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. Özlem Güney 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. Ö. Güney and S. Owa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, 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 +
∞
an z n ,
(1.1)
n =2
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 −
∞
an z n
an ≥ 0 .
(1.2)
n =2
The Hadamard product (or convolution) of two functions f given by (1.1) and g given
by
g(z) = z +
∞
bn zn
n =2
(1.3)
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Journal of Inequalities and Applications
is defined by
∞
( f ∗g)(z) = z +
an bn zn .
(1.4)
n =2
We denote the subclass ᏾(α,β) of ᏿ consisting of α-prestarlike functions of order β by
᏾(α,β) = f ∈ ᏿ : f ∗sα (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 sα is the
well-known extremal function for ᏿∗ (α) given by
sα (z) = z(1 − z)−2(1−α)
(1.6)
(cf. [1, 2]). Letting
n
cn (α) =
k=2 (k − 2α)
(n − 1)!
(n = 2,3,...),
(1.7)
sα can be written in the form
sα (z) = z +
∞
cn (α)zn .
(1.8)
n =2
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 ∈ ᏿ : z f (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] and Ꮿ[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].
The unified class ᏼ(α,β,σ) was studied by Raina and Srivastava [6].
(1.12)
H. Ö. Güney 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. A function f defined by (1.2) belongs to the class ᏼ(α,β,σ) if and only if
∞ (n − β)(1 − σ + σn)
1−β
n =2
cn (α)an ≤ 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
f1 (z) = z,
fk (z) = z −
1−β
zk
(k − β)(1 − σ + σk)ck (α)
(k = 2,3,...).
(1.14)
Then f ∈ ᏼ(α,β,σ) if and only if it can be expressed in the form
f (z) =
∞
λk fk (z),
(1.15)
λk fk (z).
(1.16)
k =1
where λk ≥ 0 and
∞
k=1 λk
= 1.
Proof. Assume that
f (z) =
∞
k =1
Then
f (z) = λ1 f1 (z) +
∞
λk fk (z)
k =2
= λ1 z +
=
∞
λk z −
k =2
∞
λk
k =1
=z−
∞
(1.17)
∞
1−β
z − λk
zk
(k
−
β)(1
−
σ
+
σk)c
(α)
k
k =2
λk
k =2
1−β
zk
(k − β)(1 − σ + σk)ck (α)
1−β
zk .
(k − β)(1 − σ + σk)ck (α)
Thus
∞
k =2
λk
1−β
(k − β)(1 − σ + σk)ck (α)
=
∞
λk =
k =2
∞
(k − β)(1 − σ + σk)ck (α)
1−β
(1.18)
λk − λ1 = 1 − λ1 ≤ 1.
k =1
Therefore, we have f ∈ ᏼ(α,β,σ).
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Journal of Inequalities and Applications
Conversely, suppose that f ∈ ᏼ(α,β,σ). Since
|ak | ≤
1−β
(k − β)(1 − σ + σk)ck (α)
(k = 2,3,...),
(1.19)
we can set
λk =
(k − β)(1 − σ + σk)ck (α)
1−β
(k = 2,3,...),
λ1 = 1 −
∞
λk .
(1.20)
k=1
Then
f (z) = z −
∞
ak z k
k =2
=z−
∞
λk
k =2
= 1−
∞
1−β
zk
(k − β)(1 − σ + σk)ck (α)
λk z +
k =2
= λ1 f1 (z) +
∞
λk fk (z)
(1.21)
k =2
∞
λk fk (z)
k =2
=
∞
λk fk (z).
k =1
This completes the assertion of Lemma 1.2.
Lemma 1.2 gives us the following.
Corollary 1.3. The extreme points of ᏼ(α,β,σ) are given by
f1 (z) = z,
fk (z) = z −
1−β
zk .
(k − β)(1 − σ + σk)ck (α)
(1.22)
We will make use of the following definitions of fractional derivatives by Owa [7] (also
by Srivastava and Owa [8]).
