Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 97135, 9 pages
doi:10.1155/2007/97135
Research Article
Integral Means Inequalities for Fractional
Derivatives of a Unified Subclass of Prestarlike Functions
with Negative Coefficients
H.
¨
Ozlem G
¨
uney and Shigeyoshi Owa
Received 24 May 2007; Revised 13 July 2007; Accepted 28 July 2007
Recommended by Narendra K. Govil
Integral means inequalities are obtained for the fractional derivatives of order p + λ(0
≤
p ≤ n,0≤ λ<1) of functions belonging to a unified subclass of prestarlike functions.
Relevant connections with various known integral means inequalities are also pointed
out.
Copyright © 2007 H.
¨
O. G
¨
uney and S. Owa. T his is an open access article distributed un-
der the Creative Commons Attribution License, which permits unrestricted use, dist ribu-
tion, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let denote the class of (normalized) functions of the form
f (z)
= z +
∞
n=2
a
n
z
n
, (1.1)
which are analytic and univalent in the open unit disk
U
={
z ∈ C : |z| < 1}. Also let ᐀
denote the subclass of consisting of functions f of the form
f (z)
= z −
∞
n=2
a
n
z
n
a
n
≥ 0
. (1.2)
The Hadamard product (or convolution) of two functions f given by (1.1)andg given
by
g(z)
= z +
∞
n=2
b
n
z
n
(1.3)
2 Journal of Inequalities and Applications
is defined by
( f
∗g)(z) = z +
∞
n=2
a
n
b
n
z
n
. (1.4)
We denote the subclass (α,β)of consisting of α-prestarlike functions of order β by
(α,β)
=
f ∈ :
f ∗s
α
(z) ∈
∗
(β), 0 ≤ α<1, 0 ≤ β<1
, (1.5)
where
∗
(β) denotes the class of starlike functions of order β(0 ≤ β<1) and s
α
is the
well-known extremal function for
∗
(α)givenby
s
α
(z) = z(1 − z)
−2(1−α)
(1.6)
(cf. [1, 2]). Letting
c
n
(α) =
n
k
=2
(k − 2α)
(n − 1)!
(n
= 2,3, ), (1.7)
s
α
can be written in the form
s
α
(z) = z +
∞
n=2
c
n
(α)z
n
. (1.8)
The class (α,β) was investigated by Sheil-Small et al. [3]. We also denote the subclass
Ꮿ(α,β)of, which was investigated by Owa and Uralegaddi [4], by
Ꮿ(α,β)
=
f ∈ : zf
(z) ∈ (α,β)
. (1.9)
In particular, the subclasses
[α,β]
= (α,β) ∩ ᐀, Ꮿ[α,β] = Ꮿ(α,β) ∩ ᐀ (1.10)
were considered earlier by Srivastava and Aouf [5]. Let us define the unified class ᏼ(α,
β,σ) of the classes [α,β]andᏯ[α,β]by
ᏼ(α,β,σ)
= (1 − σ)[α,β]+σᏯ[α, β](0≤ σ ≤ 1), (1.11)
so that
ᏼ(α,β,0)
= [α,β], ᏼ(α,β,1) = Ꮿ[α,β]. (1.12)
The unified class ᏼ(α,β, σ) was studied by Raina and Srivastava [6].
H.
¨
O. G
¨
uney and S. Owa 3
We begin by recalling the following useful characterizations of the function class
ᏼ(α,β,σ) due to Raina and Srivastava [6].
Lemma 1.1. Afunction f defined by (1.2)belongstotheclassᏼ(α,β,σ) if and only if
∞
n=2
(n − β)(1 − σ + σn)
1 − β
c
n
(α)a
n
≤ 1, (1.13)
for some α(0
≤ α<1), β(0 ≤ β<1), σ(0 ≤ σ ≤ 1).
We continue by proving the following lemma.
Lemma 1.2. Let
f
1
(z) = z, f
k
(z) = z −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
z
k
(k = 2,3, ). (1.14)
Then f
∈ ᏼ(α,β,σ) if and only if it can be expressed in the form
f (z)
=
∞
k=1
λ
k
f
k
(z), (1.15)
where λ
k
≥ 0 and
∞
k=1
λ
k
= 1.
Proof. Assume that
f (z)
=
∞
k=1
λ
k
f
k
(z). (1.16)
Then
f (z)
= λ
1
f
1
(z)+
∞
k=2
λ
k
f
k
(z)
= λ
1
z +
∞
k=2
λ
k
z −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
z
k
=
∞
k=1
λ
k
z −
∞
k=2
λ
k
1 − β
(k − β)(1 − σ + σk)c
k
(α)
z
k
= z −
∞
k=2
λ
k
1 − β
(k − β)(1 − σ + σk)c
k
(α)
z
k
.
(1.17)
Thus
∞
k=2
λ
k
1 − β
(k − β)(1 − σ + σk)c
k
(α)
(k − β)(1 − σ + σk)c
k
(α)
1 − β
=
∞
k=2
λ
k
=
∞
k=1
λ
k
− λ
1
= 1 − λ
1
≤ 1.
(1.18)
Therefore, we have f
∈ ᏼ(α,β,σ).
4 Journal of Inequalities and Applications
Conversely, suppose that f
∈ ᏼ(α,β,σ). Since
|a
k
|≤
1 − β
(k − β)(1 − σ + σk)c
k
(α)
(k
= 2,3, ), (1.19)
we can set
λ
k
=
(k − β)(1 − σ + σk)c
k
(α)
1 − β
(k
= 2,3, ), λ
1
= 1 −
∞
k=1
λ
k
. (1.20)
Then
f (z)
= z −
∞
k=2
a
k
z
k
= z −
∞
k=2
λ
k
1 − β
(k − β)(1 − σ + σk)c
k
(α)
z
k
=
1 −
∞
k=2
λ
k
z +
∞
k=2
λ
k
f
k
(z)
= λ
1
f
1
(z)+
∞
k=2
λ
k
f
k
(z)
=
∞
k=1
λ
k
f
k
(z).
(1.21)
This completes the assertion of Lemma 1.2.
Lemma 1.2 g ives us the following.
Corollar y 1.3. The extreme points of ᏼ(α,β,σ) are given by
f
1
(z) = z, f
k
(z) = z −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
z
k
. (1.22)
We will make use of the following definitions of fractional derivatives by Owa [7](also
by Srivastava and Owa [8]).
Definit ion 1.4. The fractional derivative of order λ is defined, for a function f ,by
D
λ
z
f (z) =
1
Γ(1 − λ)
d
dz
z
0
f (ξ)
(z − ξ)
λ
dξ (0 ≤ λ<1), (1.23)
where the function f is analytic in a simply connected region of the complex z-plane
containing the origin, and the multiplicity of (z
− ξ)
−λ
is removed by requiring log(z − ξ)
to be real when (z
− ξ) > 0.
H.
¨
O. G
¨
uney and S. Owa 5
Definit ion 1.5. Under the hypothesis of Definition 1.4, the fractional derivative of order
(n + λ) is defined, for a function f ,by
D
n+λ
z
f (z) =
d
n
dz
n
D
λ
z
f (z), (1.24)
where 0
≤ λ<1andn = 0,1,2,
It readily follows from (1.23)inDefinition 1.4 that
D
λ
z
z
k
=
Γ(k +1)
Γ(k − λ +1)
z
k−λ
(0 ≤ λ<1). (1.25)
We will also need the concept of subordination between analytic functions and a subor-
dination theorem of Littlewood [9] in our investigation.
Given two functions f and g, which are analytic in
U, the function f is said to be
subordinate to g in
U if there exists a function w analytic in U with
w(0)
= 0,
w(z)
< 1(z ∈ U), (1.26)
such that
f (z)
= g
w(z)
(z ∈ U). (1.27)
We denote this subordination by
f (z)
≺ g(z). (1.28)
Lemma 1.6. If the functions f and g are analytic in
U with
g(z)
≺ f (z), (1.29)
then, for μ>0 and z
= re
iθ
(0 <r<1),
2π
0
g
re
iθ
μ
dθ ≤
2π
0
f
re
iθ
μ
dθ. (1.30)
2. The main integral means inequalities
We discuss the integral means inequalities for functions f in ᏼ(α,β,σ). Our main theo-
rem is contained in the following.
Theorem 2.1. Let f
∈ ᏼ(α,β,σ) and suppose that
∞
n=2
(n − p)
p+1
a
n
≤
(1 − β)Γ(k +1)Γ(3 − λ− p)
(k − β)(1 − σ + σk)c
k
(α)Γ(k +1− λ − p)Γ(2 − p)
(k
≥ 2) (2.1)
for 0
≤ λ<1,where(n − p)
p+1
denotes the Pochhammer symbol defi ned by
(n
− p)
p+1
= (n − p)(n − p +1)···n. (2.2)
6 Journal of Inequalities and Applications
Also let the function f
k
be defined by
f
k
(z) = z −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
z
k
. (2.3)
If there exists an analytic function w defined by
w(z)
k−1
=
(k − β)(1 − σ + σk)c
k
(α)
1 − β
Γ(k +1
− λ − p)
Γ(k +1)
∞
n=2
(n − p)
p+1
Φ(n)a
n
z
n−1
(2.4)
with
Φ(n)
=
Γ(n − p)
Γ(n +1− λ− p)
(0
≤ λ<1, n = 2,3, ), (2.5)
then, for μ>0 and z
= re
iθ
(0 <r<1),
2π
0
D
p+λ
z
f (z)
μ
dθ ≤
2π
0
D
p+λ
z
f
k
(z)
μ
dθ (0 ≤ λ<1, μ>0). (2.6)
Proof. By virtue of the fractional derivative formula (1.25)andDefinition 1.5,wefind
from (1.1)that
D
p+λ
z
f (z) =
z
1−p−λ
Γ(2 − λ− p)
1 −
∞
n=2
Γ(2 − λ− p)Γ(n +1)
Γ(n +1− λ− p)
a
n
z
n−1
=
z
1−p−λ
Γ(2 − λ− p)
1 −
∞
n=2
Γ(2 − λ− p)(n − p)
p+1
Φ(n)a
n
z
n−1
,
(2.7)
where
Φ(n)
=
Γ(n − p)
Γ(n +1− λ− p)
(0
≤ λ<1, n = 2,3, ). (2.8)
Since Φ is a decreasing function of n,wehave
0 < Φ(n)
≤ Φ(2) =
Γ(2 − p)
Γ(3 − λ− p)
(0
≤ λ<1, n = 2,3, ). (2.9)
Similarly, from (2.3), (1.25), and Definition 1 .5,weobtain
D
p+λ
z
f
k
(z) =
z
1−p−λ
Γ(2 − λ− p)
1 −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
Γ(2
− λ − p)Γ(k +1)
Γ(k +1− λ − p)
z
k−1
.
(2.10)
H.
¨
O. G
¨
uney and S. Owa 7
For μ>0andz
= re
iθ
(0 <r<1), we must show that
2π
0
1 −
∞
n=2
Γ(2 − λ− p)(n − p)
p+1
Φ(n)a
n
z
n−1
μ
dθ
≤
2π
0
1 −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
Γ(2
− λ − p)Γ(k +1)
Γ(k +1− λ − p)
z
k−1
μ
dθ.
(2.11)
Thus, by applying Lemma 1.6,itwouldsuffice to show that
1
−
∞
n=2
Γ(2 − λ− p)(n − p)
p+1
Φ(n)a
n
z
n−1
≺ 1 −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
Γ(2
− λ − p)Γ(k +1)
Γ(k +1− λ − p)
z
k−1
.
(2.12)
If the subordination (2.12) holds true, then we have an analytic function w with w(0)
= 0
and
|w(z)| < 1suchthat
1
−
∞
n=2
Γ(2 − λ− p)(n − p)
p+1
Φ(n)a
n
z
n−1
= 1 −
1 − β
(k − β)(1 − σ + σk)c
k
(α)
Γ(2
− λ − p)Γ(k +1)
Γ(k +1− λ − p)
w(z)
k−1
.
(2.13)
By the condition of the theorem, we define the function w by
w(z)
k−1
=
(k − β)(1 − σ + σk)c
k
(α)
1 − β
Γ(k +1
− λ − p)
Γ(k +1)
∞
n=2
(n − p)
p+1
Φ(n)a
n
z
n−1
(2.14)
which readily yields w(0)
= 0. For such a function w,wehave
w(z)
k−1
≤
(k − β)(1 − σ + σk)c
k
(α)
1 − β
Γ(k +1
− λ − p)
Γ(k +1)
∞
n=2
(n − p)
p+1
Φ(n)a
n
|z|
n−1
≤|z|
(k − β)(1 − σ + σk)c
k
(α)
1 − β
Γ(k +1
− λ − p)
Γ(k +1)
Φ(2)
∞
n=2
(n − p)
p+1
a
n
=|z|
(k−β)(1−σ +σk)c
k
(α)
1−β
Γ(k+1
−λ− p)
Γ(k+1)
Γ(2
− p)
Γ(3−λ− p)
∞
n=2
(n− p)
p+1
a
n
=|z| < 1,
(2.15)
by means of the hypothesis of the theorem.
This means that the subordination (2.12) holds true; therefore the theorem is proved.
As special case p
= 0, Theorem 2.1 readily yields.
8 Journal of Inequalities and Applications
Corollar y 2.2. Let f
∈ ᏼ(α,β,σ) and suppose that
∞
n=2
n
a
n
≤
(1 − β)Γ(k +1)Γ(3 − λ)
(k − β)(1 − σ + σk)c
k
(α)Γ(k +1− λ)
(k
≥ 2). (2.16)
If there exists an analytic function w given by
w(z)
k−1
=
(k − β)(1 − σ + σk)c
k
(α)
1 − β
Γ(k +1
− λ)
Γ(k +1)
∞
n=2
nΦ(n)a
n
z
n−1
(2.17)
with
Φ(n) =
Γ(n)
Γ(n +1− λ)
(0
≤ λ<1, n = 2,3, ), (2.18)
then, for μ>0 and z
= re
iθ
(0 <r<1),
2π
0
D
λ
z
f (z)
μ
dθ ≤
2π
0
D
λ
z
f
k
(z)
μ
dθ (0 ≤ λ<1, μ>0). (2.19)
Letting p
= 1inTheorem 2.1, we have the following.
Corollar y 2.3. Let f
∈ ᏼ(α,β,σ) and suppose that
∞
n=2
n(n − 1)
a
n
≤
(1 − β)Γ(k +1)Γ(2 − λ)
(k − β)(1 − σ + σk)c
k
(α)Γ(k − λ)
(k
≥ 2). (2.20)
If there exists an analytic function w given by
w(z)
k−1
=
(k − β)(1 − σ + σk)c
k
(α)
1 − β
Γ(k
− λ)
Γ(k +1)
∞
n=2
(n − 1)
2
Φ(n)a
n
z
n−1
(2.21)
with
Φ(n) =
Γ(n − 1)
Γ(n − λ)
(0
≤ λ<1, n = 2, 3, ), (2.22)
then, for μ>0 and z
= re
iθ
(0 <r<1),
2π
0
D
1+λ
z
f (z)
μ
dθ ≤
2π
0
D
1+λ
z
f
k
(z)
μ
dθ (0 ≤ λ<1, μ>0). (2.23)
References
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Springer, New York, NY, USA, 1983.
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River Edge, NJ, USA, 1992.
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Journal d’Analyse Math
´
ematique, vol. 41, pp. 181–192, 1982.
H.
¨
O. G
¨
uney and S. Owa 9
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no. 11-12, pp. 71–78, 1999.
[7] S. Owa, “On the distortion theorems. I,” Kyungpook Mathematical Journal,vol.18,no.1,pp.
53–59, 1978.
[8] H.M.SrivastavaandS.Owa,Eds.,Univalent Functions, Fractional Calculus, and Their Applica-
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H.
¨
Ozlem G
¨
uney: Department of Mathematics, Faculty of Science and Letters, University of Dicle,
21280 Diyarbakır, Turkey
Email address:
Shigeyoshi Owa: Department of Mathematics, Kinki University, Osaka 577-8502,
Higashi-Osaka, Japan
Email address: