Gen. Math. Notes, Vol. 15, No. 1, March, 2013, pp. 28-43
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
On a Certain Subclass of Univalent Functions
Defined by Differential Subordination
Property
Waggas Galib Atshan1, Hadi Jabber Mustafa2 and Emad Kadhim Mouajeeb3
1
Department of Mathematics
College of Computer Science and Mathematics
University of Al-Qadisiya
Diwaniya – Iraq
E-mail: waggashnd@gmail.com; waggas_hnd@yahoo.com
2
Department of Mathematics
College of Mathematics and Computer Science
University of Kufa
Najaf - Kufa – Iraq
E-mail: hadi.mustafa@uokufa.edu-iq
3
Department of Mathematics
College of Mathematics and Computer Science
University of Kufa
Najaf - Kufa – Iraq
E-mail: eallamy@yahoo.com
(Received: 7-12-12 / Accepted: 23-1-13)
Abstract
The object of the present paper is to investigate and study certain subclass of
univalent ,
functions defined by differential subordination by using the linear
. Coefficient bounds, some properties of neighborhoods,
operator,,
convolution properties, Integral mean inequalities for the fractional integral for
this certain subclass are given.
On a Certain Subclass of Univalent Functions…
29
Keywords: Univalent Function, Differential Subordination, - neighborhood,
Convolution, Hypergeometric Function, Linear Operator, Integral Mean,
Fractional Integral.
1
Introduction
Let G be the class of all functions of the form: = + , ∈ ,
1.1
∞
which are analytic and univalent in the open unit disk = ∈ ₵ ∶ || < 1 .
Let ! denote the subclass of G containing of functions of the form:#
= − , ≥ 0 ,
∈ .
1.2
The Hadamard product (or convolution ) of two power series
#
= − in ! is defined (as usual )by
#
'
( = − ) #
∗ ( = ∗ ( = − ) .
For positive real values of - . , … , -0 ' 1. , … , 12 314 ≠ 0, −1, … , 6 =
1,2, … , 7,
the generalized hypergeometric function 0 82 is defined by
#
0 82 ≡ 0 82 α. , … , α; ; β. , … , β> ; z = E ≤ 7 + 1; E, 7GC = ∪ 0 ; ∈ ,
@C
α. @ … α; @
β. @ … β> @
where is the Pochhammer symbol defined by
@ = I
1,
+ 1 + 2 … + − 1,
=0
∈ .
z@
n!
1.3
1.4
1.5
J 1.6
The notation 0 82 is quite useful for representing many well- known functions
such as the exponential, the Binomial, the Bessel and Laguerre polynomial. Let
30
Waggas Galib Atshan et al.
HMα. , … , α; ; β. , … , β> N: ! ⟶ !
be a linear operator defined by
HM α. , … , α; ; β. , … , β> Nz = z 0 82 α. , α , … , α; ; β. , β , … , β> ; z ∗ z
#
= − Q -. ; E; 7 ,
1.7
Where,
Q -. ; E; 7 =
-. S.… -0 S.
1
.
1. S.… 12 S. − 1!
1.8
0
For notational simplicity, we use shorter notation U2
M-. N for
UM-. , … , -0 ; 1. , … , 12 N.
In the sequel. It follows from (1.7) that
UC. M1Nfz = z, UC. M2Nz = z W z.
;
Mα. N is called Dziok–Srivastava operator (see[3])
The linear operator H>
introduced by Dziok and Srivastava which was subsequently extended by Dziok
and Raina [2] by using the generalized hypergeometric function, recently
Z,[\
Srivastava et. al. ([10]) defined the linear operator ℒY,0,2
as follows:C
ℒY,[\ = \
0 M- N
0 M- NW
ℒY,0,2
= 1 − ]U2
+ ]U2
.
.
.,[
\
= ℒY,0,2
,
[
] ≥ 0,
\
\
\
ℒY,0,2
= ℒY,0,2
_ℒY,0,2
`
,[
[
and in general ,
.,[
Z,[\
[\
ZS.,[
ℒY,0,2
= ℒY,0,2
_ℒY,0,2 \ `, E ≤ 7 + 1; E, 7 ∈ C = ∪ 0 ; ∈ 1.9
1.10
1.11
If the function is given by (1.2), then we see form (1.7), (1.8), (1.9) and
(1.11) that
On a Certain Subclass of Univalent Functions…
Z,[\
ℒY,0,2
where,
QZ -. ; ]; E; 7
31
#
= − QZ -. ; ]; E; 7 ,
-. S.… -0 S. M1 + ] − 1N
=a
b ,
− 1!
1. S.… 12 S.
1.12
Z
∈ \1 , dGC .
1.13
Unless otherwise stated. We note that when d = 1 ' ] = 0 the linear
Z,[\
operator ℒY,0,2
would reduce to the familier Dziok – Srivastava linear operator
given by (see [3]), includes (as its special cases) various other linear operators
introduced and studied by Carlson and Shaffer[1], Owa[7] and Ruscheweyh[8].
For two analytic functions , ( ∈ !, we say that is subordinate to (, written
≺ ( if there exists a Schwarz function Q, which (by definition) is
analytic in U with
Q0 = 0 ' │Q│ < 1 for all z ∈ U, such that = (3Qp, ∈ U.
Furthermore, if the function ( is univalent in U, then we have the following
equivalence (see [6]):
≺ ( ⟺ 0 = (0 ' ⊂ (.
Definition 1: For any function ∈ ! ' ≥ 0, sℎu − neighborhood of is
defined as,
#
#
v = w( = − ) ∈ !: │ − ) │ ≤ y.
In particular, for the function u = , we see that,
#
#
v u = w( = − ) ∈ !: │) │ ≤ y .
1.14
1.15
The concept of neighborhoods was first introduced by Goodman [4] and then
generalized by Ruscheweyh [9].
Definition 2: For fixed parameters A and B, with −1 ≤ z < { ≤ 1, we say that
∈ ! is in the class |d, }, -. , ], E, 7, {, z if it satisfies the following
subordination condition:
32
Waggas Galib Atshan et al.
ℒY,0,2 \ Z~,[
Z,[\
ℒY,0,2
≺
1 + Az
.
1 + Bz
1.16
In view of the definition of subordination, (1.16) is equivalent to the following
condition:
ℒY,0,2 \ −1
Z,[\
‚ ℒY,0,2
‚
Z~,[
Z~,[\
‚ ℒY,0,2
‚
−A
B Z,[\
ℒY,0,2 < 1, ∈ .
For convenience, we write
|d, }, -. , ], E, 7, 1 − 2ƒ, −1 = |d, }, -. , ], E, 7, ƒ,
where |d, }, -. , ], E, 7, ƒ denotes the class of functions in ! satisfying the
inequality:
Z~,[
ℒY,0,2 \ „u w Z,[\
y > ƒ, 0 ≤ ƒ < 1; ∈ .
ℒY,0,2 Neighborhoods for the Class †, ‡, , , , , ˆ, ‰
2
Theorem 2.1: A function ∈ ! belongs to the class |d, }, -. , ], E, 7, {, z if
and only if
QZ -. ; ]; E; 7Š1 − BQ -. ; ]; E; 7 − 1 − A‹ ≤ A − B
∞
for d, }, E, 7 ∈ C , E ≤ 7 + 1 , ] ≥ 0 ' − 1 ≤ z < { ≤ 1 .
Proof: Let ∈ |d, }, -. , ], s, 7, A, B. Then,
ℒY,0,2 \ Z~,[
Z,[\
ℒY,0,2
≺
1 + Az
∈.
1 + Bz
Therefore, there exists an analytic function w such that
2.1
2.2
On a Certain Subclass of Univalent Functions…
Q =
Hence,
33
\
ℒY,0,2 \ − ℒY,0,2
Z~,[
Z,[
\
BℒY,0,2 \ − AℒY,0,2
Z~,[
Z,[
.
2.3
\
ℒY,0,2 \ − ℒY,0,2
Z~,[
Z,[
│Q│ = Œ Z~,[
Œ
Z,[\
BℒY,0,2 \ − AℒY,0,2
=
Thus,
„u 
Z

∑∞
Q -. ; ]; E; 7ŠQ -. ; ]; E; 7 − 1‹ Z

A − B + ∑∞
Q -. ; ]; E; 7ŠBQ -. ; ]; E; 7 − A‹ Z

∑∞
Q -. ; ]; E; 7ŠQ -. ; ]; E; 7 − 1‹ Z

A − B + ∑∞
Q -. ; ]; E; 7ŠBQ -. ; ]; E; 7 − A‹  < 1.
 < 1.
2.4
Taking ││ = ‘, for sufficiently small r with 0 < ‘ < 1, the denominator of
(2.4) is positive and so it is positive for all r with 0 < ‘ < 1, since Q is
analytic for ││ < 1. Then, the inequality (2.4) yields
QZ -. ; ]; E; 7ŠQ -. ; ]; E; 7 − 1‹ ‘ ∞
< A − B‘ + QZ -. ; ]; E; 7ŠBQ -. ; ]; E; 7 − A‹ ‘ .
Equivalently,
∞
QZ -. ; ]; E; 7Š1 − BQ -. ; ]; E; 7 − 1 − A‹ ‘ ≤ A − B‘,
∞
and (2.1) follows upon letting ‘ → 1 .
Conversely, for ││ = ‘ , 0 < ‘ < 1 , we have ‘ < ‘ . That is,
QZ -. ; ]; E; 7Š1 − BQ -. ; ]; E; 7 − 1 − A‹ ‘ ∞
34
Waggas Galib Atshan et al.
≤ QZ -. ; ]; E; 7Š1 − BQ -. ; ]; E; 7 − 1 − A‹ ‘ ≤ A − B‘.
∞
From (2.1), we have
Œ QZ -. ; ]; E; 7ŠQ -. ; ]; E; 7 − 1‹ Œ
∞
≤ QZ -. ; ]; E; 7ŠQ -. ; ]; E; 7 − 1‹ ‘ ∞
< A − B‘ + ŠBQ -. ; ]; E; 7 − A‹QZ -. ; ]; E; 7 ‘ ∞
< ŒA − B + ŠBQ -. ; ]; E; 7 − A‹QZ -. ; ]; E; 7 Œ.
∞
This proves that
ℒY,0,2 \ Z~,[
Z,[\
ℒY,0,2
and hence
_
1 + Az
,
1 + Bz
∈
∈ |d, }, -. , ], E, 7, A, B.
Theorem 2.2 If
=
≺
“\ \…“” \
.~Y`
•\ \…•– \
—˜\
ΑSΒ
™
“\ \ …“” \
.~Yb S.SΑN
•\ \ …•– \
M.SΒa
then |d, }, -. , ], E, 7, A, B ⊂ v u.
,
2.5
Proof: It follows from (2.1), that if ∈ |d, }, -. , ], E, 7, A, B, then
QZS. -. ; ]; E; 7Š1 − BQ -. ; ]; E; 7 − 1 − A‹ ≤ A − B,
∞
Hence
_š
[\ \ …[” \
\ \ …š– \
ZS.
1 + ]`
w1 − Β aš
1 − Α ∑∞
≤ A − B.
[\ \ …[” \
\ \ …š– \

1 + ]b −
2.6
On a Certain Subclass of Univalent Functions…
35
Which implies,
∞
≤
Α − Β

ZS.
- … -0 .
- … -0 .
1 + ]b − 1 − Αž
› . .
1 + ]œ
1 − Β a . .
1. . … 12 .
1. . … 12 .
= .
Using (1.15), we get the result.
Definition (2.1): The function ( defined by
2.7
( = − ) ∞
is said to be a member of the class |Ÿ d, }, -. , ], E, 7, A, B if there exists a
function ∈ |d, }, -. , ], E, 7, A, B such that
£¢ − 1 ≤ 1 − ¤, ∈ , 0 ≤ ¤ < 1.
¡¢
Theorem (2.3): If ∈ |d, }, -. , ], E, 7, A, B and
\
\
¦
¤ = 1 − ¥— [ ¦;Y;0;2Š.S§¥
™ [ ;Y;0;2S.S¨‹SΑSΒ ,
v¥ — [ ;Y;0;2Š.S§¥ ™ [ ;Y;0;2S.S¨‹
¦
\
\
¦
then v ⊂ |Ÿ d, }, -. , ], E, 7, A, B.
Proof: Let ( ∈ v . Then we have from (1.14) that
│ − ) │ ≤ ,
∞
which implies the coefficient inequality
│ − ) │ ≤
∞
.
2
Also since ∈ |d, }, -. , ], E, 7, A, B, we have from (2.1)
∞
≤
Α − Β
,
QZ -. ; ]; E; 7M1 − BQ -. ; ]; E; 7 − 1 − AN
2.8
2.9
36
Waggas Galib Atshan et al.
where
Z
-. . … -0 .
1 + ]ª ,
QZ -. ; ]; E; 7 = ©
1. . … 12 .
Q -. ; ]; E; 7
So that

-. . … -0 .
=›
1 + ]œ .
1. . … 12 .
∑∞
∑∞
(
− ) │ − ) │

− 1 = 

<
− ∑∞
1 − ∑∞
QZ -. ; ]; E; 7M1 − BQ -. ; ]; E; 7 − 1 − AN
≤ . Z
2 Q -. ; ]; E; 7«1 − BQ -. ; ]; E; 7 − 1 − A¬ − Α − Β
=1-y .
Thus by Definition (2.1), ( ∈ |Ÿ d, }, -. , ], E, 7, A, B for ¤ given by (2.9).
This completes the proof.
3
Convolution Properties:
Theorem 3.1: Let the functions 4 6 = 1,2 defined by
4 = − ,4 ,
∞
3,4 ≥ 0, 6 = 1,2p ,
3.1
be in the class |d, }, -. , ], E, 7, A, B .
Then . ∗ ∈ |d, }, -. , ], E, 7, A, σ, where
­≤
QZ -. ; ]; E; 7«1 − BQ -. ; ]; E; 7 − 1 − A¬ Α − Α − Β Q -. ; ]; E; 7 − 1 − Α
QZ -. ; ]; E; 7M1 − BQ -. ; ]; E; 7 − 1 − AN − Q -. ; ]; E; 7Α − Β
Proof: We must find the largest σ such that
QZ -. ; ]; E; 7«1 − σQ -. ; ]; E; 7 − 1 − A¬
,. , ≤ 1.
A−­
∞
Since 4 ∈ |d, }, -. , ], E, 7, A, B 6 = 1,2, then
On a Certain Subclass of Univalent Functions…
37
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
,4 ≤ 1,
A−Β
∞
6 = 1,2.
3.2
By Cauchy-Schwarz inequality, we get
∞
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
®,. , ≤ 1.
A−Β
3.3
We want only to show that
QZ -. ; ]; E; 7«1 − σQ -. ; ]; E; 7 − 1 − A¬
,. ,
A−­
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
≤
®,. , .
A−Β
This equivalently to
®,. , ≤
From (3.3), we have
®,. , ≤
Α − ­«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
A − Β«1 − σQ -. ; ]; E; 7 − 1 − A¬
QZ -. ; ]; E; 7«1
Thus, it is sufficient to show that
.
Α−Β
.
− ΒQ -. ; ]; E; 7 − 1 − A¬
Α−Β
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
≤
Which implies to
­≤
Α − ­«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
A − Β«1 − σQ -. ; ]; E; 7 − 1 − A¬
,
QZ -. ; ]; E; 7«1 − BQ -. ; ]; E; 7 − 1 − A¬ Α − Α − Β Q -. ; ]; E; 7 − 1 − Α
.
QZ -. ; ]; E; 7M1 − BQ -. ; ]; E; 7 − 1 − AN − Q -. ; ]; E; 7Α − Β
38
Waggas Galib Atshan et al.
Theorem (3.2): Let the functions 4 6 = 1,2 defined by (3.1) be in the class
Wτ, θ, α. , λ, ι, m, A, B. Then the function µ defined by
µ = − ,.
+ ,
3.4
∞
belong to the class |d, }, -. , ], E, 7, A, ε, where
¶
≤
ΑQZ -. ; ]; E; 7 «1 − BQ -. ; ]; E; 7 − 1 − A¬ − 2Α − Β QZ~ -. ; ]; E; 7 + 2Α − Β 1 − AQZ -. ; ]; E; 7
QZ -. ; ]; E; 7 M1 − BQ -. ; ]; E; 7 − 1 − AN − 2Α − Β QZ~ -. ; ]; E; 7
Proof: We must find the largest ¶ such that
QZ -. ; ]; E; 7«1 − εQ -. ; ]; E; 7 − 1 − A¬ ,. + ,
≤ 1.
A−¶
∞
Since 4 ∈ |d, }, -. , ], E, 7, A, B 6 = 1,2, we get
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
a
b ,.
Α−Β
#
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
≤ ©
,. ª
Α−Β
#
≤1,
and
3.5
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
a
b ,
Α−Β
#
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
≤ ©
, ª
Α−Β
#
≤ 1.
Combining the inequalities (3.5) and (3.6), gives
1 QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬
a
b 3,.
+ ,
p
2
Α−Β
#
≤ 1.
3.6
3.7
On a Certain Subclass of Univalent Functions…
39
But, k ∈ |d, }, -. , ], E, 7, A, ε, if and only if
∞
QZ -. ; ]; E; 7«1 − εQ -. ; ]; E; 7 − 1 − A¬ ,. + ,
A−¶
≤ 1.
3.8
The inequality (3.8) will be satisfied if
QZ -. ; ]; E; 7«1 − ¶Q -. ; ]; E; 7 − 1 − {¬
{−¶
Z Q . ; ]; E; 7 «1 − zQ -. ; ]; E; 7 − 1 − {¬
≤
,
2{ − º
(n=2,3,…)
»
so that
≤
¿
¼½¾ ; ; ; ¿ M − ‰½‡¾ ; ; ; − − ˆN¿ − ¿¼ − À¿ ½~‡
¾ ; ; ; + ¿¼ − À − ˆ½¾ ; ; ; ½¾ ; ; ; ¿ M − ‰½‡¾ ; ; ; − − ˆN¿ − ¿¼ − À¿ ½~‡
;
;
;
¾
4
Integral Mean Inequalities for the Fractional
Integral:
Definition (4.1) [6]: The fractional integral of order Á Á > 0 is defined for a
function by
¢
1
s
SÃ
¢ =
Ä
's ,
4.1
ΓÁ C − s.SÃ
where the function is an analytic in a simply – connected region of the complex
z-plane containing the origin, and multiplicity of − sÃS. is removed by
requiring ÅÆ( − s to be real, when − s > 0.
In 1925, Littlewood [5] proved the following subordination theorem:
Theorem 4.1 (Littlewood [5]): If ' ( are analytic in U with ≺ (, then for
Ç > 0 ' = ‘u È 0 < ‘ < 1
Ê
Ê
Ä ||É '} ≤ Ä |(|É '}.
C
C
Theorem 4.2: Let ∈ |d, }, -. , ], E, 7, {, º and suppose that is defined by
40
Waggas Galib Atshan et al.
z = z −
z @ , n ≥ 2.
4.2
Α − Β Γn + 1Γs + ƒ + 3
,
− ΒQ -. ; ]; E; 7 − 1 − ANΓn + s + ƒ + 1Γ2 − ƒ
4.3
— [ ;Y;0;2«.SΒ¥ ™ [ ;Y;0;2S.S¨¬
¥Ë
\
Ë \
ΑSΒ
Also let
Ì − ƒÍ~. È
∞
È
≤
QZ -. ; ]; E; 7M1
for 0 ≤ ƒ ≤ Ì , Á > 0 , where Ì − ƒÍ~. denote the pochhammer symbol
defined by Ì − ƒÍ~. = Ì − ƒÌ − ƒ + 1 … Ì.
If there exists an analytic function Î defined by
ÎS.
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + ƒ + 1
=
UÌÈ ÈS. ,
Ì − ƒÍ~.
Α − Β Γn + 1
∞
where Ì ≥ ƒ and
UÌ =
ÈÏ~.
ΓÌ − ƒ
,
ΓÌ + s + ƒ + 1
then, for = ‘u ÈÐ ' 0 < ‘ < 1
Ä
Ê
C
É
SÃSÍ
Ñ¢
Ñ
Ê
'Ò ≤ Ä Ñ¢
C
Á > 0 , Ì ≥ 2 ,
SÃSÍ
Ñ 'Ò,
É
Á > 0, Ç > 0.
Proof: Let
= − È È .
∞
È
For ƒ ≥ 0 and Definition(4.1), we get
SÃSÍ
¢
Γ2 Ó~Í~.
ΓÌ + 1Γs + ƒ + 2
=
©1 − ÈS. ª
Γs + ƒ + 2
Γ2ΓÌ + s + ƒ + 1 È
∞
È
(4.5)
4.6
4.7
On a Certain Subclass of Univalent Functions…
41
Γ2 Ó~Í~.
Γs + ƒ + 2
Ì − ƒÍ~. UÌÈ ÈS. ª,
=
©1 − Γs + ƒ + 2
Γ2
∞
È
where
UÌ =
ΓÌ − ƒ
,
ΓÌ + s + ƒ + 1
Since U is a decreasing function of Ì, we have
0 < UÌ ≤ U2 =
Á > 0 , Ì ≥ 2.
Γ2 − ƒ
.
Γs + ƒ + 3
Similarly, from (4.2) and Definition4.1, we get
Γ2 Ó~Í~.
=
a1
Γs + ƒ + 2
¢
SÃSÍ
−
Α − Β Γn + 1Γs + ƒ + 2
S. b.
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + ƒ + 1
For Ç > 0 and = ‘u ÈÐ 0 < ‘ < 1, we must show that
Ä
≤Ä
Ê
C
Ê
C
Ô
1−
É
Γs + ƒ + 2
Ì − ƒÍ~. UÌÈ ÈS. Œ 'Ò
Œ1 − Γ2
∞
È
QZ -. ; ]; E; 7M1
É
Α − Β Γn + 1Γs + ƒ + 2
Ô 'Ò .
S.
− ΒQ -. ; ]; E; 7 − 1 − ANΓ2Γn + s + ƒ + 1
By applying Littlewood’s subordination theorem, it would suffice to show that
1− ∞
≺1
−
È
QZ -. ; ]; E; 7«1
By setting
Γs + ƒ + 2
Ì − ƒÍ~. UÌÈ ÈS.
Γ2
Α − Β Γn + 1Γs + ƒ + 2
S. .
− ΒQ -. ; ]; E; 7 − 1 − A¬Γ2Γn + s + ƒ + 1
1− ∞
È
Γs + ƒ + 2
Ì − ƒÍ~. UÌÈ ÈS.
Γ2
42
Waggas Galib Atshan et al.
=1
−
QZ -. ; ]; E; 7«1
we find that
Α − Β Γn + 1Γs + ƒ + 2
ÎS. ,
 1
− ΒQ . ; ]; E; 7 − − A¬Γ2Γn + s + ƒ + 1
ÎS.
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + ƒ + 1
=
Ì
Α − Β Γn + 1
∞
− ƒÍ~. UÌÈ ÈS. ,
which readily yields Q0 = 0 . For such a function Î, we obtain
|Î|S.
È
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + ƒ + 1
≤
Ì
Α − Β Γn + 1
∞
È
− ƒÍ~. UÌÈ ||ÈS.
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + ƒ + 1
≤
U2││ Ì
Α − Β Γn + 1
∞
È
− ƒÍ~. UÌÈ
= ││
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + ƒ + 1Γ2 − ƒ
Ì
Α − Β Γs + ƒ + 3Γn + 1
∞
− ƒÍ~. È ≤ ││ < 1.
È
This completes the proof of the theorem .
By taking ƒ = 0 in the Theorem4.2, we have the following corollary :
Corollary 4.1: Let ∈ |d, }, -. , ], E, 7, {, z and suppose that is defined by
(4.2). Also let
ÌÈ ≤
∞
È
QZ -. ; ]; E; 7«1 −
≥ 2.
Α − Β Γn + 1Γs + 3
,n
ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + 1Γ2
If there exists an analytic function Î defined by
On a Certain Subclass of Univalent Functions…
43
ÎS.
QZ -. ; ]; E; 7«1 − ΒQ -. ; ]; E; 7 − 1 − A¬Γn + s + 1
=
ÌUÌÈ ÈS. ,
Α − Β Γn + 1
∞
È
where
UÌ =
ΓÌ
,
ΓÌ + s + 1
then, for = ‘u ÈÐ ' 0 < ‘ < 1
Ê
Ê
Á > 0 , Ì ≥ 2,
Ä |¢Sà |É 'Ò ≤ Ä |¢Sà |É 'Ò,
C
C
Á > 0, Ç > 0
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
B.C. Carlson and D.B. Shaffer, Starlike and prestarlike hypergeometric
functions, SIAM, J. Math. Anal., 15(1984), 737-745.
J. Dziok and R.K. Raina, Families of analytic functions associated with the
Wright generalized hypergeometric function, demonstration, Math., 37(3)
(2004), 533-542.
J. Dziok and H.M. Srivastava, Certain subclasses of analytic functions
associated with the generalized hypergeometric function, Integral
Transform Spec. Funct., 14(2003), 7-18.
A.W. Goodman, Univalent functions and non-analytic curves, Proc. Amer.
Math. Soc., 8(3) (1957), 598-601.
L.E. Littlewood, On inequalities in the theory of functions, Proc. London
Math. Soc., 23(1925), 481-519.
S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and
Applications, Series on Monographs and Textbooks in Pure and Applied
Mathematics (Vol. 225), Marcel Dekker, New York and Basel, (2000).
S. Owa, On the distortion the theorems –I, Kyungpook Math. J., 18(1978),
53-59.
S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math.
Soc., 49(1975), 109-115.
S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math.
Soc., 81(1981), 521-527.
H.M. Srivastava, S.H. Li and H. Tang, Certain classes of k-uniformly
close –to – convex functions and other related functions defined by using
the Dziok – Srivastava operator, Bull. Math. Anal. Appl., 1(3) (2009), 4963.