research article sharp bounds for toader mean in terms of...

Research ArticleSharp Bounds for Toader Mean in terms of Arithmetic andSecond Contraharmonic Means

Wei-Mao Qian1 Ying-Qing Song2 Xiao-Hui Zhang2 and Yu-Ming Chu2

1School of Distance Education Huzhou Broadcast and TV University Huzhou 313000 China2School of Mathematics and Computation Science Hunan City University Yiyang 413000 China

Correspondence should be addressed to Yu-Ming Chu chuyuming2005126com

Received 25 April 2015 Accepted 10 September 2015

Academic Editor Lars E Persson

Copyright copy 2015 Wei-Mao Qian et alThis 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 present the best possible parameters 1205821 1205831isin R and 120582

2 1205832isin (12 1) such that double inequalities 120582

1119862(119886 119887) + (1 minus 120582

1)119860(119886 119887) lt

119879(119886 119887) lt 1205831119862(119886 119887) + (1 minus 120583

1)119860(119886 119887) 119862[120582

2119886 + (1 minus 120582

2)119887 1205822119887 + (1 minus 120582

2)119886] lt 119879(119886 119887) lt 119862[120583

2119886 + (1 minus 120583

2)119887 1205832119887 + (1 minus 120583

2)119886] hold for

all 119886 119887 gt 0 with 119886 = 119887 where 119860(119886 119887) = (119886 + 119887)2 119862(119886 119887) = (1198863+ 1198873)(1198862+ 1198872) and 119879(119886 119887) = 2 int

1205872

0

radic1198862cos2120579 + 1198872sin2120579119889120579120587 are thearithmetic second contraharmonic and Toader means of 119886 and 119887 respectively

1 Introduction

For 119886 119887 gt 0 the Toader mean 119879(119886 119887) [1] second contrahar-monic mean 119862(119886 119887) and arithmetic mean 119860(119886 119887) of 119886 and 119887

are given by

119879 (119886 119887) =2

120587int

1205872

0

radic1198862cos2120579 + 1198872sin2120579 119889120579

=

2119886E(radic1 minus (119887119886)2)

120587 119886 gt 119887

2119887E(radic1 minus (119886119887)2)

120587 119886 lt 119887

119886 119886 = 119887

(1)

119862 (119886 119887) =1198863+ 1198873

1198862 + 1198872

119860 (119886 119887) =119886 + 119887

2

(2)

respectively where E(119903) = int1205872

0(1 minus 119903

2sin2119905)12119889119905 (119903 isin (0 1))

is the complete elliptic integral of the second kindTheToader

mean 119879(119886 119887) is well known in mathematical literature formany years it satisfies

119879 (119886 119887) = 119877119864(1198862 1198872)

119879 (1 119903) =2

120587E (radic1 minus 1199032)

(3)

for all 119886 119887 gt 0 and 0 lt 119903 lt 1 where

119877119864(119886 119887) =

1

120587int

infin

0

[119886 (119905 + 119887) + 119887 (119905 + 119886)] 119905

(119905 + 119886)32

(119905 + 119887)32

119889119905 (4)

stands for the symmetric complete elliptic integral of thesecond kind (see [2ndash4]) therefore it cannot be expressed interms of the elementary transcendental functions

Recently the Toader mean 119879(119886 119887) has been the subject ofintensive research In particular many remarkable inequali-ties for the Toader mean can be found in the literature [5ndash9]

Let 119901 isin R 119902 isin [0 1] and 119886 119887 gt 0 Then the 119901thpower mean 119872

119901(119886 119887) 119901th Gini mean 119866

119901(119886 119887) 119901th Lehmer

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 452823 5 pageshttpdxdoiorg1011552015452823

2 Journal of Function Spaces

mean 119871119901(119886 119887) and 119902th generalized Seiffert mean 119878

119902(119886 119887) are

defined by

119872119901(119886 119887) =

(119886119901+ 119887119901

2)

1119901

119901 = 0

radic119886119887 119901 = 0

119866119901(119886 119887) =

(119886119901minus1

+ 119887119901minus1

119886 + 119887)

1(119901minus2)

119901 = 2

(119886119886119887119887)1(119886+119887)

119901 = 2

119871119901(119886 119887) =

119886119901+1

+ 119887119901+1

119886119901 + 119887119901

119878119902(119886 119887)

=

119901 (119886 minus 119887)

arctan [2119901 (119886 minus 119887) (119886 + 119887)] 0 lt 119901 le 1 119886 = 119887

119886 + 119887

2 119901 = 0 119886 = 119887

119886 119886 = 119887

(5)

respectively It is well known that119872119901(119886 119887)119866

119901(119886 119887) 119871

119901(119886 119887)

and 119878119902(119886 119887) are continuous and strictly increasing with

respect to 119901 isin R and 119902 isin [0 1] for fixed 119886 119887 gt 0 with 119886 = 119887respectively

Vuorinen [10] conjectured that inequality

11987232

(119886 119887) lt 119879 (119886 119887) (6)

holds for all 119886 119887 gt 0 with 119886 = 119887 This conjecture was provedby Qiu and Shen [11] and Barnard et al [12] respectively

Alzer and Qiu [13] presented a best possible upper powermean bound for the Toader mean as follows

119879 (119886 119887) lt 119872log 2(log120587minuslog 2) (119886 119887) (7)

for all 119886 119887 gt 0 with 119886 = 119887In [14 15] the authors found the best possible parameters

120572 120573 isin [0 1] and 120582 120583 isin R such that double inequalities119878120572(119886 119887) lt 119879(119886 119887) lt 119878

120573(119886 119887) and 119866

120582(119886 119887) lt 119879(119886 119887) lt

119866120583(119886 119887) hold for all 119886 119887 gt 0 with 119886 = 119887Chu and Wang [16] proved that double inequality

119871119901(119886 119887) lt 119879 (119886 119887) lt 119871

119902(119886 119887) (8)

holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 119901 le 0 and119902 ge 14

Inequality (8) leads to

119860 (119886 119887) = 1198710(119886 119887) lt 119879 (119886 119887) lt 119871

14(119886 119887) lt 119871

2(119886 119887)

= 119862 (119886 119887)

(9)

for all 119886 119887 gt 0 with 119886 = 119887Let 119886 119887 gt 0 with 119886 = 119887 be fixed and 119891(119909) = 119862[119909119886 + (1 minus

119909)119887 119909119887 + (1 minus 119909)119886] Then it is not difficult to verify that 119891(119909)is continuous and strictly increasing on [12 1] Note that

119891(1

2) = 119860 (119886 119887) lt 119879 (119886 119887) lt 119862 (119886 119887) = 119891 (1) (10)

Motivated by inequalities (9) and (10) it is natural to askwhat are the best possible parameters 120582

1 1205831isin R and 120582

2 1205832isin

(12 1) such that double inequalities

1205821119862 (119886 119887) + (1 minus 120582

1) 119860 (119886 119887) lt 119879 (119886 119887)

lt 1205831119862 (119886 119887) + (1 minus 120583

1) 119860 (119886 119887)

119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] lt 119879 (119886 119887)

lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886]

(11)

hold for all 119886 119887 gt 0 with 119886 = 119887 The main purpose of thispaper is to answer this question

2 Main Results

In order to prove our main results we need some basicknowledge and two lemmas which we present in this section

For 119903 isin (0 1) the complete elliptic integral K(119903) of thefirst kind is defined by

K (119903) = int

1205872

0

(1 minus 1199032sin2119905)

minus12

119889119905 (12)

We clearly see that

K (0+) = E (0

+) =

120587

2

E (1minus) = 1

(13)

and K(119903) and E(119903) satisfy formulas (see [17 Appendix E p474-475])

119889K (119903)

119889119903=E (119903) minus (1 minus 119903

2)K (119903)

119903 (1 minus 1199032)

119889E (119903)

119889119903=E (119903) minusK (119903)

119903

E(2radic119903

1 + 119903) =

2E (119903) minus (1 minus 1199032)K (119903)

1 + 119903

(14)

Lemma 1 (see [17 Theorem 125]) Let minusinfin lt 119886 lt 119887 lt

infin 119891 119892 [119886 119887] rarr (minusinfininfin) be continuous on [119886 119887] anddifferentiable on (119886 119887) and 119892

1015840(119909) = 0 on (119886 119887) If 1198911015840(119909)1198921015840(119909)

is increasing (decreasing) on (119886 119887) then so are

119891 (119909) minus 119891 (119886)

119892 (119909) minus 119892 (119886)

119891 (119909) minus 119891 (119887)

119892 (119909) minus 119892 (119887)

(15)

If 1198911015840(119909)1198921015840(119909) is strictly monotone then the monotonicity inthe conclusion is also strict

Lemma 2 (see [17 Theorem 321]) (1) Function 119903 997891rarr [E(119903) minus

(1 minus 1199032)K(119903)]119903

2 is strictly increasing from (0 1) to (1205874 1)(2) Function 119903 997891rarr (1 minus 119903

2)120582K(119903) is strictly decreasing from

(0 1) to (0 1205872) if 120582 ge 14

Journal of Function Spaces 3

Theorem 3 Double inequality

1205821119862 (119886 119887) + (1 minus 120582

1) 119860 (119886 119887) lt 119879 (119886 119887)

lt 1205831119862 (119886 119887) + (1 minus 120583

1) 119860 (119886 119887)

(16)

holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 1205821le 18 and

1205831ge 4120587 minus 1 = 02732

Proof Since 119860(119886 119887) 119879(119886 119887) and 119862(119886 119887) are symmetric andhomogeneous of degree 1 without loss of generality weassume that 119886 gt 119887 gt 0 Let 119903 = (119886 minus 119887)(119886 + 119887) isin (0 1)Then (1) and (2) lead to

119879 (119886 119887) =2119886

120587E(

2radic119903

1 + 119903)

=2119860 (119886 119887)

120587[2E (119903) minus (1 minus 119903

2)K (119903)]

(17)

119862 (119886 119887) = 119860 (119886 119887)1 + 3119903

2

1 + 1199032 (18)

We clearly see that inequality (16) is equivalent to

1205821lt

119879 (119886 119887) minus 119860 (119886 119887)

119862 (119886 119887) minus 119860 (119886 119887)lt 1205831 (19)

It follows from (17) and (18) that

119879 (119886 119887) minus 119860 (119886 119887)

119862 (119886 119887) minus 119860 (119886 119887)

=1

120587

2E (119903) minus (1 minus 1199032)K (119903) minus 1205872

1199032 (1 + 1199032)

(20)

Let

119891 (119903) =2E (119903) minus (1 minus 119903

2)K (119903) minus 1205872

1199032 (1 + 1199032)

1198911(119903) = 2E (119903) minus (1 minus 119903

2)K (119903) minus

120587

2

1198912(119903) =

1199032

1 + 1199032

(21)

Then simple computations lead to

1198911(0+) = 1198912(0+) = 0

1198911015840

1(119903)

11989110158402(119903)

=(1 + 119903

2)2

2

E (119903) minus (1 minus 1199032)K (119903)

1199032

(22)

From Lemmas 1 and 2 together with (21) and (22) weknow that 119891(119903) is strictly increasing on (0 1) and

119891 (0+) =

1198911015840

1(0+)

11989110158402(0+)

=120587

8

119891 (1minus) = 4 minus 120587

(23)

Therefore Theorem 3 follows from (19)ndash(21) and (23)together with the monotonicity of 119891(119903)

Theorem 4 Let 1205822 1205832isin (12 1) Then double inequality

119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] lt 119879 (119886 119887)

lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886]

(24)

holds for all 119886 119887 gt 0with 119886 = 119887 if and only if 1205822le 12+radic28 =

06767 and 1205832ge 12 + radic(4 minus 120587)(3120587 minus 4)2 = 06988

Proof Let 120582lowast

2= 12 + radic28 and 120583

lowast

2= 12 +

radic(4 minus 120587)(3120587 minus 4)2 We first prove that

119879 (119886 119887) gt 119862 [120582lowast

2119886 + (1 minus 120582

lowast

2) 119887 120582lowast

2119887 + (1 minus 120582

lowast

2) 119886] (25)

119879 (119886 119887) lt 119862 [120583lowast

2119886 + (1 minus 120583

lowast

2) 119887 120583lowast

2119887 + (1 minus 120583

lowast

2) 119886] (26)

for all 119886 119887 gt 0 with 119886 = 119887Without loss of generality we assume that 119886 gt 119887 Let 119903 =

(119886 minus 119887)(119886 + 119887) isin (0 1) and 119901 isin (12 1) Then (2) leads to

119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]

= 119860 (119886 119887)3 (2119901 minus 1)

2

1199032+ 1

(2119901 minus 1)2

1199032 + 1

(27)


119879 (119886 119887) minus 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]

= 119860 (119886 119887) [2

120587(2E (119903) minus (1 minus 119903

2)K (119903))

+2

(2119901 minus 1)2

1199032 + 1

minus 3]

(28)

Let

119892 (119903) =2

120587(2E (119903) minus (1 minus 119903

2)K (119903))

+2

(2119901 minus 1)2

1199032 + 1

minus 3

(29)

1198921(119903) =

1

1199031198921015840(119903) (30)


Then making use of Lemma 2 and simple computations leadto

119892 (0) = 0 (31)

119892 (1) =4

120587+

2

(2119901 minus 1)2

+ 1

minus 3 (32)

1198921(119903) =

2

120587

E (119903) minus (1 minus 1199032)K (119903)

1199032

minus4 (2119901 minus 1)

2

[(2119901 minus 1)2

1199032 + 1]2

(33)

1198921(0) =

1

2minus 4 (2119901 minus 1)

2

(34)

1198921(1) =

2

120587minus

4 (2119901 minus 1)2

[(2119901 minus 1)2

+ 1]2 (35)

We divide the proof into two cases

Case 1 Consider 119901 = 120582lowast

2= 12 + radic28 Then (34) becomes

1198921(0) = 0 (36)

It follows from Lemma 2(1) and (33) together with (36) that

1198921(119903) gt 0 (37)

for all 119903 isin (0 1)Therefore inequality (25) follows easily from (28)ndash(31)

and (37)


2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then

(32) (34) and (35) lead to

119892 (1) = 0 (38)

1198921(0) = minus

36 minus 11120587

6120587 minus 8lt 0 (39)

1198921(1) =

31205872minus 14120587 + 16

1205872gt 0 (40)

It follows from Lemma 2(1) (33) (39) and (40) that thereexists 119903

0isin (0 1) such that 119892

1(119903) lt 0 for 119903 isin (0 119903

0) and

1198921(119903) gt 0 for 119903 isin (119903

0 1)Then (30) leads to the conclusion that

119892(119903) is strictly decreasing on (0 1199030] and strictly increasing on

(1199030 1]Therefore inequality (26) follows easily from (28) (29)

(31) (38) and the piecewise monotonicity of 119892(119903)Next we prove that 120582

2= 120582lowast

2= 12 + radic28 is the best

possible parameter on (12 1) such that inequality

119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] (41)

holds for all 119886 119887 gt 0 with 119886 = 119887Indeed if 120582lowast

2= 12 + radic28 lt 119901 lt 1 then (34) leads to

1198921(0) lt 0 and there exists 120575

1isin (0 1) such that

1198921(119903) lt 0 (42)

for 119903 isin (0 1205751)

Equations (28)ndash(31) and inequality (42) imply that

119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)

for (119886 minus 119887)(119886 + 119887) isin (0 1205751)

Finally we prove that 1205832= 120583lowast

2= 12+radic(4 minus 120587)(3120587 minus 4)

2 is the best possible parameter on (12 1) such that doubleinequality

119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886] (44)

for all 119886 119887 gt 0 with 119886 = 119887In fact if 12 lt 119901 lt 120583

lowast

2= 12+radic(4 minus 120587)(3120587 minus 4)2 then

(32) leads to 119892(1) gt 0 and there exists 1205752isin (0 1) such that

119892 (119903) gt 0 (45)

for 119903 isin (1 minus 1205752 1)

Equations (28) and (29) together with inequality (45)imply that

119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)

for (119886 minus 119887)(119886 + 119887) isin (1 minus 1205752 1)

Let 119903 isin (0 1) 119886 = 1 119887 = radic1 minus 1199032 1205821= 18 120583

1= 4120587 minus 1

1205822= 12 + radic28 and 120583


Theorems 3 and 4 lead to Corollary 5 as follows

Corollary 5 Double inequalities

120587 [1 + (1 minus 1199032)32

]

16 (2 minus 1199032)+

7120587 [1 + (1 minus 1199032)12

]

32lt E (119903)

lt

(4 minus 120587) [1 + (1 minus 1199032)32

]

2 (2 minus 1199032)+ (

120587

2minus 1) [1 + (1 minus 119903

2)12

]

120587

2

32 minus 211199032+ 21 (1 minus 119903

2)12

+ 11 (1 minus 1199032)32

36 minus 181199032 + 28 (1 minus 1199032)12

lt E (119903) lt120587

2

sdot3120587 minus 4 minus 3 (120587 minus 2) 119903

2+ 3 (120587 minus 2) (1 minus 119903

2)12

+ 2 (1 minus 1199032)32

120587 (2 minus 1199032) + 4 (120587 minus 2) (1 minus 1199032)12

(47)

hold for all 119903 isin (0 1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Natural Science Founda-tion of China under Grants 11371125 11401191 and 61374086the Natural Science Foundation of Zhejiang Province underGrant LY13A010004 the Natural Science Foundation ofHunan Province under Grant 12C0577 and the NaturalScience Foundation of the Zhejiang Broadcast and TV Uni-versity under Grant XKT-15G17


References

[1] G Toader ldquoSome mean values related to the arithmetic-geometric meanrdquo Journal of Mathematical Analysis and Appli-cations vol 218 no 2 pp 358ndash368 1998

[2] E Neuman ldquoBounds for symmetric elliptic integralsrdquo Journalof Approximation Theory vol 122 no 2 pp 249ndash259 2003

[3] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegralsrdquo Journal of Approximation Theory vol 146 no 2 pp212ndash226 2007

[4] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegrals IIrdquo in Special Functions and Orthogonal PolynomialsContemporary Mathematics 471 pp 127ndash138 American Math-ematical Society Providence RI USA 2008

[5] Y-M Chu M-K Wang and X-Y Ma ldquoSharp bounds forToader mean in terms of contraharmonic mean with applica-tionsrdquo Journal of Mathematical Inequalities vol 7 no 2 pp 161ndash166 2013

[6] Y-Q Song W-D Jiang Y-M Chu and D-D Yan ldquoOptimalbounds for Toader mean in terms of arithmetic and contrahar-monic meansrdquo Journal of Mathematical Inequalities vol 7 no4 pp 751ndash757 2013

[7] W-H Li and M-M Zheng ldquoSome inequalities for boundingToader meanrdquo Journal of Function Spaces and Applications vol2013 Article ID 394194 5 pages 2013

[8] Y Hua and F Qi ldquoA double inequality for bounding Toadermean by the centroidal meanrdquo ProceedingsmdashMathematical Sci-ences vol 124 no 4 pp 527ndash531 2014

[9] Y Hua and F Qi ldquoThe best bounds for Toader mean in termsof the centroidal and arithmetic meansrdquo Filomat vol 28 no 4pp 775ndash780 2014

[10] MVuorinen ldquoHypergeometric functions in geometric functiontheoryrdquo in Special Functions andDifferential Equations (Madras1977) pp 119ndash126 Allied Publishers New Delhi India 1998

[11] S-L Qiu and J-M Shen ldquoOn two problems concerningmeansrdquoJournal of Hangzhou Institute of Electronic Engineering vol 17no 3 pp 1ndash7 1997 (Chinese)

[12] R W Barnard K Pearce and K C Richards ldquoAn inequalityinvolving the generalized hypergeometric function and the arclength of an ellipserdquo SIAM Journal on Mathematical Analysisvol 31 no 3 pp 693ndash699 2000

[13] H Alzer and S-L Qiu ldquoMonotonicity theorems and inequali-ties for the complete elliptic integralsrdquo Journal of Computationaland Applied Mathematics vol 172 no 2 pp 289ndash312 2004

[14] Y-M Chu M-K Wang S-L Qiu and Y-F Qiu ldquoSharpgeneralized Seiffert mean bounds for Toader meanrdquo Abstractand Applied Analysis vol 2011 Article ID 605259 8 pages 2011

[15] Y-M Chu and M-K Wang ldquoInequalities between arithmetic-geometric Gini and Toader meansrdquo Abstract and AppliedAnalysis vol 2012 Article ID 830585 11 pages 2012

[16] Y-M Chu and M-K Wang ldquoOptimal Lehmer mean boundsfor the Toader meanrdquo Results in Mathematics vol 61 no 3-4pp 223ndash229 2012

[17] G D Anderson M K Vamanamurthy and M K VuorinenConformal Invariants Inequalities and Quasiconformal MapsJohn Wiley amp Sons New York NY USA 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


mean 119871119901(119886 119887) and 119902th generalized Seiffert mean 119878

119902(119886 119887) are

defined by

119872119901(119886 119887) =

(119886119901+ 119887119901

2)

1119901

119901 = 0

radic119886119887 119901 = 0

119866119901(119886 119887) =

(119886119901minus1

+ 119887119901minus1

119886 + 119887)

1(119901minus2)

119901 = 2

(119886119886119887119887)1(119886+119887)

119901 = 2

119871119901(119886 119887) =

119886119901+1

+ 119887119901+1

119886119901 + 119887119901

119878119902(119886 119887)

=

119901 (119886 minus 119887)

arctan [2119901 (119886 minus 119887) (119886 + 119887)] 0 lt 119901 le 1 119886 = 119887

119886 + 119887

2 119901 = 0 119886 = 119887

119886 119886 = 119887

(5)

respectively It is well known that119872119901(119886 119887)119866

119901(119886 119887) 119871

119901(119886 119887)

and 119878119902(119886 119887) are continuous and strictly increasing with

respect to 119901 isin R and 119902 isin [0 1] for fixed 119886 119887 gt 0 with 119886 = 119887respectively

Vuorinen [10] conjectured that inequality

11987232

(119886 119887) lt 119879 (119886 119887) (6)

holds for all 119886 119887 gt 0 with 119886 = 119887 This conjecture was provedby Qiu and Shen [11] and Barnard et al [12] respectively

Alzer and Qiu [13] presented a best possible upper powermean bound for the Toader mean as follows

119879 (119886 119887) lt 119872log 2(log120587minuslog 2) (119886 119887) (7)

for all 119886 119887 gt 0 with 119886 = 119887In [14 15] the authors found the best possible parameters

120572 120573 isin [0 1] and 120582 120583 isin R such that double inequalities119878120572(119886 119887) lt 119879(119886 119887) lt 119878

120573(119886 119887) and 119866

120582(119886 119887) lt 119879(119886 119887) lt

119866120583(119886 119887) hold for all 119886 119887 gt 0 with 119886 = 119887Chu and Wang [16] proved that double inequality

119871119901(119886 119887) lt 119879 (119886 119887) lt 119871

119902(119886 119887) (8)

holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 119901 le 0 and119902 ge 14

Inequality (8) leads to

119860 (119886 119887) = 1198710(119886 119887) lt 119879 (119886 119887) lt 119871

14(119886 119887) lt 119871

2(119886 119887)

= 119862 (119886 119887)

(9)

for all 119886 119887 gt 0 with 119886 = 119887Let 119886 119887 gt 0 with 119886 = 119887 be fixed and 119891(119909) = 119862[119909119886 + (1 minus

119909)119887 119909119887 + (1 minus 119909)119886] Then it is not difficult to verify that 119891(119909)is continuous and strictly increasing on [12 1] Note that

119891(1

2) = 119860 (119886 119887) lt 119879 (119886 119887) lt 119862 (119886 119887) = 119891 (1) (10)

Motivated by inequalities (9) and (10) it is natural to askwhat are the best possible parameters 120582

1 1205831isin R and 120582

2 1205832isin

(12 1) such that double inequalities

1205821119862 (119886 119887) + (1 minus 120582

1) 119860 (119886 119887) lt 119879 (119886 119887)

lt 1205831119862 (119886 119887) + (1 minus 120583

1) 119860 (119886 119887)

119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] lt 119879 (119886 119887)

lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886]

(11)

hold for all 119886 119887 gt 0 with 119886 = 119887 The main purpose of thispaper is to answer this question

2 Main Results

In order to prove our main results we need some basicknowledge and two lemmas which we present in this section

For 119903 isin (0 1) the complete elliptic integral K(119903) of thefirst kind is defined by

K (119903) = int

1205872

0

(1 minus 1199032sin2119905)

minus12

119889119905 (12)

We clearly see that

K (0+) = E (0

+) =

120587

2

E (1minus) = 1

(13)

and K(119903) and E(119903) satisfy formulas (see [17 Appendix E p474-475])

119889K (119903)

119889119903=E (119903) minus (1 minus 119903

2)K (119903)

119903 (1 minus 1199032)

119889E (119903)

119889119903=E (119903) minusK (119903)

119903

E(2radic119903

1 + 119903) =

2E (119903) minus (1 minus 1199032)K (119903)

1 + 119903

(14)

Lemma 1 (see [17 Theorem 125]) Let minusinfin lt 119886 lt 119887 lt

infin 119891 119892 [119886 119887] rarr (minusinfininfin) be continuous on [119886 119887] anddifferentiable on (119886 119887) and 119892

1015840(119909) = 0 on (119886 119887) If 1198911015840(119909)1198921015840(119909)

is increasing (decreasing) on (119886 119887) then so are

119891 (119909) minus 119891 (119886)

119892 (119909) minus 119892 (119886)

119891 (119909) minus 119891 (119887)

119892 (119909) minus 119892 (119887)

(15)

If 1198911015840(119909)1198921015840(119909) is strictly monotone then the monotonicity inthe conclusion is also strict

Lemma 2 (see [17 Theorem 321]) (1) Function 119903 997891rarr [E(119903) minus

(1 minus 1199032)K(119903)]119903

2 is strictly increasing from (0 1) to (1205874 1)(2) Function 119903 997891rarr (1 minus 119903

2)120582K(119903) is strictly decreasing from

(0 1) to (0 1205872) if 120582 ge 14



1205821119862 (119886 119887) + (1 minus 120582

1) 119860 (119886 119887) lt 119879 (119886 119887)

lt 1205831119862 (119886 119887) + (1 minus 120583

1) 119860 (119886 119887)

(16)


1205831ge 4120587 minus 1 = 02732


119879 (119886 119887) =2119886

120587E(

2radic119903

1 + 119903)

=2119860 (119886 119887)

120587[2E (119903) minus (1 minus 119903

2)K (119903)]

(17)

119862 (119886 119887) = 119860 (119886 119887)1 + 3119903

2

1 + 1199032 (18)


1205821lt

119879 (119886 119887) minus 119860 (119886 119887)

119862 (119886 119887) minus 119860 (119886 119887)lt 1205831 (19)


119879 (119886 119887) minus 119860 (119886 119887)

119862 (119886 119887) minus 119860 (119886 119887)

=1

120587


1199032 (1 + 1199032)

(20)

Let

119891 (119903) =2E (119903) minus (1 minus 119903

2)K (119903) minus 1205872

1199032 (1 + 1199032)

1198911(119903) = 2E (119903) minus (1 minus 119903

2)K (119903) minus

120587

2

1198912(119903) =

1199032

1 + 1199032

(21)


1198911(0+) = 1198912(0+) = 0

1198911015840

1(119903)

11989110158402(119903)

=(1 + 119903

2)2

2

E (119903) minus (1 minus 1199032)K (119903)

1199032

(22)


119891 (0+) =

1198911015840

1(0+)

11989110158402(0+)

=120587

8

119891 (1minus) = 4 minus 120587

(23)



119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] lt 119879 (119886 119887)

lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886]

(24)




2= 12 + radic28 and 120583

lowast

2= 12 +


119879 (119886 119887) gt 119862 [120582lowast

2119886 + (1 minus 120582

lowast

2) 119887 120582lowast

2119887 + (1 minus 120582

lowast

2) 119886] (25)

119879 (119886 119887) lt 119862 [120583lowast

2119886 + (1 minus 120583

lowast

2) 119887 120583lowast

2119887 + (1 minus 120583

lowast

2) 119886] (26)



119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]

= 119860 (119886 119887)3 (2119901 minus 1)

2

1199032+ 1

(2119901 minus 1)2

1199032 + 1

(27)



= 119860 (119886 119887) [2

120587(2E (119903) minus (1 minus 119903

2)K (119903))

+2

(2119901 minus 1)2

1199032 + 1

minus 3]

(28)

Let

119892 (119903) =2

120587(2E (119903) minus (1 minus 119903

2)K (119903))

+2

(2119901 minus 1)2

1199032 + 1

minus 3

(29)

1198921(119903) =

1

1199031198921015840(119903) (30)



119892 (0) = 0 (31)

119892 (1) =4

120587+

2

(2119901 minus 1)2

+ 1

minus 3 (32)

1198921(119903) =

2

120587

E (119903) minus (1 minus 1199032)K (119903)

1199032


2

[(2119901 minus 1)2

1199032 + 1]2

(33)

1198921(0) =

1

2minus 4 (2119901 minus 1)

2

(34)

1198921(1) =

2

120587minus

4 (2119901 minus 1)2

[(2119901 minus 1)2

+ 1]2 (35)




1198921(0) = 0 (36)


1198921(119903) gt 0 (37)


and (37)



(32) (34) and (35) lead to

119892 (1) = 0 (38)

1198921(0) = minus

36 minus 11120587

6120587 minus 8lt 0 (39)

1198921(1) =

31205872minus 14120587 + 16

1205872gt 0 (40)



1(119903) lt 0 for 119903 isin (0 119903

0) and

1198921(119903) gt 0 for 119903 isin (119903





2= 120582lowast



119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] (41)





1198921(119903) lt 0 (42)

for 119903 isin (0 1205751)


119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)

for (119886 minus 119887)(119886 + 119887) isin (0 1205751)




119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886] (44)


lowast



119892 (119903) gt 0 (45)

for 119903 isin (1 minus 1205752 1)


119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)



1= 4120587 minus 1

1205822= 12 + radic28 and 120583




120587 [1 + (1 minus 1199032)32

]

16 (2 minus 1199032)+

7120587 [1 + (1 minus 1199032)12

]

32lt E (119903)

lt

(4 minus 120587) [1 + (1 minus 1199032)32

]

2 (2 minus 1199032)+ (

120587

2minus 1) [1 + (1 minus 119903

2)12

]

120587

2

32 minus 211199032+ 21 (1 minus 119903

2)12

+ 11 (1 minus 1199032)32

36 minus 181199032 + 28 (1 minus 1199032)12

lt E (119903) lt120587

2


2+ 3 (120587 minus 2) (1 minus 119903

2)12

+ 2 (1 minus 1199032)32


(47)




Acknowledgments



References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205821119862 (119886 119887) + (1 minus 120582

1) 119860 (119886 119887) lt 119879 (119886 119887)

lt 1205831119862 (119886 119887) + (1 minus 120583

1) 119860 (119886 119887)

(16)


1205831ge 4120587 minus 1 = 02732


119879 (119886 119887) =2119886

120587E(

2radic119903

1 + 119903)

=2119860 (119886 119887)

120587[2E (119903) minus (1 minus 119903

2)K (119903)]

(17)

119862 (119886 119887) = 119860 (119886 119887)1 + 3119903

2

1 + 1199032 (18)


1205821lt

119879 (119886 119887) minus 119860 (119886 119887)

119862 (119886 119887) minus 119860 (119886 119887)lt 1205831 (19)


119879 (119886 119887) minus 119860 (119886 119887)

119862 (119886 119887) minus 119860 (119886 119887)

=1

120587


1199032 (1 + 1199032)

(20)

Let

119891 (119903) =2E (119903) minus (1 minus 119903

2)K (119903) minus 1205872

1199032 (1 + 1199032)

1198911(119903) = 2E (119903) minus (1 minus 119903

2)K (119903) minus

120587

2

1198912(119903) =

1199032

1 + 1199032

(21)


1198911(0+) = 1198912(0+) = 0

1198911015840

1(119903)

11989110158402(119903)

=(1 + 119903

2)2

2

E (119903) minus (1 minus 1199032)K (119903)

1199032

(22)


119891 (0+) =

1198911015840

1(0+)

11989110158402(0+)

=120587

8

119891 (1minus) = 4 minus 120587

(23)



119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] lt 119879 (119886 119887)

lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886]

(24)




2= 12 + radic28 and 120583

lowast

2= 12 +


119879 (119886 119887) gt 119862 [120582lowast

2119886 + (1 minus 120582

lowast

2) 119887 120582lowast

2119887 + (1 minus 120582

lowast

2) 119886] (25)

119879 (119886 119887) lt 119862 [120583lowast

2119886 + (1 minus 120583

lowast

2) 119887 120583lowast

2119887 + (1 minus 120583

lowast

2) 119886] (26)



119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]

= 119860 (119886 119887)3 (2119901 minus 1)

2

1199032+ 1

(2119901 minus 1)2

1199032 + 1

(27)



= 119860 (119886 119887) [2

120587(2E (119903) minus (1 minus 119903

2)K (119903))

+2

(2119901 minus 1)2

1199032 + 1

minus 3]

(28)

Let

119892 (119903) =2

120587(2E (119903) minus (1 minus 119903

2)K (119903))

+2

(2119901 minus 1)2

1199032 + 1

minus 3

(29)

1198921(119903) =

1

1199031198921015840(119903) (30)



119892 (0) = 0 (31)

119892 (1) =4

120587+

2

(2119901 minus 1)2

+ 1

minus 3 (32)

1198921(119903) =

2

120587

E (119903) minus (1 minus 1199032)K (119903)

1199032


2

[(2119901 minus 1)2

1199032 + 1]2

(33)

1198921(0) =

1

2minus 4 (2119901 minus 1)

2

(34)

1198921(1) =

2

120587minus

4 (2119901 minus 1)2

[(2119901 minus 1)2

+ 1]2 (35)




1198921(0) = 0 (36)


1198921(119903) gt 0 (37)


and (37)



(32) (34) and (35) lead to

119892 (1) = 0 (38)

1198921(0) = minus

36 minus 11120587

6120587 minus 8lt 0 (39)

1198921(1) =

31205872minus 14120587 + 16

1205872gt 0 (40)



1(119903) lt 0 for 119903 isin (0 119903

0) and

1198921(119903) gt 0 for 119903 isin (119903





2= 120582lowast



119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] (41)





1198921(119903) lt 0 (42)

for 119903 isin (0 1205751)


119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)

for (119886 minus 119887)(119886 + 119887) isin (0 1205751)




119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886] (44)


lowast



119892 (119903) gt 0 (45)

for 119903 isin (1 minus 1205752 1)


119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)



1= 4120587 minus 1

1205822= 12 + radic28 and 120583




120587 [1 + (1 minus 1199032)32

]

16 (2 minus 1199032)+

7120587 [1 + (1 minus 1199032)12

]

32lt E (119903)

lt

(4 minus 120587) [1 + (1 minus 1199032)32

]

2 (2 minus 1199032)+ (

120587

2minus 1) [1 + (1 minus 119903

2)12

]

120587

2

32 minus 211199032+ 21 (1 minus 119903

2)12

+ 11 (1 minus 1199032)32

36 minus 181199032 + 28 (1 minus 1199032)12

lt E (119903) lt120587

2


2+ 3 (120587 minus 2) (1 minus 119903

2)12

+ 2 (1 minus 1199032)32


(47)




Acknowledgments



References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892 (0) = 0 (31)

119892 (1) =4

120587+

2

(2119901 minus 1)2

+ 1

minus 3 (32)

1198921(119903) =

2

120587

E (119903) minus (1 minus 1199032)K (119903)

1199032


2

[(2119901 minus 1)2

1199032 + 1]2

(33)

1198921(0) =

1

2minus 4 (2119901 minus 1)

2

(34)

1198921(1) =

2

120587minus

4 (2119901 minus 1)2

[(2119901 minus 1)2

+ 1]2 (35)




1198921(0) = 0 (36)


1198921(119903) gt 0 (37)


and (37)



(32) (34) and (35) lead to

119892 (1) = 0 (38)

1198921(0) = minus

36 minus 11120587

6120587 minus 8lt 0 (39)

1198921(1) =

31205872minus 14120587 + 16

1205872gt 0 (40)



1(119903) lt 0 for 119903 isin (0 119903

0) and

1198921(119903) gt 0 for 119903 isin (119903





2= 120582lowast



119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582

2) 119887 1205822119887 + (1 minus 120582

2) 119886] (41)





1198921(119903) lt 0 (42)

for 119903 isin (0 1205751)


119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)

for (119886 minus 119887)(119886 + 119887) isin (0 1205751)




119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583

2) 119887 1205832119887 + (1 minus 120583

2) 119886] (44)


lowast



119892 (119903) gt 0 (45)

for 119903 isin (1 minus 1205752 1)


119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)



1= 4120587 minus 1

1205822= 12 + radic28 and 120583




120587 [1 + (1 minus 1199032)32

]

16 (2 minus 1199032)+

7120587 [1 + (1 minus 1199032)12

]

32lt E (119903)

lt

(4 minus 120587) [1 + (1 minus 1199032)32

]

2 (2 minus 1199032)+ (

120587

2minus 1) [1 + (1 minus 119903

2)12

]

120587

2

32 minus 211199032+ 21 (1 minus 119903

2)12

+ 11 (1 minus 1199032)32

36 minus 181199032 + 28 (1 minus 1199032)12

lt E (119903) lt120587

2


2+ 3 (120587 minus 2) (1 minus 119903

2)12

+ 2 (1 minus 1199032)32


(47)




Acknowledgments



References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article sharp bounds for toader mean in terms of...

Documents