sub R

119899 be a domain andlet 119865 denote the family of all qr mappings withmax119870119874(119909 119891)119870119868(119909 119891) le 119870 and 119891 119866 rarr R119899 1198861119891 1198862119891 119886119898119891 where119886119894119891 are points in R

119899 such that 119902(119886119894119891 119886119895119891) gt 120598 119894 = 119895 Then 119865forms a normal family in 119866

Now we proveTheorem 37

Proof If we suppose that this result is not true then thereis a sequence of positive numbers 119886119899 which converges tozero and a sequence of 119870-quasiregular functions 119891119899 suchthat 119891119899(B) does not intersect [119886119899 +infin) 119899 ge 1 Next thefunctions 119892119899 = 119891119899119886119899 map B into 119866 = R119899 [1 +infin) andhence by Lemma 38 the sequence 119892119899 is equicontinuous andtherefore forms normal family Thus there is a subsequencewhich we denote again by 119892119899 which converges uniformlyon compact subsets of B to a quasiregular function 119892 Since119863(119892119899) converges to119863(119892) and |119863(119892119899)| = |119863(119891119899)|119886119899 convergesto infinity we have a contradiction by Lemma 35

A path-connected topological space 119883 with a trivial fun-damental group 1205871(119883) is said to be simply connected We saythat a domain119881 inR3 is spatially simply connected if the fun-damental group 1205872(119881) is trivial

As an application of Theorem 37 we immediately obtainthe following result which we call the Koebe theorem forquasiregular mappings

Theorem 39 (second version of Koebe theorem for 119870-qua-siregular functions) Let 119861 = 119861(119886 119903) let 119891 be 119870-quasiregularfunction on 119861 sub 119877

119899 119863 = 119891(119861) 119891(119886) = 119887 and let theunbounded component 119863infin of 119863119888 be not empty and let 119889infin =

119889infin(119887) = dist(119887 119863infin) Then there exists an absolute constant 119888

(a1) 119903|119863119891(119886)| le 119888119889infin

(a2) if in addition 119863 sub R3 and 119863 is spatially simply con-nected then 119863 contains the disk 119861(119887 120588) of radius 120588where 120588 = 120588119891(119886) = 119903|119863119891(119886)|119888

(A1) Now using Theorem 39 and Lemma 20 we willestablish the characterization of Lipschitz-type spaces forquasiregular mappings by the average Jacobian and in par-ticular an extension of Dyakonovrsquos theorem for quasiregularmappings in space (without Dyakonovrsquos hypothesis that itis a quasiregular local homeomorphism) In particular ourapproach is based on the estimate (a3) below

Theorem 40 (Theorem-DyMa) Let 0 lt 120572 le 1 119870 ge 1 and119899 ge 2 Suppose 119866 is a 119871120572-extension domain in R119899 and 119891 is a119870-quasiregular mapping of 119866 onto 119891(119866) sub R119899 The followingare equivalent

(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)

(iii4) |119863119891(119903 119909)| le 119888119903120572minus1 for every ball 119861 = 119861(119909 119903) sub 119866


Proof Suppose that 119891 is a119870-quasiregular mapping of119866 onto119891(119866) sub R119899 We first establish that for every ball 119861 = 119861(119909 119903) sub119866 one has the following(a3) 119903|119863119891(119903 119909)| le 119888120596|119891|(119903 119909) where119863119891(119903 119909) = 119863119891119861(119909)

Let 119897 be line throughout 0 and 119891(119909) and denote by 119863infin119903the unbounded component of 119863119888 where 119863 = 119891(119861(119909 119903))Then using a similar procedure as in the proof ofTheorem 22the part (iii1) one can show that there is 1199111015840 isin 120597119863infin119903 cap 119897 suchthat 119889infin(119909

1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)

Take a point 119911 isin 119891minus1(1199111015840) Then 1199111015840 = 119891(119911) and |119911 minus 119909| = 119903Since ||119891(119911)| minus |119891(119909)|| le 120596|119891|(119909 119903) Then using Theorem 39we find (a3)

Now we suppose (ii4) that is |119891| isin Lip(120572 119871 119866)Thus we have 120596|119891|(119909 119903) le 119871119903

120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888

It is clear that (iii4) is denoted here as (a4) The implica-tion (ii4) rArr (iii4) is thus established

If we suppose (iii4) then an application of Lemma 20shows that 119891 isin locΛ120572(119866) and since 119866 is a 119871120572-extensiondomain we conclude that 119891 isin Λ

120572(119866) Thus (iii4) implies

(i4)Finally the implication (i4) rArr (ii4) is a clear corollary

of the triangle inequality

(A2) Nowwe give another outline that (i4) is equivalent to(ii4)

Here we use approach as in [15] In particular (a4)implies that the condition (91) holds

We consider two cases(1) 119889(119909) le 2|119909 minus 119910|(2) 119904 = |119909 minus 119910| le 119889(119909)2Then we apply Lemma 33 on 119866 and 119860 = 119861(119909 2119904) in Case

(1) and Case (2) respectivelyInmore detail if |119891| isin 119871120572(119866) then for every ball119861(119909 119903) sub

119866 by (a3) 119903|119863119891(119909)| le 119888119903120572minus1 and the condition (91) holds

Then Lemma 33 tells us that (92) holds with a fixed constantfor all balls 119861 sub 119866 and all pairs of points 1199091 1199092 isin 119861 Nextwe pick two points 119909 isin 119866 with |119909 minus 119910| lt 119889(119909 120597119866) and apply(92) with 119866 = 119860 where 119860 = 119861(119909 |119909 minus 119910|) letting 1199091 = 119909 and1199092 = 119910 The resulting inequality

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)

shows that 119891 isin locΛ120572(119866) and since 119866 is a 119871120572-extensiondomain we conclude that 119891 isin Λ

120572(119866)

The implication (ii4) rArr (i4) is thus established Theconverse being trivially true

The consideration in (A1) shows that (i4) and (ii4) areequivalent with (a4)

Appendices

A Distortion of Harmonic Maps

Recall by D and T = 120597D we denote the unit disc and the unitcircle respectively and we also use notation 119911 = 119903119890

119894120579 For

a function ℎ we denote by ℎ1015840119903 ℎ1015840119909 and ℎ

1015840119910 (or sometimes by

120597119903ℎ 120597119909ℎ and 120597119909ℎ) partial derivatives with respect to 119903 119909 and119910 respectively Let ℎ = 119891 + 119892 be harmonic where 119891 and 119892are analytic and every complex valued harmonic function ℎon simply connected set 119863 is of this form Then 120597ℎ = 119891

1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|

2 If ℎ isunivalent then |1198921015840| lt |1198911015840| and therefore |ℎ1015840119903| le |119891

1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|

After writing this paper and discussion with some col-leagues (see Remark A11 below) the author found out thatit is useful to add this section For origins of this section seealso [29]

Theorem A1 Suppose that

(a) ℎ is an euclidean univalent harmonic mapping from anopen set119863 which contains D into C

(b) ℎ(D) is a convex set in C(c) ℎ(D) contains a disc 119861(119886 119877) ℎ(0) = 119886 and ℎ(T)

belongs to the boundary of ℎ(D)

Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587

Ageneralization of this result to several variables has beencommunicated at Analysis Belgrade Seminar cf [32 33]

Proposition A2 Suppose that

(a1015840) ℎ is an euclidean harmonic orientation preserving uni-valent mapping from an open set 119863 which contains Dinto C

(b1015840) ℎ(D) is a convex set in C(c1015840) ℎ(D) contains a disc 119861(119886 119877) and ℎ(0) = 119886

Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T

Since ℎ is an euclidean univalent harmonicmapping1198911015840 =

0 Using (e) and applying Maximum Principle to the analyticfunction 1198911015840 = 120597ℎ we obtain Proposition A2

Proof of Theorem A1 Without loss of generality we can sup-pose that ℎ(0) = 0 Let 0 le 120593 le 2120587 be arbitrary Sinceℎ(D) is a bounded convex set in C there exists 120591 isin [0 2120587]

such that harmonic function 119906 defined by 119906 = Re119867 where119867(119911) = 119890

119894120591ℎ(119911) has a maximum on D at 119890119894120593

Define 1199060(119911) = 119906(119890119894120593) minus 119906(119911) 119911 isin D By the mean value

theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877

Since Poisson kernel for D satisfies

119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)


using Poisson integral representation of the function 1199060(119911) =119906(119890

119894120593) minus 119906(119911) 119911 isin D we obtain

119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)

Now we derive a slight generalization of Proposition A2More precisely we show that we can drop the hypothesis (a1015840)and suppose weaker hypothesis (a10158401015840)

Proposition A3 Suppose that (11988610158401015840) ℎ is an euclidean har-monic orientation preserving univalent mapping of the unitdisc onto convex domain Ω If Ω contains a disc 119861(119886 119877) andℎ(0) = 119886 then

|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)

A proof of the proposition can be based onProposition A2 and the hereditary property of convexfunctions (i) if an analytic function maps the unit diskunivalently onto a convex domain then it also maps eachconcentric subdisk onto a convex domain It seems that wecan also use the approach as in the proof of Proposition A7but an approximation argument for convex domain119866 whichwe outline here is interesting in itself

(ii) approximation of convex domain 119866 with smoothconvex domains

Let 120601 be conformal mapping ofD onto119866 1206011015840(0) gt 0119866119899 =120601(119903119899D) 119903119899 = 119899(119899+1) ℎ is univalent mapping of the unit disconto convex domainΩ and119863119899 = ℎ

minus1(119866119899)

(iii) Let120593119899 be conformalmapping ofU onto119863119899120593119899(0) = 01205931015840119899(0) gt 0 and ℎ119899 = ℎ ∘ 120593119899 Since 119863119899 sub 119863119899+1 andcup119863119899 = D we can apply the Caratheodory theorem120593119899tends to 119911 uniformly on compacts whence 1205931015840119899(119911) rarr1 (119899 rarr infin) By the hereditary property 119866119899 is convex

(iv) Since the boundary of119863119899 is an analytic Jordan curvethe mapping 120593119899 can be continued analytically acrossT which implies that ℎ119899 has a harmonic extensionacross T

Thus we have the following

(v) ℎ119899 are harmonic onD119866119899 = ℎ119899(D) are smooth convexdomains and ℎ119899 tends to ℎ uniformly on compactssubset of D

Using (v) an application of Proposition A2 to ℎ119899 givesthe proof

As a corollary of Proposition A3 we obtain (A4)

Proposition A4 Let ℎ be an euclidean harmonic orientationpreserving119870-qcmapping of the unit discD onto convex domainΩ If Ω contains a disc 119861(119886 119877) and ℎ(0) = 119886 then

120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)

(i0) In particular 119891minus1 is Lipschitz on Ω

It is worthy to note that (A4) holds (ie 119891minus1 is Lipschitz)under assumption thatΩ is convex (without any smoothnesshypothesis)

Example A5 119891(119911) = (119911 minus 1)2 is univalent on D Since

1198911015840(119911) = 2(119911 minus 1) it follows that 1198911015840(119911) tends 0 if 119911 tends to 1

This example shows that we cannot drop the hypothesis that119891(D) is a convex domain in Proposition A4

Proof Let 1198881015840 = ((1 minus 119896)4)119877 Since10038161003816100381610038161003816119863ℎ (119911)

10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)

it follows that 120582ℎ(119911) ge 1198880 = ((1minus119896)4)119877 and therefore |ℎ(1199112)minusℎ(1199111)| ge 119888

1015840|1199112 minus 1199111|

Hall see [43 pages 66ndash68] proved the following

Lemma A6 (Hall Lemma) For all harmonic univalent map-pings 119891 of the unit disk onto itself with 119891(0) = 0

100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839

Set 1205910 = radic1198880 = 3radic32120587 Now we derive a slight gener-alization of Proposition A3 More precisely we show that wecan drop the hypothesis that the image of the unit disc isconvex

Proposition A7 Let ℎ be an euclidean harmonic orientationpreserving univalent mapping of the unit disc into C such that119891(D) contains a disc 119861119877 = 119861(119886 119877) and ℎ(0) = 119886

(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)

(i2) The constant 14 in inequality (A8) can be replacedwith sharp constant 1205911 = 3radic322120587

(i3) If in addition ℎ is119870-qc mapping and 119896 = (119870minus 1)(119870+

1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)

Proof Let 0 lt 119903 lt 119877 and 119881 = 119881119903 = ℎminus1(119861119903) and let 120593 be a

conformalmapping of the unit discD onto119881 such that120593(0) =0 and let ℎ119903 = ℎ ∘ 120593 By Schwarz lemma

100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)

The function ℎ119903 has continuous partial derivatives onD Since120597ℎ119903(0) = 120597ℎ(0)120593

1015840(0) by Proposition A3 we get |120597ℎ119903(0)| =

|120597ℎ(0)||1205931015840(0)| ge 1199034 Hence using (A10) we find |120597ℎ(0)| ge

1199034 and if 119903 tends to 119877 we get (A8)

(i2) If 1199030 = max|119911| 119911 isin 119881119877 and 1199020 = 11199030 then

(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|


Hence by the Hall lemma 2|1198861|2gt |1198861|

2+ |1198871|

2ge 1198880 and

therefore

(vii) |1198861| ge 1205911 where 1205911 = 3radic322120587 Combining (vi)and (vii) we prove (i2)

(i3) If ℎ = 119891+119892 where 119891 and 119892 are analytic then 120582ℎ(119911) =|1198911015840(119911)| minus |119892

1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus

119896)|1198911015840(119911)| Set 1198861 = 119863ℎ(0) and 1198871 = 119863ℎ(0) By the Hall

sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge

11988801198772 then |1198861| ge 119877(1205910radic1 + 1198962) and therefore (A9)

Also as a corollary of Proposition A3 we obtain the fol-lowing

Proposition A8 (see [27 44]) Let ℎ be an euclidean har-monic diffeomorphism of the unit disc onto convex domain ΩIf Ω contains a disc 119861(119886 119877) and ℎ(0) = 119886 then

119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2

The following example shows thatTheorem A1 and Prop-ositions A2 A3 A4 A7 and A8 are not true if we omit thecondition ℎ(0) = 119886

Example A9 Themapping

120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)

is a conformal automorphism of the unit disc onto itself and

100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)

In particular 1205931015840119887(0) = 1 minus |119887|2

Heinz proved (see [45]) that if ℎ is a harmonic diffeomor-phism of the unit disc onto itself such that ℎ(0) = 0 then

119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)

Using Proposition A3 we can prove another Heinz the-orem

Theorem A10 (Heinz) There exists no euclidean harmonicdiffeomorphism from the unit disc D onto C

Note that this result was a key step in his proof of theBernstein theorem for minimal surfaces in R3

Remark A11 Professor Kalaj turned my attention to the factthat in Proposition 12 the constant 14 can be replaced withsharp constant 1205911 = 3radic322120587 which is approximately0584773

Thus Hall asserts the sharp form |1198861|2+ |1198871|

2ge 1198880 =

2741205872 where 1198880 = 274120587

2= 06839 and therefore if

1198871 = 0 then |1198861| ge 1205910 where 1205910 = radic1198880 = 3radic32120587If we combine Hallrsquos sharp form with the Schwarz lemma

for harmonic mappings we conclude that if 1198861 is real then3radic32120587 le 1198861 le 1

Concerning general codomains the author using Hallrsquossharp result communicated around 1990 at Seminar Uni-versity of Belgrade a proof of Corollary 16 and a version ofProposition 13 cf [32 33]

A1 Characterization of Harmonic qcMappings (See [25]) By120594 we denote restriction of ℎ on R If ℎ isin 1198671198761198620(H) it iswell-known that 120594 R rarr R is a homeomorphism andRe ℎ = 119875[120594] Now we give characterizations of ℎ isin 1198671198761198620(H)in terms of its boundary value 120594

Suppose that ℎ is an orientation preserving diffeomor-phism ofH onto itself continuous onHcupR such that ℎ(infin) =

infin and 120594 the is restriction of ℎ on R Recall ℎ isin 1198671198761198620(H) ifand only if there is analytic function 120601 H rarr Π

+ such that120601(H) is relatively compact subset ofΠ+ and 1205941015840(119909) = Re120601lowast(119909)ae

We give similar characterizations in the case of the unitdisk and for smooth domains (see below)

Theorem A12 Let 120595 be a continuous increasing function onR such that 120595(119905 + 2120587) minus 120595(119905) = 2120587 120574(119905) = 119890119894120595(119905) and ℎ = 119875[120574]Then ℎ is qc if and only if the following hold

(1) ess inf 1205951015840 gt 0

(2) there is analytic function 120601 D rarr Π+ such that

120601(119880) is relatively compact subset of Π+ and 1205951015840(119909) =Re120601lowast(119890119894 119909) ae

In the setting of this theorem we write ℎ = ℎ120601 The readercan use the above characterization and functions of the form120601(119911) = 2 + 119872(119911) where119872 is an inner function to produceexamples of HQCmappings ℎ = ℎ120601 of the unit disk onto itselfso the partial derivatives of ℎ have no continuous extensionto certain points on the unit circle In particular we can take119872(119911) = exp((119911+1)(119911minus1)) for the subject of this subsectionscf [25 28] and references cited therein

Remark A13 Because of lack of space in this paper we couldnot consider some basic concepts related to the subject andin particular further distortion properties of qc maps asGehring and Osgood inequality [12] For an application ofthis inequality see [25 28]

B Quasi-Regular Mappings

(A) The theory of holomorphic functions of one complexvariable is the central object of study in complex analysis It isone of the most beautiful and most useful parts of the wholemathematics

Holomorphic functions are also sometimes referred to asanalytic functions regular functions complex differentiablefunctions or conformal maps


This theory deals only with maps between two-dimen-sional spaces (Riemann surfaces)

For a function which has a domain and range in the com-plex plane and which preserves angles we call a conformalmap The theory of functions of several complex variableshas a different character mainly because analytic functionsof several variables are not conformal

Conformal maps can be defined between Euclideanspaces of arbitrary dimension but when the dimension isgreater than 2 this class of maps is very small A theorem ofJ Liouville states that it consists of Mobius transformationsonly relaxing the smoothness assumptions does not help asproved by Reshetnyak This suggests the search of a gener-alization of the property of conformality which would give arich and interesting class of maps in higher dimension

The general trend of the geometric function theory inR119899

is to generalize certain aspects of the analytic functions of onecomplex variable The category of mappings that one usu-ally considers in higher dimensions is the mappings withfinite distortion thus in particular quasiconformal and qua-siregular mappings

For the dimensions 119899 = 2 and119870 = 1 the class of119870-quasi-regular mappings agrees with that of the complex-analyticfunctions Injective quasi-regular mappings in dimensions119899 ge 2 are called quasiconformal If 119866 is a domain in R119899119899 ge 2 we say that a mapping 119891 119866 rarr R119899 is discrete ifthe preimage of a point is discrete in the domain 119866 Planarquasiregular mappings are discrete and open (a fact usuallyproved via Stoılowrsquos Theorem)

Theorem A (Stoılowrsquos Theorem) For 119899 = 2 and 119870 ge 1 a 119870-quasi-regular mapping 119891 119866 rarr R2 can be represented in theform 119891 = 120601 ∘ 119892 where 119892 119866 rarr 119866

1015840 is a 119870-quasi-conformalhomeomorphism and 120601 is an analytic function on 1198661015840

There is no such representation in dimensions 119899 ge 3 ingeneral but there is representation of Stoılowrsquos type for qua-siregular mappings 119891 of the Riemann 119899-sphere S119899 = R

119899 cf[46]

Every quasiregular map 119891 S119899 rarr S119899 has a factorization119891 = 120601 ∘ 119892 where 119892 S119899 rarr S119899 is quasiconformal and 120601

S119899 rarr S119899 is uniformly quasiregularGehring-Lehto Lemma let 119891 be a complex continuous

and open mapping of a plane domainΩ which has finite par-tial derivatives ae inΩ Then 119891 is differentiable ae inΩ

Let 119880 be an open set in R119899 and let 119891 119880 rarr R119898 be amapping Set

119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

If 119891 is differentiable at 119909 then 119871(119909 119891) = |1198911015840(119909)| The theoremof Rademacher-Stepanov states that if 119871(119909 119891) lt infin ae then119891 is differentiable ae

Note that Gehring-Lehto Lemma is used in dimension119899 = 2 and Rademacher-Stepanov theorem to show the fol-lowing

For all dimensions 119899 ge 2 a quasiconformal mapping 119891 119866 rarr R119899 where 119866 is a domain in R119899 is differentiable ae in119866

Therefore the set 119878119891 of those points where it is not dif-ferentiable has Lebesgue measure zero Of course 119878119891 may benonempty in general and the behaviour of the mapping maybe very interesting at the points of this set Thus there is asubstantial difference between the two cases 119870 gt 1 and 119870 =

1 This indicates that the higher dimensions theory of quasi-regular mappings is essentially different from the theory inthe complex plane There are several reasons for this

(a) there are neither general representation theorems ofStoılowrsquos type nor counterparts of power series expan-sions in higher dimensions

(b) the usual methods of function theory based on Cau-chyrsquos Formula Morerarsquos Theorem Residue TheoremThe Residue Calculus and Consequences LaurentSeries Schwarzrsquos Lemma Automorphisms of the UnitDisc Riemann Mapping Theorem and so forth arenot applicable in the higher-dimensional theory

(c) in the plane case the class of conformal mappings isvery rich while in higher dimensions it is very small(J Liouville proved that for 119899 ge 3 and 119870 = 1 suffi-ciently smooth quasiconformal mappings are restric-tions of Mobius transformations)

(d) for dimensions 119899 ge 3 the branch set (ie the setof those points at which the mapping fails to be alocal homeomorphism) is more complicated than inthe two-dimensional case for instance it does notcontain isolated points

Injective quasiregular maps are called quasiconformalUsing the interaction between different coordinate systemsfor example spherical coordinates (120588 120579 120601) isin 119877+ times [0 2120587) times

[0 120587] and cylindrical coordinates (119903 120579 119905) one can constructcertain qc maps

Define119891 by (120588 120579 120601) 997891rarr (120601 120579 ln 120588)Then119891maps the cone119862(1206010) = (120588 120579 120601) 0 le 120601 lt 1206010 for 0 lt 1206010 le 120587 ontothe infinite cylinder (119903 120579 119905) 0 le 119903 lt 1206010 We leave it tothe reader as an exercise to check that the linear distortion119867depends only on 120601 and that119867(120588 120579 120601) = 120601 sin120601 le 1206010 sin1206010

For 1206010 = 1205872 we obtain a qc map of the half-space ontothe cylinder with the linear distortion bounded by 1205872

Since the half space and ball are conformally equivalentwe find that there is a qc map of the unit ball onto the infinitecylinder with the linear distortion bounded by 1205872

A simple example of noninjective quasiregular map 119891 isgiven in cylindrical coordinates in 3-space by the formula(119903 120579 119911) 997891rarr (119903 2120579 119911) This map is two-to-one and it is quasi-regular on any bounded domain in R3 whose closure doesnot intersect the 119911-axis The Jacobian 119869(119891) is different from 0

except on the 119911-axis and it is smooth everywhere except onthe 119911-axis

Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=

(119909 119910 119911) 119911 ge 0 The Zorich map 119885 R3 rarr R3 0 is aquasiregular analogue of the exponential function It can bedefined as follows

(1) Choose a bi-Lipschitz map ℎ [minus1205872 1205872]2 rarr 119878+


(2) Define 119885 [minus1205872 1205872]2times 119877 rarr 119867

3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)

(3) Extend 119885 to all of R3 by repeatedly reflecting inplanes

Quasiregularmaps ofR119899 which generalize the sine and cosinefunctions have been constructed by Drasin by Mayer and byBergweiler and Eremenko see in Fletcher and Nicks paper[47]

(B)There are somenewphenomena concerning quasireg-ular maps which are local homeomorphisms in dimensions119899 ge 3 A remarkable fact is that all smooth quasiregularmaps are local homeomorphisms Even more remarkable isthe following result of Zorich [48]

Theorem B1 Every quasiregular local homeomorphismR119899 rarr R119899 where 119899 ge 3 is a homeomorphism

The result was conjectured by M A Lavrentev in 1938The exponential function exp shows that there is no suchresult for 119899 = 2 Zorichrsquos theorem was generalized by Martioet al cf [49] (for a proof see also eg [35 Chapter III Section3])

Theorem B2 There is a number 119903 = 119903(119899 119870) (which onecalls the injectivity radius) such that every119870-quasiregular localhomeomorphism 119892 B rarr R119899 of the unit ball B sub 119877

119899 isactually homeomorphic on 119861(0 119903)

An immediate corollary of this is Zorichrsquos resultThis explains why in the definition of quasiregular maps

it is not reasonable to restrict oneself to smooth maps allsmooth quasiregular maps ofR119899 to itself are quasiconformal

In each dimension 119899 ge 3 there is a positive number 120598119899such that every nonconstant quasiregular mapping 119891 119863 rarr

R119899 whose distortion function satisfies119870120572120573(119909 119891) le 1 + 120598119899 forsome 1 le 120572 120573 le 119899 minus 1 is locally injective

(C) Despite the differences between the two theoriesdescribed in parts (A) and (B) many theorems about geo-metric properties of holomorphic functions of one complexvariable have been extended to quasiregular maps Theseextensions are usually highly nontrivial

(e) In a pioneering series of papers Reshetnyak provedin 1966ndash1969 that these mappings share the fun-damental topological properties of complex-analyticfunctions nonconstant quasi-regular mappings arediscrete open and sense-preserving cf [50] Here westate only the following

OpenMappingTheorem If119863 is open inR119899 and119891 is a noncon-stant qr function from119863 toR119899 we have that119891(119863) is open set(Note that this does not hold for real analytic functions)

An immediate consequence of the open mapping theo-rem is the maximum modulus principle It states that if 119891 isqr in a domain 119863 and |119891| achieves its maximum on 119863 then119891 is constant This is clear Namely if 119886 isin 119863 then the openmapping theorem says that 119891(119886) is an interior point of 119891(119863)and hence there is a point in119863 with larger modulus

(f) Reshetnyak also proved important convergence the-orems for these mappings and several analytic prop-erties they preserve sets of zero Lebesgue-measureare differentiable almost everywhere and are Holdercontinuous The Reshetnyak theory (which uses thephrase mapping with bounded distortion for ldquoquasi-regular mappingrdquo) makes use of Sobolev spacespotential theory partial differential equations cal-culus of variations and differential geometry Thosemappings solve important first-order systems of PDEsanalogous in many respects to the Cauchy-Riemannequation The solutions of these systems can beviewed as ldquoabsoluterdquo minimizers of certain energyfunctionals

(g) We have mentioned that all pure topological resultsabout analytic functions (such as theMaximumMod-ulus Principle and Rouchersquos theorem) extend to qua-siregularmaps Perhaps themost famous result of thissort is the extension of Picardrsquos theorem which is dueto Rickman cf [35]

A119870-quasiregularmapR119899 rarr R119899 can omit atmost a finiteset

When 119899 = 2 this omitted set can contain at most twopoints (this is a simple extension of Picardrsquos theorem) Butwhen 119899 gt 2 the omitted set can contain more than twopoints and its cardinality can be estimated from above interms of 119899 and 119870 There is an integer 119902 = 119902(119899 119870) such thatevery119870-qr mapping 119891 R119899 rarr R

119899 1198861 1198862 119886119902 where 119886119895

are disjoint is constant It was conjectured for a while that119902(119899 119870) = 2 Rickman gave a highly nontrivial example toshow that it is not the case for every positive integer 119901 thereexists a nonconstant119870-qr mapping 119891 R3 rarr R3 omitting 119901points

(h) It turns out that thesemappings havemany propertiessimilar to those of plane quasiconformal mappingsOn the other hand there are also striking differencesProbably the most important of these is that thereexists no analogue of the Riemann mapping theoremwhen 119899 gt 2 This fact gives rise to the following twoproblems Given a domain 119863 in Euclidean 119899-spacedoes there exist a quasiconformal homeomorphism119891 of 119863 onto the 119899-dimensional unit ball B119899 Next ifsuch a homeomorphism 119891 exists how small can thedilatation of 119891 be

Complete answers to these questions are knownwhen 119899 =2 For a plane domain 119863 can be mapped quasiconformallyonto the unit disk 119861 if and only if 119863 is simply connectedand has at least two boundary points The Riemann mappingtheorem then shows that if 119863 satisfies these conditionsthere exists a conformal homeomorphism 119891 of 119863 onto DThe situation is very much more complicated in higherdimensions and the Gehring-Vaisala paper [51] is devoted tothe study of these two questions in the case where 119899 = 3

(D) We close this subsection with short review of Dya-konovrsquos approach [15]

The main result of Dyakonovrsquos papers [6] (published inActa Math) is as follows


Theorem B Lipschitz-type properties are inherited from itsmodulus by analytic functions

A simple proof of this result was also published in ActaMath by Pavlovic cf [52]

(D1) In this item we shortly discuss the Lipschitz-typeproperties for harmonic functions The set of har-monic functions on a given open set119880 can be seen asthe kernel of the Laplace operator Δ and is thereforea vector space over R sums differences and scalarmultiples of harmonic functions are again harmonicIn several ways the harmonic functions are realanalogues to holomorphic functions All harmonicfunctions are real analytic that is they can be locallyexpressed as power series they satisfy the mean valuetheorem there is Liouvillersquos type theorem for themand so forth

Harmonic quasiregular (briefly hqr) mappings in theplane were studied first by Martio in [53] for a review of thissubject and further results see [25 54] and the referencescited there The subject has grown to include study of hqrmaps in higher dimensions which can be considered as anatural generalization of analytic function in plane and goodcandidate for a generalization of Theorem B

For example Chen et al [19] and the author [55] haveshown that Lipschitz-type properties are inherited from itsmodulo for119870-quasiregular andharmonicmappings in planarcase These classes include analytic functions in planar caseso this result is a generalization of Theorem A

(D2) Quasiregular mappings Dyakonov [15] made furtherimportant step and roughly speaking showed thatLipschitz-type properties are inherited from its mod-ulus by qc mappings (see two next results)

Theorem B3 (Theorem Dy see [15 Theorem 4]) Let 0 lt

120572 le 1 119870 ge 1 and 119899 ge 2 Suppose 119866 is a Λ120572-extension

domain inR119899 and 119891 is a119870-quasiconformal mapping of119866 onto119891(119866) sub R119899 Then the following conditions are equivalent

(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)

If in addition 119866 is a uniform domain and if 120572 le 1205720 = 120572119899119870 =

1198701(1minus119899) then (i5) and (ii5) are equivalent to

(iii5) |119891| isin Λ120572(119866 120597119866)

An example in Section 2 [15] shows that the assumption120572 le 1205720 = 120572

119899119870 = 119870

1(1minus119899) cannot be dropped For 119899 ge 3we have the following generalizationmdashbut also a consequenceof Theorem B3 dealing with quasiregular mappings that arelocal homeomorphisms (ie have no branching points)

Theorem B4 (see [15 Proposition 3]) Let 0 lt 120572 le 1 119870 ge 1and 119899 ge 3 Suppose 119866 is a Λ120572-extension domain in R119899 and 119891is a119870-quasiregular locally injective mapping of 119866 onto 119891(119866) subR119899 Then 119891 isin Λ


We remark that a quasiregular mapping 119891 in dimension119899 ge 3 will be locally injective (or equivalently locallyhomeomorphic) if the dilatation 119870(119891) is sufficiently closeto 1 For more sophisticated local injectivity criteria see theliterature cited in [10 15]

Here we only outline how to reduce the proof ofTheorem B4 to qc case

It is known that by Theorem B2 for given 119899 ge 3 and119870 ge 1 there is a number 119903 = 119903(119899 119870) such that every 119870-quasiregular local homeomorphism 119892 B rarr 119877

119899 of the unitball B sub R119899 is actually homeomorphic on 119861(0 119903) Applyingthis to themappings 119892119909(119908) = 119891(119909+119889(119909)119908)119908 isin B where 119909 isin119866 and 119889(119909) = 119889(119909 120597119866) we see that 119891 is homeomorphic (andhence quasiconformal) on each ball 119861119903(119909) = 119861(0 119903119889(119909))Theconstant 119903 = 119903(119899 119870) isin (0 1) coming from the precedingstatement depends on 119899 and119870 = 119870(119891) but not on 119909

Note also the following(i0) Define 119898119891(119909 119903) = min|119891(1199091015840) minus 119891(119909)| |1199091015840 minus 119909| = 119903

and 119872119891(119909 119903) = max|119891(1199091015840) minus 119891(119909)| |1199091015840minus 119909| = 119903 We

can express the distortion property of qc mappings in thefollowing useful form

Proposition A Suppose that 119866 is a domain in R119899 and 119891

119866 rarr R119899 is 119870-quasiconformal and 1198661015840 = 119891(119866) There is aconstant 119888 such that119872119891(119909 119903) le 119888119898119891(119909 119903) for 119909 isin 119866 and 119903 =119889(119909)2 This form shows in an explicit way that the maximaldilatation of a qc mapping is controlled by minimal and it isconvenient for some applications For example Proposition Aalso yields a simple proof of Theorem B3

Recall the following (i1) Roughly speaking quasiconformaland quasiregular mappings in R119899 119899 ge 3 are natural gener-alizations of conformal and analytic functions of one complexvariable

(i2) Dyakonovrsquos proof of Theorem 64 in [15] (as we haveindicated above) is reduced to quasiconformal case but it seemslikely that he wanted to consider whether the theorem holdsmore generally for quasiregular mappings

(i3) Quasiregular mappings are much more general thanquasiconformal mappings and in particular analytic functions

(E) Taking into account the above discussion it is naturalto explore the following research problem

Question A Is it possible to drop local homeomorphismhypothesis in Theorem B4

It seems that using the approach from [15] we cannotsolve this problem and that we need new techniques

(i4) However we establish the second version of Koebetheorem for 119870-quasiregular functions Theorem 39 and thecharacterization of Lipschitz-type spaces for quasiregularmappings by the average Jacobian Theorem 40 Using thesetheorems we give a positive solution to Question A

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

Theauthor is grateful to the anonymous referees for a numberof helpful suggestions and in particular to the Editor in ChiefProfessor Ciric for patiently reading through the whole textat several stages of its revision This research was partiallysupported by MNTRS Serbia Grant no 174 032

References

[1] K Astala and FWGehring ldquoInjectivity the BMOnormand theuniversal Teichmuller spacerdquo Journal drsquoAnalyse Mathematiquevol 46 pp 16ndash57 1986

[2] F W Gehring and O Martio ldquoLipschitz classes and quasi-conformal mappingsrdquo Annales Academiaelig Scientiarum FennicaeligSeries A I Mathematica vol 10 pp 203ndash219 1985

[3] C A Nolder ldquoA quasiregular analogue of a theorem of Hardyand Littlewoodrdquo Transactions of the American MathematicalSociety vol 331 no 1 pp 215ndash226 1992

[4] K Astala and FWGehring ldquoQuasiconformal analogues of the-orems of Koebe and Hardy-Littlewoodrdquo The Michigan Mathe-matical Journal vol 32 no 1 pp 99ndash107 1985

[5] K Kim ldquoHardy-Littlewood property with the inner lengthmet-ricrdquoKoreanMathematical Society vol 19 no 1 pp 53ndash62 2004

[6] KMDyakonov ldquoEquivalent norms on Lipschitz-type spaces ofholomorphic functionsrdquo Acta Mathematica vol 178 no 2 pp143ndash167 1997

[7] A Abaob M Arsenovic M Mateljevic and A ShkheamldquoModuli of continuity of harmonic quasiregular mappings onbounded domainsrdquo Annales Academiaelig Scientiarum Fennicaeligvol 38 no 2 pp 839ndash847 2013

[8] MMateljevic andM Vuorinen ldquoOn harmonic quasiconformalquasi-isometriesrdquo Journal of Inequalities and Applications vol2010 Article ID 178732 2010

[9] J Qiao and X Wang ldquoLipschitz-type spaces of pluriharmonicmappingsrdquo Filomat vol 27 no 4 pp 693ndash702 2013

[10] T Iwaniec and G Martin Geometric FunctionTheory and Non-linear Analysis Syraccuse Aucland New Zealand 2000

[11] M VuorinenConformal Geometry and Quasiregular Mappingsvol 1319 of Lecture Notes in Mathematics Springer BerlinGermany 1988

[12] F W Gehring and B G Osgood ldquoUniform domains and thequasihyperbolic metricrdquo Journal drsquoAnalyse Mathematique vol36 pp 50ndash74 1979

[13] M Arsenovic V Manojlovic and M Mateljevic ldquoLipschitz-type spaces and harmonic mappings in the spacerdquo AnnalesAcademiaelig Scientiarum FennicaeligMathematica vol 35 no 2 pp379ndash387 2010

[14] Sh ChenMMateljevic S Ponnusamy andXWang ldquoLipschitztype spaces and Landau-Bloch type theorems for harmonicfunctions and solutions to Poisson equationsrdquo to appear

[15] K M Dyakonov ldquoHolomorphic functions and quasiconformalmappings with smooth modulirdquo Advances in Mathematics vol187 no 1 pp 146ndash172 2004

[16] A Hinkkanen ldquoModulus of continuity of harmonic functionsrdquoJournal drsquoAnalyse Mathematique vol 51 pp 1ndash29 1988

[17] V Lappalainen ldquoLipℎ-extension domainsrdquo Annales AcademiaeligScientiarumFennicaelig Series A IMathematicaDissertationes no56 p 52 1985

[18] O Martio and R Nakki ldquoBoundary Holder continuity andquasiconformalmappingsrdquo Journal of the LondonMathematicalSociety Second Series vol 44 no 2 pp 339ndash350 1991

[19] S Chen S Ponnusamy andXWang ldquoOn planar harmonic Lip-schitz and planar harmonic Hardy classesrdquo Annales AcademiaeligScientiarum Fennicaelig Mathematica vol 36 no 2 pp 567ndash5762011

[20] Sh Chen S Ponnusamy and X Wang ldquoEquivalent moduliof continuity Blochrsquos theorem for pluriharmonic mappings in119861119899rdquo Proceedings of Indian Academy of SciencesmdashMathematical

Sciences vol 122 no 4 pp 583ndash595 2012[21] Sh Chen S PonnusamyM Vuorinen and XWang ldquoLipschitz

spaces and bounded mean oscillation of harmonic mappingsrdquoBulletin of the AustralianMathematical Society vol 88 no 1 pp143ndash157 2013

[22] S Chen S Ponnusamy and A Rasila ldquoCoefficient estimatesLandaursquos theorem and Lipschitz-type spaces on planar har-monic mappingsrdquo Journal of the Australian Mathematical Soci-ety vol 96 no 2 pp 198ndash215 2014

[23] F W Gehring ldquoRings and quasiconformal mappings in spacerdquoTransactions of the American Mathematical Society vol 103 pp353ndash393 1962

[24] J VaisalaLectures on n-DimensionalQuasiconformalMappingsSpringer New York NY USA 1971

[25] M Mateljevic ldquoQuasiconformality of harmonic mappingsbetween Jordan domainsrdquo Filomat vol 26 no 3 pp 479ndash5102012

[26] D Kalaj and M S Mateljevic ldquoHarmonic quasiconformal self-mappings and Mobius transformations of the unit ballrdquo PacificJournal of Mathematics vol 247 no 2 pp 389ndash406 2010

[27] D Kalaj ldquoOn harmonic diffeomorphisms of the unit disc ontoa convex domainrdquo Complex Variables vol 48 no 2 pp 175ndash1872003

[28] M Mateljevic Topics in Conformal Quasiconformal and Har-monic Maps Zavod za Udzbenike Belgrade Serbia 2012

[29] M Mateljevic ldquoDistortion of harmonic functions and har-monic quasiconformal quasi-isometryrdquo Revue Roumaine deMathematiques Pures et Appliquees vol 51 no 5-6 pp 711ndash7222006

[30] M Knezevic and M Mateljevic ldquoOn the quasi-isometries ofharmonic quasiconformal mappingsrdquo Journal of MathematicalAnalysis and Applications vol 334 no 1 pp 404ndash413 2007

[31] V Manojlovic ldquoBi-Lipschicity of quasiconformal harmonicmappings in the planerdquo Filomat vol 23 no 1 pp 85ndash89 2009

[32] M Mateljevic Communications at Analysis Seminar Universityof Belgrade 2012

[33] M Mateljevic ldquoQuasiconformal and Quasiregular harmonicmappings and Applicationsrdquo Annales Academiaelig ScientiarumFennicaelig Mathematica vol 32 pp 301ndash315 2007

[34] I Anic V Markovic and M Mateljevic ldquoUniformly boundedmaximal 120593-disks Bers space and harmonic mapsrdquo Proceedingsof the AmericanMathematical Society vol 128 no 10 pp 2947ndash2956 2000

[35] S Rickman Quasiregular Mappings vol 26 Springer BerlinGermany 1993

[36] O Lehto and K I Virtanen Quasiconformal Mappings in thePlane Springer New York NY USA 2nd edition 1973

[37] DGilbarg andN S TrudingerElliptic Partial Differential Equa-tions of Second Order vol 224 of Grundlehren der Mathematis-chen Wissenschaften [Fundamental Principles of MathematicalSciences] Springer Berlin Germany 2nd edition 1983


[38] B Burgeth ldquoA Schwarz lemma for harmonic and hyperbolic-harmonic functions in higher dimensionsrdquoManuscripta Math-ematica vol 77 no 2-3 pp 283ndash291 1992

[39] S Axler P Bourdon andW RameyHarmonic FunctionTheoryvol 137 of Graduate Texts in Mathematics Springer New YorkNY USA 1992

[40] W Rudin Function Theory in the Unit Ball of C119899 SpringerBerlin Germany 1980

[41] R Nakki and B Palka ldquoLipschitz conditions and quasiconfor-mal mappingsrdquo Indiana University Mathematics Journal vol 31no 3 pp 377ndash401 1982

[42] M Mateljevic ldquoVersions of Koebe 14 theorem for analyticand quasiregular harmonic functions and applicationsrdquo InstitutMathematique Publications Nouvelle Serie vol 84 pp 61ndash722008

[43] P Duren Harmonic Mappings in the Plane vol 156 of Cam-bridge Tracts in Mathematics Cambridge University PressCambridge UK 2004

[44] D Kalaj Harmonic functions and quasiconformal mappings[MS thesis] Harmonijske Funkcije i Kvazikonformna Preslika-vanja Belgrade Serbia 1998

[45] E Heinz ldquoOn one-to-one harmonic mappingsrdquo Pacific Journalof Mathematics vol 9 no 1 pp 101ndash105 1959

[46] G Martin and K Peltonen ldquoStoılow factorization for quasireg-ular mappings in all dimensionsrdquo Proceedings of the AmericanMathematical Society vol 138 no 1 pp 147ndash151 2010

[47] A N Fletcher and D A Nicks ldquoIteration of quasiregular tan-gent functions in three dimensionsrdquo Conformal Geometry andDynamics vol 16 pp 1ndash21 2012

[48] V A Zorich ldquoThe global homeomorphism theorem for spacequasiconformal mappings its development and related openproblemsrdquo in Quasiconformal Space Mappings vol 1508 ofLecture Notes in Mathematics pp 132ndash148 Springer BerlinGermany 1992

[49] O Martio S Rickman and J Vaisala ldquoTopological and metricproperties of quasiregular mappingsrdquoAnnales Academiaelig Scien-tiarum Fennicaelig Mathematica vol 488 pp 1ndash31 1971

[50] Yu G Reshetnyak ldquoSpace mappings with bounded distortionrdquoSiberian Mathematical Journal vol 8 no 3 pp 466ndash487 1967

[51] F W Gehring and J Vaisala ldquoThe coefficients of quasiconfor-mality of domains in spacerdquo Acta Mathematica vol 114 pp 1ndash70 1965

[52] M Pavlovic ldquoOn Dyakonovrsquos paper lsquoEquivalent norms onlipschitz-type spaces of holomorphic functionsrsquordquo Acta Mathe-matica vol 183 no 1 pp 141ndash143 1999

[53] O Martio ldquoOn harmonic quasiconformal mappingsrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 425 pp 3ndash10 1968

[54] M Arsenovic V Kojic and M Mateljevic ldquoOn Lipschitz conti-nuity of harmonic quasiregular maps on the unit ball in 119877

119899rdquoAnnales Academiaelig Scientiarum Fennicaelig Mathematica vol 33no 1 pp 315ndash318 2008

[55] M Mateljevic ldquoA version of Blochrsquos theorem for quasiregularharmonic mappingsrdquo Revue Roumaine de Mathematiques Pureset Appliquees vol 47 no 5-6 pp 705ndash707 2002

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Stochastic AnalysisInternational Journal of


for quasiregular mappings by the average Jacobian is alsoestablished inTheorem 40

By R119899 = 119909 = (1199091 119909119899) 1199091 119909119899 isin R denotethe real vector space of dimension 119899 For a domain 119863 in R119899

with nonempty boundary we define the distance function119889 = 119889(119863) = dist(119863) by 119889(119909) = 119889(119909 120597119863) = dist(119863)(119909) =inf|119909minus119910| 119910 isin 120597119863 and if 119891maps119863 onto1198631015840 sub R119899 in somesettings it is convenient to use short notation 119889lowast = 119889119891(119909) for119889(119891(119909) 120597119863

1015840) It is clear that 119889(119909) = dist(119909 119863119888) where 119863119888 is

the complement of119863 in R119899Let 119866 be an open set in R119899 A mapping 119891 119866 rarr R119898

is differentiable at 119909 isin 119866 if there is a linear mapping 1198911015840(119909) R119899 rarr R119899 called the derivative of 119891 at 119909 such that

119891 (119909 + ℎ) minus 119891 (119909) = 1198911015840(119909) ℎ + |ℎ| 120576 (119909 ℎ) (1)

where 120576(119909 ℎ) rarr 0 as ℎ rarr 0 For a vector-valued function119891 119866 rarr R119899 where 119866 sub R119899 is a domain we define100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816= max|ℎ|=1

100381610038161003816100381610038161198911015840(119909) ℎ

10038161003816100381610038161003816 119897 (119891

1015840(119909)) = min

|ℎ|=1

100381610038161003816100381610038161198911015840(119909) ℎ

10038161003816100381610038161003816

(2)

when 119891 is differentiable at 119909 isin 119866In Section 3 we review some results from [7 8] For

example in [7] under some conditions concerning amajorant120596 we showed the following

Let 119891 isin 1198621(119863R119899) and let 120596 be a continuous majorant

such that 120596lowast(119905) = 120596(119905)119905 is nonincreasing for 119905 gt 0Assume 119891 satisfies the following property (which we call

Hardy-Littlewood (119888 120596)-property)

(HL (119888 120596)) 119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 119888120596 (119889 (119909))

119909 isin 119863 where 119889 (119909) = dist (119909119863119888) (3)

Then

(locΛ) 119891 isin locΛ 120596 (119866) (4)

If in addition 119891 is harmonic in 119863 or more generally119891 isin 119874119862

2(119863) then (HL(119888 120596)) is equivalent to (locΛ) If 119863

is a Λ 120596-extension domain then (HL(119888 120596)) is equivalent to119891 isin Λ 120596(119863)

In Section 3 we also consider Lipschitz-type spaces ofpluriharmonic mappings and extend some results from [9]

In Appendices A and B we discuss briefly distortion ofharmonic qc maps background of the subject and basicproperty of qr mappings respectively For more details onrelated qr mappings we refer the interested reader to [10]

2 Quasiconformal Analogue of KoebersquosTheorem and Applications

Throughout the paper we denote byΩ 119866 and119863 open subsetof R119899 119899 ge 1

Let119861119899(119909 119903) = 119911 isin R119899 |119911minus119909| lt 119903 119878119899minus1(119909 119903) = 120597119861119899(119909 119903)(abbreviated 119878(119909 119903)) and let B119899 119878119899minus1 stand for the unit balland the unit sphere in R119899 respectively Sometimes we writeD and T instead ofB2 and 120597D respectively For a domain119866 sub

R119899 let 120588 119866 rarr [0infin) be a continuous function We say that120588 is a weight function or a metric density if for every locallyrectifiable curve 120574 in 119866 the integral

119897120588 (120574) = int120574120588 (119909) 119889119904 (5)

exists In this case we call 119897120588(120574) the 120588-length of 120574 A metricdensity defines a metric 119889120588 119866 times 119866 rarr [0infin) as follows For119886 119887 isin 119866 let

119889120588 (119886 119887) = inf120574119897120588 (120574) (6)

where the infimum is taken over all locally rectifiable curvesin 119866 joining 119886 and 119887

For the modern mapping theory which also considersdimensions 119899 ge 3 we do not have a Riemann mappingtheorem and therefore it is natural to look for counterparts ofthe hyperbolicmetric So-called hyperbolic typemetrics havebeen the subject of many recent papers Perhaps the mostimportant metrics of these metrics are the quasihyperbolicmetric120581119866 and the distance ratiometric 119895119866 of a domain119866 sub R119899

(see [11 12]) The quasihyperbolic metric 120581 = 120581119866 of 119866 is aparticular case of the metric 119889120588 when 120588(119909) = 1119889(119909 120597119866) (see[11 12])

Given a subset 119864 of C119899 or R119899 a function 119891 119864 rarr C (ormore generally a mapping 119891 from 119864 into C119898 or R119898) is saidto belong to the Lipschitz space Λ 120596(119864) if there is a constant119871 = 119871(119891) = 119871(119891 119864) = 119871(119891 120596 119864) which we call Lipschitzconstant such that

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le 119871120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (7)

for all 119909 119910 isin 119864 The norm 119891Λ120596(119864) is defined as the smallest

119871 in (7)There has been much work on Lipschitz-type properties

of quasiconformal mappings This topic was treated amongother places in [1ndash5 7 13ndash22]

As in most of those papers we will currently restrictourselves to the simplest majorants 120596120572(119905) = 119905

120572(0 lt 120572 le 1)

The classes Λ 120596(119864) with 120596 = 120596120572 will be denoted by Λ120572 (or byLip(120572 119864) = Lip(120572 119871 119864)) 119871(119891) and 120572 are called respectivelyLipschitz constant and exponent (of 119891 on 119864) We say that adomain 119866 sub 119877

119899 is uniform if there are constants 119886 and 119887 suchthat each pair of points 1199091 1199092 isin 119866 can be joined by rectifiablearc 120574 in 119866 for which

119897 (120574) le 11988610038161003816100381610038161199091 minus 1199092

1003816100381610038161003816

min 119897 (120574119895) le 119887119889 (119909 120597119866)(8)

for each 119909 isin 120574 here 119897(120574) denotes the length of 120574 and 1205741 1205742the components of 120574 119909 We define 119862(119866) = max119886 119887The smallest 119862(119866) for which the previous inequalities holdis called the uniformity constant of 119866 and we denote it by119888lowast= 119888

lowast(119866) Following [2 17] we say that a function119891 belongs

to the local Lipschitz space locΛ 120596(119866) if (7) holds with a fixed119862 gt 0 whenever 119909 isin 119866 and |119910 minus 119909| lt (12)119889(119909 120597119866) We saythat 119866 is a Λ 120596-extension domain if Λ 120596(119866) = locΛ 120596(119866) In





120574120588(119909)|119889119909| ge 1 for


119860119901 (120588) = intR119899

1003816100381610038161003816120588 (119909)

1003816100381610038161003816

119901119889119881 (119909) (9)


119872119901 (Γ) = inf 119860119901 (120588) 120588 isin F (Γ) (10)




119870119868 (119891) = sup119872(Γ

lowast)

119872 (Γ)

119870119874 (119891) = sup 119872(Γ)

119872 (Γlowast)

(11)









119867(119909 119891) = lim sup119903rarr0

119871 (119909 119891 119903)

119897 (119909 119891 119903)

(12)







10038161003816100381610038161003816

119899le 119870119869119891 (119909) ae (13)




119899119870(119903) = 0

(3) lim119903rarr1120579119899119870(119903) = infin



1003816100381610038161003816119891 (119910) minus 119891 (119909)

1003816100381610038161003816

119889 (119891 (119909) 120597Ω1015840)

le 120579119899119870 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816

119889 (119909 120597Ω)

) (14)


119886119891 (119909) = 119886119891119866 (119909) = exp( 1

11989910038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

log 119869119891 (119911) 119889119911) 119909 isin 119866

(15)





1

119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

le 119886119891119866 (119909) le 119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

119909 isin 119866

(16)


Set 1205720 = 120572119899119870 = 119870

1(1minus119899)


119886119891119863 (119909) le 119898119889(119909 120597119863)120572minus1 for 119909 isin 119863 (17)


1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119863))

120572 (18)


lowast(119863)


1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(19)


119909 119886 = 1198701(1minus119899) is 119870-


|119909 minus 0|119886= |119909|




1

119888

119892 (119909) le 119891 (119909) le 119888119892 (119909) 119909 isin Ω (20)




Δ119906 = 0 (21)





we define 120582ℎ(119911) = 119863minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892

1015840(119911)|






(1 minus 1198962)

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2le 119869ℎ le 119870

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2 (22)


log 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816=

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578 (23)

Hence

log 119886ℎ119863 (119911) le1

2

log119870 +

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578

= logradic119870 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816

(24)



Λ (ℎ 119911) ≍

119889 (ℎ119911 1205971198631015840)

119889 (119911 120597119863)

≍ 120582 (ℎ 119911) (25)


1198961198631015840 (ℎ (1199111) ℎ (1199112)) ≍ 119896119863 (1199111 1199112) 1199111 1199112 isin 119863 (26)







119864119891119866 (119909) =1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (27)



Define

119869119891= 119869

119891119866=119899

radic119864119891119866 (28)




1

119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

le 119869119891119866

(119909) le 119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

119909 isin 119866

(29)




(30)

where 119889lowast(119909) = 119889(119891(119909)) = 119889(119891(119909) 1205971198661015840) and 119889(119909) = 119889(119909

120597119866) Hence

1003816100381610038161003816119861 (119891 (119909) 119888lowast119889lowast)

1003816100381610038161003816le int

119861119909

119869119891 (119911) 119889119911 le1003816100381610038161003816119861 (119891 (119909) 119862lowast119889lowast)

1003816100381610038161003816 (31)



R119899



hold






119896 (119909 1199090) = 119889(1199090)minus1 1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816(1 + 119900 (1)) 119909 997888rarr 1199090

119896 (119891 (119909) 119891 (1199090)) =

1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816

119889lowast (119891 (1199090))(1 + 119900 (1))

when 119909 997888rarr 1199090

(32)






(c1) 119899radic119869119891 asymp 119886119891

(d1) 119899radic119869119891 asymp 119869119891







119869119891 (119909) gt 0 119909 isin Ω (33)













1015840119909 minus 119894ℎ

1015840119910) and 120597ℎ = (12)(ℎ

1015840119909 + 119894ℎ

1015840119910)



|120597ℎ (0)| ge

119877

4

(34)


120582ℎ (0) ge1 minus 119896

4

119877 (35)



minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892


KoebeTheorem then

1

16

119863minus(0) le 119889ℎ (0) (36)

and therefore

(1 minus |119911|2)

1

16

119863minus(119911) le 119889ℎ (119911) (37)


(1 minus |119911|2)119863

+(119911) le 16119870119889ℎ (119911) (38)


(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911) (39)




(1 minus |119911|2)Λ ℎ (119911) le 16119870119889ℎ (119911)

(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911)

(40)


119889 (119911) Λ ℎ (119911) le 16119870119889ℎ (119911)

119889 (119911) 120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911)

(41)

and1 minus 119896

4

120581119863 (119911 1199111015840) le 1205811198631015840 (ℎ119911 ℎ119911

1015840)

le 16119870120581119863 (119911 1199111015840)

(42)

where 119911 1199111015840 isin 119863







|119889119908|

|Re119908|=

|119889119911|

|119911| ln (1 |119911|) (43)




1015840) when

1199111015840rarr 119911


(A2) 120581D1015840(119911 1199111015840) le 119888120581Πminus(119908 119908


119911 = 119890119908 and 1199111015840 = 119890119908

1015840



1

2119889

le 120588 le

2

119889

(44)


1 + 119900 (1)

119889 ln (1119889)le 120588 le

2

119889

(45)

Hence 119889hyp(119911 1199111015840) le 2120581119863(119911 119911

1015840) 119911 1199111015840 isin 119863








(1 minus |119911|2) 120582ℎlowast (120577) ge

1 minus 119896

4

119889ℎ (119911) (46)


1015840(120577)| and 120588hyp119863(119911)|120601

1015840(120577)| =

120588D(120577)

120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911) 120588119863 (119911) (47)


(1 minus |119911|2) 120582ℎ (119911) ge (1 minus 119896) 1205910119889ℎ (119911) (48)

where 1205910 = 3radic32120587








1003816100381610038161003816le 119888119889 (119891 (119911)) for every 1199111 isin 119861119911 (49)



osc119881119906 = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

1003816100381610038161003816 119909 119910 isin 119881

120596119906 (119903 119909) = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

10038161003816100381610038161003816100381610038161003816119909 minus 119910

1003816100381610038161003816= 119903

(50)


1(119866) = 119874119862


1(119866) such

that

119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 1198881osc119861

119909

119891 (51)



1 Then119891 isin 119874119862








119864119891119866 (119909) 1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (52)




119899119870 = 119870

1(1minus119899) 119862lowast = 120579119899119870(12) 120579

119899119870(119888lowast) = 12

and

119869119891119866

=119899

radic119864119891119866 (53)


119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (54)

and 119869119886V119891 (119861 119909) = 119899



⟨119909 119910⟩ = 11990911199101 + sdot sdot sdot + 119909119899119910119899 (55)


|119909| = ⟨119909 119909⟩12

= (100381610038161003816100381611990911003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199091198991003816100381610038161003816

2)

12 (56)






100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119874 (119909)

10038161003816100381610038161003816119869119891 (119909)

10038161003816100381610038161003816ae (57)




119869119891 (119909) le 1198701015840119897(119891

1015840(119909))

119899ae (58)






119863119901 (119891 119861) = (1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119901)

1119901

le 119872119903120574minus1

(59)







119869119891 (119886 119903) le 119872119903120574minus1

(60)




proof



1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)











119869119891 (119909 119903) le 119888 119903120572minus1 (61)








minus1(119910119896) 119896 = 1 2






the following




1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572


1015840le 1198881119903

120572 where 1198881 =








1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)






(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)


119869119891 (119909 119903) le 1198883119903120572minus1 (63)




(iv2) |119891| isin Λ120572(119866 120597119866)







1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120574120588(119909)|119889119909| ge 1 for


119860119901 (120588) = intR119899

1003816100381610038161003816120588 (119909)

1003816100381610038161003816

119901119889119881 (119909) (9)


119872119901 (Γ) = inf 119860119901 (120588) 120588 isin F (Γ) (10)




119870119868 (119891) = sup119872(Γ

lowast)

119872 (Γ)

119870119874 (119891) = sup 119872(Γ)

119872 (Γlowast)

(11)









119867(119909 119891) = lim sup119903rarr0

119871 (119909 119891 119903)

119897 (119909 119891 119903)

(12)







10038161003816100381610038161003816

119899le 119870119869119891 (119909) ae (13)




119899119870(119903) = 0

(3) lim119903rarr1120579119899119870(119903) = infin



1003816100381610038161003816119891 (119910) minus 119891 (119909)

1003816100381610038161003816

119889 (119891 (119909) 120597Ω1015840)

le 120579119899119870 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816

119889 (119909 120597Ω)

) (14)


119886119891 (119909) = 119886119891119866 (119909) = exp( 1

11989910038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

log 119869119891 (119911) 119889119911) 119909 isin 119866

(15)





1

119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

le 119886119891119866 (119909) le 119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

119909 isin 119866

(16)


Set 1205720 = 120572119899119870 = 119870

1(1minus119899)


119886119891119863 (119909) le 119898119889(119909 120597119863)120572minus1 for 119909 isin 119863 (17)


1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119863))

120572 (18)


lowast(119863)


1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(19)


119909 119886 = 1198701(1minus119899) is 119870-


|119909 minus 0|119886= |119909|




1

119888

119892 (119909) le 119891 (119909) le 119888119892 (119909) 119909 isin Ω (20)




Δ119906 = 0 (21)





we define 120582ℎ(119911) = 119863minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892

1015840(119911)|






(1 minus 1198962)

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2le 119869ℎ le 119870

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2 (22)


log 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816=

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578 (23)

Hence

log 119886ℎ119863 (119911) le1

2

log119870 +

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578

= logradic119870 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816

(24)



Λ (ℎ 119911) ≍

119889 (ℎ119911 1205971198631015840)

119889 (119911 120597119863)

≍ 120582 (ℎ 119911) (25)


1198961198631015840 (ℎ (1199111) ℎ (1199112)) ≍ 119896119863 (1199111 1199112) 1199111 1199112 isin 119863 (26)







119864119891119866 (119909) =1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (27)



Define

119869119891= 119869

119891119866=119899

radic119864119891119866 (28)




1

119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

le 119869119891119866

(119909) le 119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

119909 isin 119866

(29)




(30)

where 119889lowast(119909) = 119889(119891(119909)) = 119889(119891(119909) 1205971198661015840) and 119889(119909) = 119889(119909

120597119866) Hence

1003816100381610038161003816119861 (119891 (119909) 119888lowast119889lowast)

1003816100381610038161003816le int

119861119909

119869119891 (119911) 119889119911 le1003816100381610038161003816119861 (119891 (119909) 119862lowast119889lowast)

1003816100381610038161003816 (31)



R119899



hold






119896 (119909 1199090) = 119889(1199090)minus1 1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816(1 + 119900 (1)) 119909 997888rarr 1199090

119896 (119891 (119909) 119891 (1199090)) =

1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816

119889lowast (119891 (1199090))(1 + 119900 (1))

when 119909 997888rarr 1199090

(32)






(c1) 119899radic119869119891 asymp 119886119891

(d1) 119899radic119869119891 asymp 119869119891







119869119891 (119909) gt 0 119909 isin Ω (33)













1015840119909 minus 119894ℎ

1015840119910) and 120597ℎ = (12)(ℎ

1015840119909 + 119894ℎ

1015840119910)



|120597ℎ (0)| ge

119877

4

(34)


120582ℎ (0) ge1 minus 119896

4

119877 (35)



minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892


KoebeTheorem then

1

16

119863minus(0) le 119889ℎ (0) (36)

and therefore

(1 minus |119911|2)

1

16

119863minus(119911) le 119889ℎ (119911) (37)


(1 minus |119911|2)119863

+(119911) le 16119870119889ℎ (119911) (38)


(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911) (39)




(1 minus |119911|2)Λ ℎ (119911) le 16119870119889ℎ (119911)

(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911)

(40)


119889 (119911) Λ ℎ (119911) le 16119870119889ℎ (119911)

119889 (119911) 120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911)

(41)

and1 minus 119896

4

120581119863 (119911 1199111015840) le 1205811198631015840 (ℎ119911 ℎ119911

1015840)

le 16119870120581119863 (119911 1199111015840)

(42)

where 119911 1199111015840 isin 119863







|119889119908|

|Re119908|=

|119889119911|

|119911| ln (1 |119911|) (43)




1015840) when

1199111015840rarr 119911


(A2) 120581D1015840(119911 1199111015840) le 119888120581Πminus(119908 119908


119911 = 119890119908 and 1199111015840 = 119890119908

1015840



1

2119889

le 120588 le

2

119889

(44)


1 + 119900 (1)

119889 ln (1119889)le 120588 le

2

119889

(45)

Hence 119889hyp(119911 1199111015840) le 2120581119863(119911 119911

1015840) 119911 1199111015840 isin 119863








(1 minus |119911|2) 120582ℎlowast (120577) ge

1 minus 119896

4

119889ℎ (119911) (46)


1015840(120577)| and 120588hyp119863(119911)|120601

1015840(120577)| =

120588D(120577)

120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911) 120588119863 (119911) (47)


(1 minus |119911|2) 120582ℎ (119911) ge (1 minus 119896) 1205910119889ℎ (119911) (48)

where 1205910 = 3radic32120587








1003816100381610038161003816le 119888119889 (119891 (119911)) for every 1199111 isin 119861119911 (49)



osc119881119906 = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

1003816100381610038161003816 119909 119910 isin 119881

120596119906 (119903 119909) = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

10038161003816100381610038161003816100381610038161003816119909 minus 119910

1003816100381610038161003816= 119903

(50)


1(119866) = 119874119862


1(119866) such

that

119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 1198881osc119861

119909

119891 (51)



1 Then119891 isin 119874119862








119864119891119866 (119909) 1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (52)




119899119870 = 119870

1(1minus119899) 119862lowast = 120579119899119870(12) 120579

119899119870(119888lowast) = 12

and

119869119891119866

=119899

radic119864119891119866 (53)


119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (54)

and 119869119886V119891 (119861 119909) = 119899



⟨119909 119910⟩ = 11990911199101 + sdot sdot sdot + 119909119899119910119899 (55)


|119909| = ⟨119909 119909⟩12

= (100381610038161003816100381611990911003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199091198991003816100381610038161003816

2)

12 (56)






100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119874 (119909)

10038161003816100381610038161003816119869119891 (119909)

10038161003816100381610038161003816ae (57)




119869119891 (119909) le 1198701015840119897(119891

1015840(119909))

119899ae (58)






119863119901 (119891 119861) = (1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119901)

1119901

le 119872119903120574minus1

(59)







119869119891 (119886 119903) le 119872119903120574minus1

(60)




proof



1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)











119869119891 (119909 119903) le 119888 119903120572minus1 (61)








minus1(119910119896) 119896 = 1 2






the following




1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572


1015840le 1198881119903

120572 where 1198881 =








1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)






(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)


119869119891 (119909 119903) le 1198883119903120572minus1 (63)




(iv2) |119891| isin Λ120572(119866 120597119866)







1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

119888

119892 (119909) le 119891 (119909) le 119888119892 (119909) 119909 isin Ω (20)




Δ119906 = 0 (21)





we define 120582ℎ(119911) = 119863minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892

1015840(119911)|






(1 minus 1198962)

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2le 119869ℎ le 119870

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

2 (22)


log 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816=

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578 (23)

Hence

log 119886ℎ119863 (119911) le1

2

log119870 +

1

210038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

log 100381610038161003816100381610038161198911015840(120577)

10038161003816100381610038161003816

2119889120585 119889120578

= logradic119870 100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816

(24)



Λ (ℎ 119911) ≍

119889 (ℎ119911 1205971198631015840)

119889 (119911 120597119863)

≍ 120582 (ℎ 119911) (25)


1198961198631015840 (ℎ (1199111) ℎ (1199112)) ≍ 119896119863 (1199111 1199112) 1199111 1199112 isin 119863 (26)







119864119891119866 (119909) =1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (27)



Define

119869119891= 119869

119891119866=119899

radic119864119891119866 (28)




1

119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

le 119869119891119866

(119909) le 119888

119889 (119891 (119909) 1205971198661015840)

119889 (119909 120597119866)

119909 isin 119866

(29)




(30)

where 119889lowast(119909) = 119889(119891(119909)) = 119889(119891(119909) 1205971198661015840) and 119889(119909) = 119889(119909

120597119866) Hence

1003816100381610038161003816119861 (119891 (119909) 119888lowast119889lowast)

1003816100381610038161003816le int

119861119909

119869119891 (119911) 119889119911 le1003816100381610038161003816119861 (119891 (119909) 119862lowast119889lowast)

1003816100381610038161003816 (31)



R119899



hold






119896 (119909 1199090) = 119889(1199090)minus1 1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816(1 + 119900 (1)) 119909 997888rarr 1199090

119896 (119891 (119909) 119891 (1199090)) =

1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816

119889lowast (119891 (1199090))(1 + 119900 (1))

when 119909 997888rarr 1199090

(32)






(c1) 119899radic119869119891 asymp 119886119891

(d1) 119899radic119869119891 asymp 119869119891







119869119891 (119909) gt 0 119909 isin Ω (33)













1015840119909 minus 119894ℎ

1015840119910) and 120597ℎ = (12)(ℎ

1015840119909 + 119894ℎ

1015840119910)



|120597ℎ (0)| ge

119877

4

(34)


120582ℎ (0) ge1 minus 119896

4

119877 (35)



minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892


KoebeTheorem then

1

16

119863minus(0) le 119889ℎ (0) (36)

and therefore

(1 minus |119911|2)

1

16

119863minus(119911) le 119889ℎ (119911) (37)


(1 minus |119911|2)119863

+(119911) le 16119870119889ℎ (119911) (38)


(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911) (39)




(1 minus |119911|2)Λ ℎ (119911) le 16119870119889ℎ (119911)

(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911)

(40)


119889 (119911) Λ ℎ (119911) le 16119870119889ℎ (119911)

119889 (119911) 120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911)

(41)

and1 minus 119896

4

120581119863 (119911 1199111015840) le 1205811198631015840 (ℎ119911 ℎ119911

1015840)

le 16119870120581119863 (119911 1199111015840)

(42)

where 119911 1199111015840 isin 119863







|119889119908|

|Re119908|=

|119889119911|

|119911| ln (1 |119911|) (43)




1015840) when

1199111015840rarr 119911


(A2) 120581D1015840(119911 1199111015840) le 119888120581Πminus(119908 119908


119911 = 119890119908 and 1199111015840 = 119890119908

1015840



1

2119889

le 120588 le

2

119889

(44)


1 + 119900 (1)

119889 ln (1119889)le 120588 le

2

119889

(45)

Hence 119889hyp(119911 1199111015840) le 2120581119863(119911 119911

1015840) 119911 1199111015840 isin 119863








(1 minus |119911|2) 120582ℎlowast (120577) ge

1 minus 119896

4

119889ℎ (119911) (46)


1015840(120577)| and 120588hyp119863(119911)|120601

1015840(120577)| =

120588D(120577)

120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911) 120588119863 (119911) (47)


(1 minus |119911|2) 120582ℎ (119911) ge (1 minus 119896) 1205910119889ℎ (119911) (48)

where 1205910 = 3radic32120587








1003816100381610038161003816le 119888119889 (119891 (119911)) for every 1199111 isin 119861119911 (49)



osc119881119906 = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

1003816100381610038161003816 119909 119910 isin 119881

120596119906 (119903 119909) = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

10038161003816100381610038161003816100381610038161003816119909 minus 119910

1003816100381610038161003816= 119903

(50)


1(119866) = 119874119862


1(119866) such

that

119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 1198881osc119861

119909

119891 (51)



1 Then119891 isin 119874119862








119864119891119866 (119909) 1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (52)




119899119870 = 119870

1(1minus119899) 119862lowast = 120579119899119870(12) 120579

119899119870(119888lowast) = 12

and

119869119891119866

=119899

radic119864119891119866 (53)


119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (54)

and 119869119886V119891 (119861 119909) = 119899



⟨119909 119910⟩ = 11990911199101 + sdot sdot sdot + 119909119899119910119899 (55)


|119909| = ⟨119909 119909⟩12

= (100381610038161003816100381611990911003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199091198991003816100381610038161003816

2)

12 (56)






100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119874 (119909)

10038161003816100381610038161003816119869119891 (119909)

10038161003816100381610038161003816ae (57)




119869119891 (119909) le 1198701015840119897(119891

1015840(119909))

119899ae (58)






119863119901 (119891 119861) = (1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119901)

1119901

le 119872119903120574minus1

(59)







119869119891 (119886 119903) le 119872119903120574minus1

(60)




proof



1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)











119869119891 (119909 119903) le 119888 119903120572minus1 (61)








minus1(119910119896) 119896 = 1 2






the following




1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572


1015840le 1198881119903

120572 where 1198881 =








1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)






(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)


119869119891 (119909 119903) le 1198883119903120572minus1 (63)




(iv2) |119891| isin Λ120572(119866 120597119866)







1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











hold






119896 (119909 1199090) = 119889(1199090)minus1 1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816(1 + 119900 (1)) 119909 997888rarr 1199090

119896 (119891 (119909) 119891 (1199090)) =

1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816

119889lowast (119891 (1199090))(1 + 119900 (1))

when 119909 997888rarr 1199090

(32)






(c1) 119899radic119869119891 asymp 119886119891

(d1) 119899radic119869119891 asymp 119869119891







119869119891 (119909) gt 0 119909 isin Ω (33)













1015840119909 minus 119894ℎ

1015840119910) and 120597ℎ = (12)(ℎ

1015840119909 + 119894ℎ

1015840119910)



|120597ℎ (0)| ge

119877

4

(34)


120582ℎ (0) ge1 minus 119896

4

119877 (35)



minus(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| and Λ ℎ(119911) =

119863+(119911) = |119891

1015840(119911)| + |119892


KoebeTheorem then

1

16

119863minus(0) le 119889ℎ (0) (36)

and therefore

(1 minus |119911|2)

1

16

119863minus(119911) le 119889ℎ (119911) (37)


(1 minus |119911|2)119863

+(119911) le 16119870119889ℎ (119911) (38)


(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911) (39)




(1 minus |119911|2)Λ ℎ (119911) le 16119870119889ℎ (119911)

(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911)

(40)


119889 (119911) Λ ℎ (119911) le 16119870119889ℎ (119911)

119889 (119911) 120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911)

(41)

and1 minus 119896

4

120581119863 (119911 1199111015840) le 1205811198631015840 (ℎ119911 ℎ119911

1015840)

le 16119870120581119863 (119911 1199111015840)

(42)

where 119911 1199111015840 isin 119863







|119889119908|

|Re119908|=

|119889119911|

|119911| ln (1 |119911|) (43)




1015840) when

1199111015840rarr 119911


(A2) 120581D1015840(119911 1199111015840) le 119888120581Πminus(119908 119908


119911 = 119890119908 and 1199111015840 = 119890119908

1015840



1

2119889

le 120588 le

2

119889

(44)


1 + 119900 (1)

119889 ln (1119889)le 120588 le

2

119889

(45)

Hence 119889hyp(119911 1199111015840) le 2120581119863(119911 119911

1015840) 119911 1199111015840 isin 119863








(1 minus |119911|2) 120582ℎlowast (120577) ge

1 minus 119896

4

119889ℎ (119911) (46)


1015840(120577)| and 120588hyp119863(119911)|120601

1015840(120577)| =

120588D(120577)

120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911) 120588119863 (119911) (47)


(1 minus |119911|2) 120582ℎ (119911) ge (1 minus 119896) 1205910119889ℎ (119911) (48)

where 1205910 = 3radic32120587








1003816100381610038161003816le 119888119889 (119891 (119911)) for every 1199111 isin 119861119911 (49)



osc119881119906 = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

1003816100381610038161003816 119909 119910 isin 119881

120596119906 (119903 119909) = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

10038161003816100381610038161003816100381610038161003816119909 minus 119910

1003816100381610038161003816= 119903

(50)


1(119866) = 119874119862


1(119866) such

that

119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 1198881osc119861

119909

119891 (51)



1 Then119891 isin 119874119862








119864119891119866 (119909) 1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (52)




119899119870 = 119870

1(1minus119899) 119862lowast = 120579119899119870(12) 120579

119899119870(119888lowast) = 12

and

119869119891119866

=119899

radic119864119891119866 (53)


119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (54)

and 119869119886V119891 (119861 119909) = 119899



⟨119909 119910⟩ = 11990911199101 + sdot sdot sdot + 119909119899119910119899 (55)


|119909| = ⟨119909 119909⟩12

= (100381610038161003816100381611990911003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199091198991003816100381610038161003816

2)

12 (56)






100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119874 (119909)

10038161003816100381610038161003816119869119891 (119909)

10038161003816100381610038161003816ae (57)




119869119891 (119909) le 1198701015840119897(119891

1015840(119909))

119899ae (58)






119863119901 (119891 119861) = (1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119901)

1119901

le 119872119903120574minus1

(59)







119869119891 (119886 119903) le 119872119903120574minus1

(60)




proof



1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)











119869119891 (119909 119903) le 119888 119903120572minus1 (61)








minus1(119910119896) 119896 = 1 2






the following




1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572


1015840le 1198881119903

120572 where 1198881 =








1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)






(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)


119869119891 (119909 119903) le 1198883119903120572minus1 (63)




(iv2) |119891| isin Λ120572(119866 120597119866)







1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1 minus |119911|2)Λ ℎ (119911) le 16119870119889ℎ (119911)

(1 minus |119911|2) 120582ℎ (119911) ge

1 minus 119896

4

119889ℎ (119911)

(40)


119889 (119911) Λ ℎ (119911) le 16119870119889ℎ (119911)

119889 (119911) 120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911)

(41)

and1 minus 119896

4

120581119863 (119911 1199111015840) le 1205811198631015840 (ℎ119911 ℎ119911

1015840)

le 16119870120581119863 (119911 1199111015840)

(42)

where 119911 1199111015840 isin 119863







|119889119908|

|Re119908|=

|119889119911|

|119911| ln (1 |119911|) (43)




1015840) when

1199111015840rarr 119911


(A2) 120581D1015840(119911 1199111015840) le 119888120581Πminus(119908 119908


119911 = 119890119908 and 1199111015840 = 119890119908

1015840



1

2119889

le 120588 le

2

119889

(44)


1 + 119900 (1)

119889 ln (1119889)le 120588 le

2

119889

(45)

Hence 119889hyp(119911 1199111015840) le 2120581119863(119911 119911

1015840) 119911 1199111015840 isin 119863








(1 minus |119911|2) 120582ℎlowast (120577) ge

1 minus 119896

4

119889ℎ (119911) (46)


1015840(120577)| and 120588hyp119863(119911)|120601

1015840(120577)| =

120588D(120577)

120582ℎ (119911) ge1 minus 119896

4

119889ℎ (119911) 120588119863 (119911) (47)


(1 minus |119911|2) 120582ℎ (119911) ge (1 minus 119896) 1205910119889ℎ (119911) (48)

where 1205910 = 3radic32120587








1003816100381610038161003816le 119888119889 (119891 (119911)) for every 1199111 isin 119861119911 (49)



osc119881119906 = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

1003816100381610038161003816 119909 119910 isin 119881

120596119906 (119903 119909) = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

10038161003816100381610038161003816100381610038161003816119909 minus 119910

1003816100381610038161003816= 119903

(50)


1(119866) = 119874119862


1(119866) such

that

119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 1198881osc119861

119909

119891 (51)



1 Then119891 isin 119874119862








119864119891119866 (119909) 1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (52)




119899119870 = 119870

1(1minus119899) 119862lowast = 120579119899119870(12) 120579

119899119870(119888lowast) = 12

and

119869119891119866

=119899

radic119864119891119866 (53)


119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (54)

and 119869119886V119891 (119861 119909) = 119899



⟨119909 119910⟩ = 11990911199101 + sdot sdot sdot + 119909119899119910119899 (55)


|119909| = ⟨119909 119909⟩12

= (100381610038161003816100381611990911003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199091198991003816100381610038161003816

2)

12 (56)






100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119874 (119909)

10038161003816100381610038161003816119869119891 (119909)

10038161003816100381610038161003816ae (57)




119869119891 (119909) le 1198701015840119897(119891

1015840(119909))

119899ae (58)






119863119901 (119891 119861) = (1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119901)

1119901

le 119872119903120574minus1

(59)







119869119891 (119886 119903) le 119872119903120574minus1

(60)




proof



1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)











119869119891 (119909 119903) le 119888 119903120572minus1 (61)








minus1(119910119896) 119896 = 1 2






the following




1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572


1015840le 1198881119903

120572 where 1198881 =








1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)






(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)


119869119891 (119909 119903) le 1198883119903120572minus1 (63)




(iv2) |119891| isin Λ120572(119866 120597119866)







1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816le 119888119889 (119891 (119911)) for every 1199111 isin 119861119911 (49)



osc119881119906 = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

1003816100381610038161003816 119909 119910 isin 119881

120596119906 (119903 119909) = sup 1003816100381610038161003816119906 (119909) minus 119906 (119910)

10038161003816100381610038161003816100381610038161003816119909 minus 119910

1003816100381610038161003816= 119903

(50)


1(119866) = 119874119862


1(119866) such

that

119889 (119909)

100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816le 1198881osc119861

119909

119891 (51)



1 Then119891 isin 119874119862








119864119891119866 (119909) 1

10038161003816100381610038161198611199091003816100381610038161003816

int

119861119909

119869119891 (119911) 119889119911 119909 isin 119866 (52)




119899119870 = 119870

1(1minus119899) 119862lowast = 120579119899119870(12) 120579

119899119870(119888lowast) = 12

and

119869119891119866

=119899

radic119864119891119866 (53)


119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (54)

and 119869119886V119891 (119861 119909) = 119899



⟨119909 119910⟩ = 11990911199101 + sdot sdot sdot + 119909119899119910119899 (55)


|119909| = ⟨119909 119909⟩12

= (100381610038161003816100381611990911003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199091198991003816100381610038161003816

2)

12 (56)






100381610038161003816100381610038161198911015840(119909)

10038161003816100381610038161003816

119899le 119870119874 (119909)

10038161003816100381610038161003816119869119891 (119909)

10038161003816100381610038161003816ae (57)




119869119891 (119909) le 1198701015840119897(119891

1015840(119909))

119899ae (58)






119863119901 (119891 119861) = (1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119901)

1119901

le 119872119903120574minus1

(59)







119869119891 (119886 119903) le 119872119903120574minus1

(60)




proof



1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)











119869119891 (119909 119903) le 119888 119903120572minus1 (61)








minus1(119910119896) 119896 = 1 2






the following




1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572


1015840le 1198881119903

120572 where 1198881 =








1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)






(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)


119869119891 (119909 119903) le 1198883119903120572minus1 (63)




(iv2) |119891| isin Λ120572(119866 120597119866)







1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119869119891 (119886 119903) le 119872119903120574minus1

(60)




proof



1015840119891B = 119889(119891(119909) 120597B1015840) and 119877(119891(119909)B1015840) = 119888lowast119889

1015840where B1015840 = 119891(B)











119869119891 (119909 119903) le 119888 119903120572minus1 (61)








minus1(119910119896) 119896 = 1 2






the following




1015840| = 119903 then 119888lowast119889

1015840le |1199102| minus |1199101| le 2

120572119871119903120572


1015840le 1198881119903

120572 where 1198881 =








1198991119903

119899 119869119891(119861 119909) asymp 1198891119903and by (iii1) 1198891 ⪯ 119903

120572 on 119866 Hence

119869119891 (119861 119909) ⪯ 119903120572minus1

(62)






(i2) 119891 isin Λ120572(119866)

(ii2) |119891| isin Λ120572(119866)


119869119891 (119909 119903) le 1198883119903120572minus1 (63)




(iv2) |119891| isin Λ120572(119866 120597119866)







1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (64)


120572(119866)



119864119891 (119861 119909) =1

|119861|

int

119861119869119891 (119911) 119889119911 119909 isin 119866 (65)




(i3) 119891 isin Λ120572(119863)


119869119891 (119861 119911) le 119888119903120572minus1

(119911) (66)




Let 1199110 isin 119863 1205770 = 119892(1199110) 1198660 = 119892(1198611199110

) 1198820 = 120601(1198660) 119889 =

119889(1199110) = 119889(1199110 120597119863) and 1198891 = 1198891(1205770) = 119889(1205770 120597119866) Let 119861 =

119861(119911 119903) sub 1198611199110


1015840(1205770)| If 1199111 = 119892



1015840(120577)| le 1198881119903


119864119891 (119861 119911) =1

10038161003816100381610038161198611199111003816100381610038161003816

int

119861119911

100381610038161003816100381610038161206011015840(120577)

10038161003816100381610038161003816119869119892 (119911) 119889119909 119889119910 (67)

we find

119869119891119863 (1199110) le 1198882

119903120572

1198891 (1205770)119869119892119866 (1199110) (68)


2 we get

119869119891 (119861 119911) le 1198883119903120572minus1

(69)







119872lowast119886 = sup 100381610038161003816

1003816ℎ (119910) minus ℎ (119886)

1003816100381610038161003816 119910 isin 119878

119899minus1(119886 119903) (70)


119903

10038161003816100381610038161003816nablaℎ

119896(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 119896 = 1 119898 (71)



119903

10038161003816100381610038161003816ℎ1015840(119886)

10038161003816100381610038161003816le 119899119872

lowast119886 (72)


and ℎ(0) = 0 Let

119870(119909 119910) = 119870119910 (119909) =1199032minus |119909|

2

1198991205961198991199031003816100381610038161003816119909 minus 119910

1003816100381610038161003816

119899 (73)


120597

120597119909119895

119870 (0 120585) =

120585119895

120596119899119903119899+1

(74)



Then

1198701015840120585 (0) 120578 =

1

120596119899119903119899+1

(120585 120578) (75)



100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











100381610038161003816100381610038161198701015840120585 (0) 120578

10038161003816100381610038161003816le

1

120596119899119903119899 and therefore 1003816

1003816100381610038161003816nabla119870120585 (0)

10038161003816100381610038161003816le

1

120596119899119903119899 (76)


10038161003816100381610038161003816ℎ1015840(0) (120578)

10038161003816100381610038161003816le int

119878119899minus1(119886119903)

1003816100381610038161003816nabla119870

119910(0)1003816100381610038161003816

1003816100381610038161003816ℎ (119910)

1003816100381610038161003816119889120590 (119910)

le

119872lowast0 119899120596119899119903

119899minus1

120596119899119903119899

=

119872lowast0 119899

119903

(77)




⟨119911 119886⟩ = 11991111198861 + sdot sdot sdot + 119911119899119886119899 (78)


|119911| = ⟨119911 119911⟩12

= (100381610038161003816100381611991111003816100381610038161003816

2+ sdot sdot sdot +

10038161003816100381610038161199111198991003816100381610038161003816

2)

12 (79)


B119899(1199111015840 119903) = 119911 isin C

119899

10038161003816100381610038161003816119911 minus 119911

101584010038161003816100381610038161003816lt 119903 (80)






int

120575

0

120596 (119905)

119905

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750(81)

120575int

infin

120575

120596 (119905)

1199052

119889119905 le 119862120596 (120575) 0 lt 120575 lt 1205750 (82)




1003816100381610038161003816119891 (119911) minus 119891 (119908)

1003816100381610038161003816le 119862120596 (|119911 minus 119908|) (83)








and 119876 = 119906119909119896

define form

120596119896 = 119875119896119889119909119896 + 119876119896119889119910119896 = minus119906119910119896

119889119909119896 + 119906119909119896

119889119910119896 (84)

120596 = sum119899119896=1 120596119896 and V(119909 119910) = int

(119909119910)

(11990901199100)120596 Then (i) holds for 1198910 =







Define 119863119891 = (1198631119891 119863119899119891) and 119863119891 = (1198631119891 119863119899119891)where 119911119896 = 119909119896 + 119894119910119896 119863119896119891 = (12)(120597119909

119896

119891 minus 119894120597119910119896

119891) and 119863119896119891 =

(12)(120597119909119896

119891 + 119894120597119910119896

119891) 119896 = 1 2 119899


119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596|119891| (119889Ω (119911)) (85)




1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1015840| le |119863119892| + |119863ℎ|




1015840(0)| le 4|119908119896 minus 119908| 119896 = 1 2


119889Ω(119911)|1198651015840(0)| le 4120596|119891|(119889Ω(119911))

Define 1198911015840(119911) = (119891

10158401199111

(119911) 1198911015840119911119899

(119911)) Since 1198651015840(0) =

sum119899119896=1119863119896119891(119911) 119886119896 = (119891


119886 = (1198861 119886119899) Hence

119889Ω (119911)1003816100381610038161003816nabla119891 (119911)

1003816100381610038161003816le 4120596119906 (119889Ω (119911)) le 4120596|119891| (119889Ω (119911)) (86)


4119862120596(119889Ω(119911))








sup119861119909

1198892(119909)

1003816100381610038161003816Δ119891 (119909)

1003816100381610038161003816le 119888 osc119861

119909

119891 (87)


1(119866) In [7] we







1015840(119910)| le 119889




1015840(119909)| le 119888119889(119909)





119899 is the number119902 (119886 119887) =

1003816100381610038161003816119901 (119886) minus 119901 (119887)

1003816100381610038161003816 (88)


defined by

119901 (119909) = 119890119899+1 +119909 minus 119890119899+1

1003816100381610038161003816119909 minus 119890119899+1

1003816100381610038161003816

2 (89)


119902 (119886 119887) = |119886 minus 119887| (1 + |119886|2)

minus12(1 + |119887|

2)

minus12 (90)


119863119891119866(119909) = (1|119861| int119861|1198911015840|

119899)

1119899 where 119861 = 119861119909 = 119861119909119866 If there is







119863119891 (119909) le 1198621120596 (119889 (119909)) 119889(119909 120597119863)minus1

119891119900119903 119909 isin 119866 (91)


1003816100381610038161003816le 1198622120596 (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816+ 119889 (1199091 120597119866)) (92)








1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















1003816100381610038161003816119891 (1199091) minus 119891 (1199092)

1003816100381610038161003816le 119862(

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)120572

(1199091 1199092 isin 119863 infin)

(93)






119861(0 119903) then int119861119903

|1198911015840(119911)|119889119909 119889119910 = int

119903

0int

2120587

01205882(119886minus1)

120588119889 120588119889120579 asymp 1199032119886

and119863119891119861119903

(0) asymp 119903119886minus1



120572



(

1

|119861|

int

119861

10038161003816100381610038161003816119891101584010038161003816100381610038161003816

119899)

1119899

le 119862119870

120590

120590 minus 1

(

1

|120590119861|

int

120590119861

10038161003816100381610038161003816119891119895 minus 119886

10038161003816100381610038161003816

119899)

1119899

(94)




















(a1) 119903|119863119891(119886)| le 119888119889infin




(i4) 119891 isin Λ120572(119866)

(ii4) |119891| isin Λ120572(119866)





1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1015840) le |119911

1015840minus 119891(119909)| = ||119911

1015840| minus |119891(119909)|| where 1199091015840 = 119891(119909)



120572 Hence we get

(a4) 119903|119863119891(119903 119909)| le 119888119903120572 where 119888 = 119871119888












1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le const100381610038161003816

1003816119909 minus 119910

1003816100381610038161003816

120572 (95)


120572(119866)



Appendices



119894120579 For




1015840ℎ1015840119903 = 119891

1015840119903 + 119892

1015840119903 119891

1015840119903 = 119891

1015840(119911)119890

119894120579 and 119869ℎ = |1198911015840|

2minus |119892

1015840|


1015840119903 | + |119892

1015840119903| and

|ℎ1015840119903| lt 2|119891

1015840|






Then

(d) |ℎ1015840119903(119890119894120593)| ge 1198772 0 le 120593 le 2120587





Then

(d1015840) |120597ℎ(119911)| ge 1198774 119911 isin D

By (d) we have

(e) |1198911015840| ge 1198774 on T







theorem (12120587) int212058701199060(119890

119894120579) 119889120579 = 119906(119890

119894120593) minus 119906(0) ge 119877


119875119903 (120579) ge1 minus 119903

1 + 119903

(A1)




119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119906 (119890119894120593) minus 119906 (119903119890

119894120593) ge

1 minus 119903

1 + 119903

(119906 (119890119894120593) minus 119906 (0)) (A2)

and hence (d)



|120597ℎ (119911)| ge

119877

4

119911 isin D (A3)




minus1(119866119899)








120582ℎ (119911) ge1 minus 119896

4

119877 (A4)

1003816100381610038161003816ℎ (1199112) minus ℎ (1199111)

1003816100381610038161003816ge 119888

1015840 10038161003816100381610038161199112 minus 1199111

1003816100381610038161003816 1199111 1199112 isin D (A5)







10038161003816100381610038161003816le 119896 |119863ℎ (119911)| (A6)


1015840|1199112 minus 1199111|



100381610038161003816100381611988611003816100381610038161003816

2+100381610038161003816100381611988711003816100381610038161003816

2ge 1198880 =

27

41205872 (A7)

where 1198861 = 119863119891(0) 1198871 = 119863119891(0) and 1198880 = 2741205872 = 06839



(i1) Then

|120597ℎ (0)| ge

119877

4

(A8)



1) then

120582ℎ (0) ge 1198771205910 (1 minus 119896)

radic1 + 1198962 (A9)



100381610038161003816100381610038161205931015840(0)

10038161003816100381610038161003816le 1 (A10)






(vi) 1199020|120597ℎ119903(0)| le |120597ℎ(0)|



2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2+ |1198871|

2ge 1198880 and

therefore



1015840(119911)| and 120582ℎ(119911) = |119891

1015840(119911)| minus |119892

1015840(119911)| ge (1 minus


sharp form |1198861|2+ |1198871|

2ge 1198880119877

2 we get |1198861|2(1 + 119896

2) ge




119890 (ℎ) (119911) ge

1

16

1198772 119911 isin D (A11)

where 119890(ℎ)(119911) = |120597ℎ(119911)|2 + |120597ℎ(119911)|2



120593119887 (119911) =119911 minus 119887

1 minus 119887119911

|119887| lt 1 (A12)


100381610038161003816100381610038161205931015840119887 (119911)

10038161003816100381610038161003816=

1 minus |119887|2

100381610038161003816100381610038161 minus 119887119911

10038161003816100381610038161003816

2 119911 isin D (A13)



119890 (ℎ) (119911) ge

1

1205872 119911 isin D (A14)






2ge 1198880 =

2741205872 where 1198880 = 274120587











(1) ess inf 1205951015840 gt 0


















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119899 cf[46]





119871 (119909 119891) = lim sup1003816100381610038161003816119891 (119909 + ℎ) minus 119891 (119909)

1003816100381610038161003816

ℎ

(B1)

















Set 119878+ = (119909 119910 119911) 1199092+ 119910

2+ 119911

2= 1 119911 ge 0 and 1198673

=





3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











3 by 119885(119909 119910 119911) =119890119911ℎ(119909 119910)




















119899 1198861 1198862 119886119902 where 119886119895
















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















(i5) 119891 isin Λ120572(119866)

(ii5) |119891| isin Λ120572(119866)



(iii5) |119891| isin Λ120572(119866 120597119866)


119899119870 = 119870























Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article distortion of quasiregular mappings and...

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