Definition 1.4. The fractional derivative of order λ is defined, for a function f , by
Dzλ f (z) =
d
Γ(1 − λ) dz
1
z
f (ξ)
λ dξ
0 (z − ξ)
(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. Ö. Güney and S. Owa 5
Definition 1.5. Under the hypothesis of Definition 1.4, the fractional derivative of order
(n + λ) is defined, for a function f , by
Dzn+λ f (z) =
dn λ
D f (z),
dzn z
(1.24)
where 0 ≤ λ < 1 and n = 0,1,2,....
It readily follows from (1.23) in Definition 1.4 that
Dzλ zk =
Γ(k + 1) k−λ
z
Γ(k − λ + 1)
(0 ≤ λ < 1).
(1.25)
We will also need the concept of subordination between analytic functions and a subordination 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 = reiθ (0 < r < 1),
2π
0
iθ μ
g re dθ ≤
2π
0
iθ μ
f re 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 theorem is contained in the following.
Theorem 2.1. Let f ∈ ᏼ(α,β,σ) and suppose that
∞
(n − p) p+1 an ≤
n =2
(1 − β)Γ(k + 1)Γ(3 − λ − p)
(k − β)(1 − σ + σk)ck (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 defined by
(n − p) p+1 = (n − p)(n − p + 1) · · · n.
(2.2)
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Journal of Inequalities and Applications
Also let the function fk be defined by
fk (z) = z −
1−β
zk .
(k − β)(1 − σ + σk)ck (α)
(2.3)
If there exists an analytic function w defined by
w(z)
k −1
∞
(k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p) (n − p) p+1 Φ(n)an zn−1
1−β
Γ(k + 1)
n =2
=
(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 = reiθ (0 < r < 1),
2π
0
p+λ
μ
Dz f (z) dθ ≤
2π
0
p+λ
μ
Dz fk (z) dθ
(0 ≤ λ < 1, μ > 0).
(2.6)
Proof. By virtue of the fractional derivative formula (1.25) and Definition 1.5, we find
from (1.1) that
p+λ
Dz f (z) =
∞
∞
Γ(2 − λ − p)Γ(n + 1)
z1− p−λ
1−
a n z n −1
Γ(2 − λ − p)
Γ(n
+
1
−
λ
−
p)
n =2
z1− p−λ
1 − Γ(2 − λ − p)(n − p) p+1 Φ(n)an zn−1 ,
=
Γ(2 − λ − p)
n =2
(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, we have
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, we obtain
p+λ
Dz
fk (z) =
1−β
Γ(2 − λ − p)Γ(k + 1) k−1
z1− p−λ
1−
.
z
Γ(2 − λ − p)
(k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p)
(2.10)
H. Ö. Güney and S. Owa 7
For μ > 0 and z = reiθ (0 < r < 1), we must show that
μ
2π ∞
n
−
1
1 −
Γ(2
− λ − p)(n − p) p+1 Φ(n)an z dθ
0
n =2
μ
2π 1−β
Γ(2 − λ − p)Γ(k + 1) k−1 1 −
≤
dθ.
z (k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p)
0 (2.11)
Thus, by applying Lemma 1.6, it would suffice to show that
1−
∞
Γ(2 − λ − p)(n − p) p+1 Φ(n)an zn−1
n =2
≺ 1−
1−β
Γ(2 − λ − p)Γ(k + 1) k−1
z .
(k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p)
(2.12)
If the subordination (2.12) holds true, then we have an analytic function w with w(0) = 0
and |w(z)| < 1 such that
1−
∞
Γ(2 − λ − p)(n − p) p+1 Φ(n)an zn−1
n =2
= 1−
k−1
1−β
Γ(2 − λ − p)Γ(k + 1) w(z)
.
(k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p)
(2.13)
By the condition of the theorem, we define the function w by
w(z)
k −1
=
∞
(k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p) (n − p) p+1 Φ(n)an zn−1 (2.14)
1−β
Γ(k + 1)
n =2
which readily yields w(0) = 0. For such a function w, we have
∞
w(z)k−1 ≤ (k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p) (n − p) Φ(n)an |z|n−1
p+1
1−β
Γ(k + 1)
n =2
≤ |z|
∞
(k − β)(1 − σ + σk)ck (α) Γ(k + 1 − λ − p)
Φ(2) (n − p) p+1 an
1−β
Γ(k + 1)
n=2
= |z|
∞
(k − β)(1 − σ +σk)ck (α) Γ(k+1 − λ − p) Γ(2 − p) (n − p) p+1 an
1−β
Γ(k+1)
Γ(3 − λ − p) n=2
= |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.
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Journal of Inequalities and Applications
Corollary 2.2. Let f ∈ ᏼ(α,β,σ) and suppose that
∞
n an ≤
n =2
(1 − β)Γ(k + 1)Γ(3 − λ)
(k − β)(1 − σ + σk)ck (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)ck (α) Γ(k + 1 − λ) nΦ(n)an zn−1
1−β
Γ(k + 1) n=2
(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 = reiθ (0 < r < 1),
2π
0
λ
D f (z)μ dθ ≤
z
2π
0
λ
D fk (z)μ dθ
z
(0 ≤ λ < 1, μ > 0).
(2.19)
Letting p = 1 in Theorem 2.1, we have the following.
Corollary 2.3. Let f ∈ ᏼ(α,β,σ) and suppose that
∞
n(n − 1)an ≤
(1 − β)Γ(k + 1)Γ(2 − λ)
(k − β)(1 − σ + σk)ck (α)Γ(k − λ)
(k ≥ 2).
(2.20)
∞
(k − β)(1 − σ + σk)ck (α) Γ(k − λ) (n − 1)2 Φ(n)an zn−1
1−β
Γ(k + 1) n=2
(2.21)
n =2
If there exists an analytic function w given by
w(z)
k −1
=
with
Φ(n) =
Γ(n − 1)
Γ(n − λ)
(0 ≤ λ < 1, n = 2,3,...),
(2.22)
then, for μ > 0 and z = reiθ (0 < r < 1),
2π
0
1+λ
D f (z)μ dθ ≤
z
2π
0
1+λ
D fk (z)μ dθ
z
(0 ≤ λ < 1, μ > 0).
(2.23)
References
[1] P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften,
Springer, New York, NY, USA, 1983.
[2] H. M. Srivastava and S. Owa, Eds., Current Topics in Analytic Function Theory, World Scientific,
River Edge, NJ, USA, 1992.
[3] T. Sheil-Small, H. Silverman, and E. Silvia, “Convolution multipliers and starlike functions,”
Journal d’Analyse Mathématique, vol. 41, pp. 181–192, 1982.
H. Ö. Güney and S. Owa 9
[4] S. Owa and B. A. Uralegaddi, “A class of functions α-prestarlike of order β,” Bulletin of the Korean
Mathematical Society, vol. 21, no. 2, pp. 77–85, 1984.
[5] H. M. Srivastava and M. K. Aouf, “Some applications of fractional calculus operators to certain
subclasses of prestarlike functions with negative coefficients,” Computers & Mathematics with
Applications, vol. 30, no. 1, pp. 53–61, 1995.
[6] R. K. Raina and H. M. Srivastava, “A unified presentation of certain subclasses of prestarlike functions with negative coefficients,” Computers & Mathematics with Applications, vol. 38,
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. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK;
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H. Özlem Güney: Department of Mathematics, Faculty of Science and Letters, University of Dicle,
21280 Diyarbakır, Turkey
Email address: ozlemg@dicle.edu.tr
Shigeyoshi Owa: Department of Mathematics, Kinki University, Osaka 577-8502,
Higashi-Osaka, Japan
Email address: owa@math.kindai.ac.jp