bessel function with linear differential operator · 2021. 1. 22. · bessel function with linear...
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Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 16, Number 5 (2020), pp. 623-639 © Research India Publications http://www.ripublication.com/gjpam.htm
Bessel Function with Linear Differential Operator
S. Lakshmi1*, K. R. Karthikeyan2, S. Varadharajan3 and C. Selvaraj4
2 National University of Science and Technology, Muscat, Sultanate of Oman.
3 University of Technology and Applied Science – Al Musanna, Sultanate of Oman.
4 Presidency College (Autonoumous), Chennai, India.
Abstract
In this paper, a class of analytic functions f defined on the open unit disc satisfying
2
1)(
)(1>
2
1)(
)(11
zf
z'zf
zf
z'zfieRe
is studied, among other results, inclusion relations and applications involving a certain class of integral operator are also considered.
Keywords: Analytic function, univalent function, starlike function, convex function, subordination, bessel function
2010 Mathematics Subject Classification: 30C45, 30C50.
1 INTRODUCTION
Let A denote the class of all analytic function of the form
n
n
n
zazzf
2=
=)( (1.1)
in the open unit disc 1|
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624 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
of univalent functions in U . We say that the function f is convex when )(Uf is a convex set. Also, we say that a function f is starlike with respect to the origin when
)(Uf is a starlike set with respect to 0. By K or *S we denote the subclasses of A
consisting of all functions which are convex or starlike respectively, while by )(S* we denote the class of starlike functions of order 0,1, .
In 1991, Goodman [7] introduced the class UCV of uniformly convex functions. A function CVf is the class UCV if for every circular are U , with center in U , the arc )(f is convex. A more useful characterization of class UCV was given by Ma and Minda [11] (see also [17]) as:
).(,)(
)(>
)(
)(1 Uz
z'f
z''zf
z'f
z''zfReandff
AUCV
In 1999, Kanas and Wisniowska [8] (see also [9]) introduced the class of k -uniformaly convex functions, 0k , denoted by k - UCV and the class k - ST related to k -UCV by Alexandar type relation, i.e.,
).(,)(
)(>
)(
)(1 Uz
z'f
z''zfk
z'f
z''zfReandfk'zfkf
ASTUCV
In [8] and [9] respectively, their geometric definitions and connections with the conic domains were also considered. For a fixed 0k , the class k -UCV is defined purely geometrically as a subclass of univalent functions which map the intersection of U with any disk centered at k||, , onto a convex domain. The notion of k -uniform convexity is a natural extension of the classical convexity. Observe that, if 0=k then the center is the origin and the class k - UCV reduces to the class Vk . Moreover for 1=k it coincides with the class of uniformly convex functions UCV introduced by Goodman [7] and studied extensively by R ø nning [17] and independently by Ma and Minda [11]. The class k - UCV started much earlier in papers [5, 6] with some additional conditions but without the geometric interpretation.
We say that a function Af is in the {0}\0,,*, CS kk , if and only if
).(,1)(
)(1>1
)(
)(11 Uz
zf
z'zfk
zf
z'zfRe
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Bessel Function with Linear Differential Operator 625
A lot of authors investigated the properties of the class * ,kS and their generalizations
in several directions, e.g., see [1, 2, 6, 9, 15, 17, 19]. An analytic function f is said to be subordinate to an analytic function g (written as gf ) if and only if there exists an analytic function with
Uzforzand 1|
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626 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
defined by
).(21!
1)(=)(
2
0=
CJ
zz
nvnz
vnn
n
v
(1.4)
ii) For 1=b and 1= d in (1.3), we obtain the modified Bessel function of the first kind of order v defined by
).(21!
1)(=)(
2
0=
CI
zz
nvnz
vnn
n
v
(1.5)
iii) For 2=b and 1=d in (1.3), the function )(,, zdbv reduces to )(2
zvS
,
where vS is the spherical Bessel function of the first kind of order v , defined by
).(2
2
3!
1)(
2=)(
2
0=
CS
zz
nvn
z
vnn
n
v
(1.6)
Now, consider the function CC:)(,, zu dbv , defined by the transformation
).(2
12=)( ,,
2,, zz
bvzu dbv
v
v
dbv
(1.7)
By using the well-known Pochhammer symbol (or the shifted factorial) )( defined,
for C, and in terms of the Euler function, by
),;=(1)(1)(
{0}),\0;=(1=
)(
)(:=)(
CN
C
nn
and 1=)( 0 , we obtain for the function )(,, zu dbv the following representation
,!
2
1
4=)(
0
,,n
z
bv
d
zun
n
n
n
dbv
where 2,1,0,2
1=
bvk . This function is analytic on C and satisfies the
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Bessel Function with Linear Differential Operator 627
second order linear differential equation
0.=)()(1)2(2)(24 zdzuz'zubvz''uz
Now, we introduce the function )(,, zdbv defined in terms of generalized Bessel
function )(,, zdbv , defined by
)(=)( ,,,, zzuz dbvdbv
)(2
12= ,,
21
zzb
v dbv
v
v
2
1=,
)(!4
)(=
1
1=
bvkwhere
kn
zdz
n
n
nn
n
),,(= zdkg
Motivated by Selvaraj and Karthikeyan [14], we define the following UUzfdkDm :)(),( by
),,()(=)(),( zdkgzfzfdkD (1.8)
'zdkgzfzzdkgzfzfdkD )),,()((),,()()(1=)(),(1 (1.9)
)(),(=)(),( 11 zfdkDDzfdkD mm (1.10)
If Af , then from (1.9) and (1.10) we may easily deduce that
1
1
1
2= )(1)!(4
)(1)(1=)(),(
n
n
n
n
nm
n
m
kn
zadnzzfdkD
(1.11)
where 0=0 NNm and 0 .
It can be easily verified from definition of (1.11) that
)(),()(1)(),(1=)(),( zfdkmDzfdkmD'
zfdkmDz
(1.12)
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628 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
In the special cases of the )(),( zfdkDm , we obtain the following operators related to the Bessel function:
i) Choosing 0=m in (1.11) we get the Deniz operator
1
1
1
2= )(1)!(4
)(==
n
n
n
n
n
n
c
kkn
zadzB .
ii) Choosing 0=m and 1== db in (1.11) we obtain the operator AAJ :v related with Bessel function, defined by
.1)(1)!(4
1)(=)(
1
1
1
2=
n
n
n
n
n
n
vvn
zazzfJ
iii) Choosing 0=m , 1=b and 1= d in (1.11) we obtain the operator AAJ :v related with modified Bessel function, defined by
.1)(1)!(4
(1)=)(
1
1
1
2=
n
n
n
n
n
n
vvn
zazzfJ
iv) Choosing 0=m , 2=b and 1=d in (1.11) we obtain the operator AAS :v related with spherical Bessel function, defined by
.
2
31)!(4
1)(=)(
1
1
1
2=
n
n
n
n
n
n
v
vn
zazzfS
Definition 1.1 A function A)(zf is in the class ,,S , {0}\C if and only if
2
1)(
)(1>
2
1)(
)(11
zf
z'zf
zf
z'zfieRe
. (1.13)
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Bessel Function with Linear Differential Operator 629
Remark 1 The above differential inequality can be equivalently written as
)(=sincos)(1)(
)(11 zKizhie
zf
z'zf
,
where
2
2
)(1
)(1
)(1
)(1
2
cos1=)(
z
z
z
z
logzh
, 2
1
1=
e
e and
cos2= .
Definition 1.2 A function A)(zf is in the class ,,,,dkQm ,if and only if
2
1,,,,1
>1,,,,1
1
2
zdkmzdkmeRe i
JJ ,
where
.)(),(
)(),(
=,,,,zfdkmD
'zfdkmDz
zdkm
J (1.14)
Since sincos)(=)( izhezKi is a convex univalent function, we can write
Definition 1.1 in subordination form
).()(,,,,,,,, UzzKzdkmandfdkQf m JA
Special Cases
For 0= and ,,,,dkQfm we have ,,)(),(
*SzfdkDm .
2 PRELIMINARY RESULTS
Lemma 1 [13] Let h be a convex univalent function in U with 0>)( zhRe , where Uz {0},\, CC . If p is analytic in U with (0)=(0) hp , then
).()(
)()( zh
zp
z'zpzp
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630 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
Lemma 2 [12] Let d be a complex number with 0>dRe . Suppose that CC 2:
is continuous and satisfies the conditions 0),( yixRe , when x is real and
).)/(2||( 2 dReixdy
If p is analytic in U with dp =(0) and 0>)(),(
z'zpzpRe for Uz ,
then 0>)(zpRe in U .
Lemma 3 [18] Let f and g be in the class Vk and *S respectively. Then, for
every function F analytic in U , we have
,,)()()(
)()()(UzUFco
zgzf
zFzgzf
where )(UFco , denotes the closed convex hull of the set )(UF .
Lemma 4 [16] Let the function )(z given by
nnn
zBz
1=
=)(
be convex in U . Suppose also that the function )(zh given by
nnn
zhzh
1=
=)(
is holomorphic in U . If )()( zzh , Uz , then 1,2,3,=|,||| 1 NnBhn .
3 COEFFICIENT INEQUALITIES
A function A)(zf is said to be bi-univalent in U if both )(zf and )(1 zf are
univalent in U . Let denote the class of bi-univalent functions defined in the unit disk U .
Definition 3.1 Let CUh : be a convex univalent function such that 1=(0)h and
)(=)( zhzh , for Uz and 0>))(( zhRe . A function f is said to be in the class
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Bessel Function with Linear Differential Operator 631
{0}\),,( CS
if the following conditions are satisfied:
Uzizhzf
zzfef
'i
,sincos)(1
)(
)(11,
(3.1)
and
,,sincos)(1)(
)(11 Uih
g
zge
'i
(3.2)
where
2,
2,)5(5)(2=)(=)( 4432
3
2
3
3
2
2
2
2
1 aaaaaaafg .
Definition 3.2 A function f given by (1.1) is said to be in the class
),,,,(, hdkm
S if the following conditions are satisfied:
UzizhzfdkmD
'zfdkmDz
ie
,s inco s)(1)(),(
)(),(1
1
(3.3)
and
,,s inco s)(1)(),(
)(),(1
1 UihgdkmD
'gdkmD
ie
(3.4)
where Uzfg ,1,{0},0\),(=)( 1 C on specializing the parameter .
Theorem 1 Let f given by (1.1) be in the class ),(
S , then
cos|||||| 12 Ba
and
.cos||||4|| 13 Ba
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632 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
Theorem 2 Let f given by (1.1) be in the class ),,,,(, hdkm
S , then
2
1
22
2
2
12
)8(
)(1
)8(
)2(1
cos||||2||
k
d
k
d
Ba
mm
and
.
)8(
)(1
)8(
)2(1
cos||||2||
2
1
22
2
2
1
2
3
k
d
k
d
Ba
mm
Proof. It follows from (3.3) and (3.4) that
UzizpzfdkmD
'zfdkmDz
ie
,sincos)(=1)(),(
)(),(1
1
(3.5)
and
,,sincos)(=1)(),(
)(),(1
1 UiqgdkmD
'gdkmD
ie
(3.6)
where )()( zhzp and )()( hq have the forms
2211=)( zpzpzp
and
2211=)( qqq
respectively. It follows from (3.5) and (3.6) that
cos=)(4(1)!
)()(112
1
pak
de mi (3.7)
cos=)(4(1)!
)()(1
)((2)!4
)2(2)(12
2
2
2
1
3
2
2
2
pak
da
k
de mmi
(3.8)
cos=)(4(1)!
)()(112
1
qak
de mi (3.9)
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Bessel Function with Linear Differential Operator 633
and
.cos=)(4(1)!
)()(1
)((2)!4
)2(2)(1
)((2)!4
)2(4)(12
2
2
2
1
3
2
2
22
2
2
2
2
qak
da
k
da
k
de mmmi
(3.10)
From (3.7) and (3.9), we obtain
.= 11 qp
Adding (3.8) and (3.10), we get
.cos)(=)8(
)(1
)8(
)2(122
2
22
1
22
2
2
qpak
d
k
de mmi
(3.11)
Since )(, Uhqp , applying Lemma 4, we have
NmBm
pp
m
m ,!
(0)= 1
)(
(3.12)
NmBm
qq
m
m ,!
(0)= 1
)(
. (3.13)
Applying (3.12), (3.13) and Lemma 4 for the coefficients 121 ,, qpp and 2q , we get
21)8(
22)(1
2)8(
2)2(1
cos|1|||2|2|
k
dm
k
dm
Ba
. (3.14)
Subtracting (3.10) from (3.8), we get
cos)(=)8(
)2(1
)8(
)2(122
2
2
2
2
3
2
2
qpak
da
k
de mmi
(3.15)
or, equivalently
21)8(
22)(1
2)8(
2)2(1
cos)22(2
2)2(1
cos)22(2)8(=3
k
dm
k
dm
qp
iedm
qpk
iea
.
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634 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
Applying (3.12), (3.13) and Lemma 4 once again for the coefficients 121 ,, qpp and 2q, we get
.
)8(
)(1
)8(
)2(1
cos||||2||
2
1
22
2
2
1
2
3
k
d
k
d
Ba
mm
This complete the proof of Theorem 2.
Corollary 1 Let Af be bi convex function of order then 1|| 2a and
1|| 3a .
Corollary 2 Let Af satisfy the condition )()(
)(1 zh
z'f
z''zf and
)()(
)(1 wh
w'g
w''zg then
|22|2||
21
2
1
11
2
BBB
BBa
and |)12|(
2
1|| 13 BBBa .
Corollary 3 Let f be given by (1.1) and 1= fg . If f and g satisfies the
condition )()(
)(1
)(
)()(1 zh
z'f
z''zf
zf
z'zf
and )()(
)(1
)(
)()(1 wg
w'g
w''wg
wg
w'wg
,
then
|))((1|)(1
||
21
2
1
11
2
BBB
BBa
(3.16)
and
.1
|12||| 13
BBBa (3.17)
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Bessel Function with Linear Differential Operator 635
Remark 2 Put 1= in corollary 3, we get the result |22|2
||
21
2
1
11
2
BBB
BBa
and
|)12|(2
1|| 13 BBBa of corollary 2.
4 CLOSURE PROPERTY
Theorem 3 Let
).(),(=))(,,,,( zfdkDzfdkmF m (4.1)
Then ,,,,dkQfm if and only if ,,))(,,,,( *SzfdkmF .
Proof. Let ,,))(,,,,( *SzfdkmF , then
.
2
1))(,,,,(
)))((,,,,(1>
2
1))(,,,,(
)))((,,,,(11
zfdkmF
zzfdkm'zF
zfdkmF
zzfdkm'zFieRe
(4.2)
Thus (4.1) together with (4.2) implies
,2
1),,,,(1
>
2
1),,,,(1
1
zdkmzdkmieRe
JJ
where ),,,,( zdkmJ is given by (1.14). Therefore ,,,,)( dkQzfm .
Converse is immediate.
Theorem 4 For 1m , ,,,,,,,,1 dkQdkQ mm .
Proof. Let ,,,,1 dkQf m and
)(=)(),(
)(),(
zpzfdkmD
'zfdkmDz
, (4.3)
where p is analytic in U and 1=(0)p .
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636 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
From (1.12) and (4.3) and after some simplification, we obtain
).(1)(=)(),(
)(),(1
zpzfdkD
zfdkDm
m
(4.4)
By logarithmic differentiation of (4.4), we have
.)()(1
)()(=
)(),(1
)(),(1
zp
z'zpzp
zfdkmD
'zfdkmDz
(4.5)
Since ,,,,1 dkQf m , so
).()()(1
)()( zK
zp
z'zpzp
Thus by using Lemma 1, )()( zKzp and hence ,,,,dkQfm .
Let us consider the Bernardi integral operator F given by
.)(1=)( 10
dttftz
zfFz
(4.6)
For with 1> Re the operator has the property AA :F , (see, for instance,
[10], p.11).
Theorem 5 Let 1> and ,,,,dkQfm , then ,,,,dkQfF
m .
Proof. Let ,,,,dkQfm and
)(=))((),(
))((),(
zpzfFdkmD
'zfFdkmDz
, (4.7)
where the function )(zp is analytic in U and 1=(0)p . From (4.6), we have
)(1)(=))(()()( zfzfFz'fFz
and so
))((),()(),(1)((=))((),( zfFdkmDzfdkmD'zfFdkmDz
. (4.8)
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Bessel Function with Linear Differential Operator 637
Then, by using equations (4.7) and (4.8), we obtain
)(=))((),(
)(),(1)((zp
zfFdkD
zfdkDm
m
(4.9)
By logarithmic differentiation of(4.9), we have
).(,)(
)(=)(),(
)(),(
zzp
'zpzp
zfdkmD
'zfdkmDz
Hence by Lemma1, we conclude that )()( zKzp in U which implies that
,,,,dkQfFm .
Theorem 6 Let ,,,,dkQfm and Vk . If 1
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638 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj
By Theorem(3), the function ,,)(),(=)(*SzfdkDzF m .
Hence, by Lemma (3) , we have
).()]([)(),(
)(),(
zKUFcozFdkmD
'zFdkmDz
Since )(zK is a convex univalent and )()( zKzp .
Hence ,,,,= dkQfFm .
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Bessel Function with Linear Differential Operator 639
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640 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj