Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 134932, 12 pages
doi:10.1155/2008/134932
Research Article
A New Subclass of Analytic Functions Defined by
Generalized Ruscheweyh Differential Operator
Serap Bulut
Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 İzmit-Kocaeli, Turkey
Correspondence should be addressed to Serap Bulut, serap.bulut@kocaeli.edu.tr
Received 1 July 2008; Accepted 3 September 2008
Recommended by Narendra Kumar Govil
We investigate a new subclass of analytic functions in the open unit disk U which is defined by
generalized Ruscheweyh differential operator. Coefficient inequalities, extreme points, and the
integral means inequalities for the fractional derivatives of order p η 0 ≤ p ≤ n, 0 ≤ η < 1
of functions belonging to this subclass are obtained.
Copyright q 2008 Serap Bulut. 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
Throughout this paper, we use the following notations:
N : {1, 2, 3, . . .},
N0 : N ∪ {0},
R−1 : {u ∈ R : u > −1},
1.1
R0−1 : R−1 \ {0}.
Let A denote the class of all functions of the form
fz z ∞
an zn ,
1.2
n2
which are analytic in the open unit disk U : {z ∈ C : |z| < 1}.
For fj ∈ A given by
fj z z ∞
an,j zn
n2
j 1, 2,
1.3
2
Journal of Inequalities and Applications
the Hadamard product or convolution f1 ∗f2 of f1 and f2 is defined by
f1 ∗f2 z z ∞
an,1 an,2 zn .
1.4
n2
Using the convolution 1.4, Shaqsi and Darus 1 introduced the generalization of the
Ruscheweyh derivative as follows.
For f ∈ A, λ ≥ 0, and u ∈ R−1 , we consider
Ruλ fz z
1 − zu1
∗Rλ fz
z ∈ U,
1.5
where Rλ fz 1 − λfz λzf z, z ∈ U.
If f ∈ A is of the form 1.2, then we obtain the power series expansion of the form
Ruλ fz z ∞
1 n − 1λCu, nan zn ,
1.6
n2
where
Cu, n 1 un−1
n − 1!
n ∈ N,
1.7
and where an is the Pochhammer symbol or shifted factorial defined in terms of the
Gamma function by
Γa n
an :
Γa
1,
aa 1 · · · a n − 1,
if n 0, a ∈ C \ {0},
if n ∈ N, a ∈ C.
1.8
In the case m ∈ N0 , we have
Rm
λ fz zzm−1 fz
m!
m
,
1.9
m
and for λ 0, we obtain uth Ruscheweyh derivative introduced in 2, Rm
0 R .
u
Using the generalized Ruscheweyh derivative operator Rλ , we define the following
classes.
Definition 1.1. Let Sλ u, v; α be the class of functions f ∈ A satisfying
u
Rλ fz
Re
>α
Rvλ fz
for some 0 ≤ α < 1, u ∈ R0−1 , v ∈ R−1 , λ ≥ 0, and all z ∈ U.
1.10
Serap Bulut
3
In this paper, basic properties of the class Sλ u, v; α are studied, such as coefficient
bounds, extreme points, and integral means inequalities for the fractional derivative.
2. Coefficient inequalities
Theorem 2.1. Let 0 ≤ α < 1, u ∈ R0−1 , v ∈ R−1 , and λ ≥ 0. If f ∈ A satisfies
∞
Bn u, v, α|an | ≤ 21 − α,
2.1
n2
where
Bn u, v, α : 1 n − 1λ{|Cu, n − 1 αCv, n| Cu, n 1 − αCv, n},
2.2
then f ∈ Sλ u, v; α.
Proof. Let 2.1 be true for 0 ≤ α < 1, u ∈ R0−1 , v ∈ R−1 , and λ ≥ 0. For f ∈ A, define the
function F by
Fz :
Ruλ fz
− α.
Rvλ fz
2.3
It is sufficient to show that
Fz − 1 Fz 1 < 1
2.4
for z ∈ U.
So, we have
Fz − 1 Ruλ fz − 1 αRvλ fz Fz 1 Ru fz 1 − αRv fz λ
λ
≤
n−1
α− ∞
n2 1 n − 1λCu, n − 1 αCv, nan z
∞
n−1
2 − α n2 1 n − 1λCu, n 1 − αCv, nan z α
2 − α
∞
n−1
n2 1 n − 1λ|Cu, n − 1 αCv, n||an ||z|
∞
− n2 1 n − 1λCu, n 1 − αCv, n|an ||z|n−1
α ∞
n2 1 n − 1λ|Cu, n − 1 αCv, n||an |
<
2 − α − ∞
n2 1 n − 1λCu, n 1 − αCv, n|an |
< 1 by 2.1.
Therefore, f ∈ Sλ u, v; α.
2.5
4
Journal of Inequalities and Applications
Theorem 2.2. If f ∈ Sλ u, v; α, then
|an | ≤
n−1
21 − α
1 n − u − 1λCv, n − u|an−u |
1 n − 1λ|Cu, n − Cv, n| u1
2.6
for n ≥ 2, with a1 1.
Proof. Define the function
Gz :
u
∞
Rλ fz
1
−
α
:
1
a
n zn .
1 − α Rvλ fz
n1
2.7
Since Re{Gz} > 0, we get
|
an | ≤ 2
2.8
for n 1, 2, . . ..
From the definition of Gz, we obtain
∞
Ruλ fz − αRvλ fz
v
n
R fz 1 a
n z .
1−α
n1
2.9
So, by 1.6, we have
z
1λ
1 2λ
Cu, 2 − αCv, 2a2 z2 Cu, 3 − αCv, 3a3 z3 · · ·
1−α
1−α
z a
1 z2 a
2 z3 a
3 z4 · · ·
2.10
1 λCv, 2a2 z2 1 λCv, 2a2 a
1 z3 1 λCv, 2a2 a
2 z4 · · ·
1 2λCv, 3a3 z3 1 2λCv, 3a3 a
1 z4 · · ·
or
z
1λ
1 2λ
Cu, 2 − Cv, 2a2 z2 Cu, 3 − Cv, 3a3 z3 · · ·
1−α
1−α
z a
1 z2 1 λCv, 2a2 a
1 a
2 z3
1 2λCv, 3a3 a
1 1 λCv, 2a2 a
2 a
3 z4 · · ·
2.11
Serap Bulut
5
or, equivalently,
z
∞
1 n − 1λ
n2
1−α
z
Cu, j − Cv, jan zn
∞
n−1
n2
2.12
1 n − u − 1λCv, n − uan−u a
u zn .
u1
When we consider the coefficients of zn of both series in the above equality, we have
an n−1
1−α
1 n − u − 1λCv, n − uan−u a
u .
1 n − 1λCu, n − Cv, n u1
2.13
Therefore,
|an | ≤
n−1
1−α
1 n − u − 1λCv, n − u|an−u ||
au |
1 n − 1λ|Cu, n − Cv, n| u1
n−1
21 − α
≤
1 n − u − 1λCv, n − u|an−u |,
1 n − 1λ|Cu, n − Cv, n| u1
2.14
since |
au | ≤ 2, u 1, 2, . . ..
3. Extreme points
Definition 3.1. Let Sλ u, v; α be the subclass of Sλ u, v; α which consists of function
fz z ∞
an zn
an ≥ 0
3.1
n2
whose coefficients satisfy inequality 2.1.
Theorem 3.2. Let f1 z z and
fk z z where Bk u, v, α is given by 2.2.
21 − α k
z
Bk u, v, α
k 2, 3, . . .,
3.2
6
Journal of Inequalities and Applications
Then f ∈ Sλ u, v; α if and only if it can be expressed in the form
∞
δk fk z,
fz 3.3
k1
where δk ≥ 0 and
∞
k1 δk
1.
Proof. Assume that
∞
δk fk z.
fz 3.4
k1
Then
∞
δk fk z
fz δ1 f1 z k2
δ1 z ∞
∞
21 − α k
z
δk z Bk u, v, α
k2
δk
k1
z
∞
δk
k2
∞
3.5
21 − α k
z δk
z
B
k u, v, α
k2
21 − α k
z .
Bk u, v, α
Thus
∞
∞
21 − α
Bk u, v, α 21 − α δk 21 − α1 − δ1 ≤ 21 − α.
δk
Bk u, v, α
k2
k2
3.6
Therefore, we have f ∈ Sλ u, v; α.
Conversely, suppose that f ∈ Sλ u, v; α. Since
ak ≤
21 − α
Bk u, v, α
k 2, 3, . . .,
3.7
we can set
δk :
Bk u, v, α
ak
21 − α
∞
δ1 : 1 − δk .
k2
k 2, 3, . . .,
3.8
Serap Bulut
7
Then
fz z ∞
ak zk
k2
∞
δk z k1
∞
k2
δk
21 − α k
z
Bk u, v, α
∞
21 − α k
z
δ1 z δk z Bk u, v, α
k2
δ1 f1 z 3.9
∞
δk fk z
k2
∞
δk fk z.
k1
This completes the proof of Theorem 3.2.
Corollary 3.3. The extreme points of Sλ u, v; α are given by
f1 z z,
fk z z 21 − α k
z
Bk u, v, α
k 2, 3, . . .,
3.10
where Bk u, v, α is given by 2.2.
4. The main integral means inequalities for the fractional derivative
We discuss the integral means inequalities for functions f ∈ Sλ u, v; α.
The following definitions of fractional derivatives by Owa 3 also by Srivastava and
Owa 4 will be required in our investigation.
Definition 4.1. The fractional derivative of order η is defined, for a function f, by
η
Dz fz d
1
Γ1 − η dz
z
fξ
η dξ
0 z − ξ
0 ≤ η < 1,
4.1
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−η is removed by requiring logz − ξ
to be real when z − ξ > 0.
Definition 4.2. Under the hypothesis of Definition 4.1, the fractional derivative of order p η
is defined, for a function f, by
pη
Dz fz where 0 ≤ η < 1 and p ∈ N0 .
dp η
Dz fz,
dzp
4.2
8
Journal of Inequalities and Applications
It readily follows from 4.1 in Definition 4.1 that
η
Dz zk Γk 1 k−η
z
Γk 1 − η
0 ≤ η < 1, k ∈ N.
4.3
We will also need the concept of subordination between analytic functions and a
subordination theorem of Littlewood 5 in our investigation.
Definition 4.3. 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
w0 0,
|wz| < 1
z ∈ U,
4.4
such that
fz gwz z ∈ U.
4.5
fz ≺ gz.
4.6
We denote this subordination by
Lemma 4.4. If the functions f and g are analytic in U with
fz ≺ gz,
4.7
then, for μ > 0 and z reiθ 0 < r < 1,
2π
μ
|fz| dθ ≤
2π
0
|gz|μ dθ.
4.8
0
Our main theorem is contained in the following.
Theorem 4.5. Let f ∈ Sλ u, v; α and suppose that
∞
n − pp1 an ≤
n2
21 − αΓk 1Γ3 − η − p
Bk u, v, αΓk 1 − η − pΓ2 − p
4.9
for 0 ≤ p ≤ n, k ≥ p, 0 ≤ η < 1, where n − pp1 denotes the Pochhammer symbol defined by
n − pp1 n − pn − p 1 · · · n.
4.10
Also let the function fk be defined by
fk z z 21 − α k
z
Bk u, v, α
k 2, 3, . . ..
4.11
Serap Bulut
9
If there exists an analytic function w defined by
wzk−1 :
∞
Bk u, v, αΓk 1 − η − p n − pp1 Ψnan zn−1
21 − αΓk 1
n2
4.12
with
Ψn Γn − p
,
Γn 1 − η − p
0 ≤ η < 1, n 2, 3, . . .,
4.13
then, for μ > 0 and z reiθ 0 < r < 1,
2π
pη
Dz fzμ dθ ≤
0
2π
pη
Dz fk zμ dθ,
0 ≤ η < 1.
4.14
0
Proof. By means of 4.3 and Definition 4.2, we find from 3.1 that
pη
Dz fz
∞
Γ2 − η − pΓn 1
z1−η−p
n−1
an z
1
Γ2 − η − p
Γn 1 − η − p
n2
1−η−p
z
Γ2 − η − p
∞
n−1
,
1 Γ2 − η − pn − pp1 Ψnan z
4.15
n2
where
Ψn Γn − p
,
Γn 1 − η − p
0 ≤ η < 1, n 2, 3, . . ..
4.16
Since Ψ is a decreasing function of n, we get
0 < Ψn ≤ Ψ2 Γ2 − p
.
Γ3 − η − p
4.17
Similarly, from 4.11, 4.3, and Definition 4.2, we have
pη
Dz fk z
z1−η−p
21 − α Γ2 − η − pΓk 1 k−1
z
.
1
Γ2 − η − p
Bk u, v, α Γk 1 − η − p
4.18
10
Journal of Inequalities and Applications
For μ > 0 and z reiθ 0 < r < 1, we want to show that
μ
2π ∞
n−1 1 Γ2 − η − pn − pp1 Ψnan z dθ
0 n2
2π μ
1 21 − α Γ2 − η − pΓk 1 zk−1 dθ.
≤
Bk u, v, α Γk 1 − η − p
0
4.19
So, by applying Lemma 4.4, it is enough to show that
1
∞
21 − α Γ2 − η − pΓk 1 k−1
z .
Γ2 − η − pn − pp1 Ψnan zn−1 ≺ 1 B
Γk 1 − η − p
k u, v, α
n2
4.20
If the above subordination holds true, then we have an analytic function w with w0 0 and
|wz| < 1 such that
1
∞
21 − α Γ2 − η − pΓk 1
wzk−1 .
Γ2 − η − pn − pp1 Ψnan zn−1 1 B
u,
v,
α
Γk
1
−
η
−
p
k
n2
4.21
By the condition of the theorem, we define the function w by
wzk−1 ∞
Bk u, v, αΓk 1 − η − p n − pp1 Ψnan zn−1 ,
21 − αΓk 1
n2
4.22
which readily yields w0 0. For such a function w, we have
|wz|k−1 ≤
∞
Bk u, v, αΓk 1 − η − p n − pp1 Ψnan |z|n−1
21 − αΓk 1
n2
≤ |z|
|z|
∞
Bk u, v, αΓk 1 − η − p
Ψ2 n − pp1 an
21 − αΓk 1
n2
∞
Bk u, v, αΓk 1 − η − p Γ2 − p
n − pp1 an
21 − αΓk 1
Γ3 − η − p n2
≤ |z| < 1
by means of the hypothesis of the theorem.
Thus the theorem is proved.
As a special case p 0, we have the following result from Theorem 4.5.
4.23
Serap Bulut
11
Corollary 4.6. Let f ∈ Sλ u, v; α and suppose that
∞
21 − αΓk 1Γ3 − η
Bk u, v, αΓk 1 − η
nan ≤
n2
k 2, 3, . . ..
4.24
If there exists an analytic function w defined by
wzk−1 ∞
Bk u, v, αΓk 1 − η nΨnan zn−1
21 − αΓk 1
n2
4.25
with
Ψn Γn
,
Γn 1 − η
0 ≤ η < 1, n 2, 3, . . .,
4.26
then, for μ > 0 and z reiθ 0 < r < 1,
2π
η
Dz fzμ dθ ≤
0
2π
η
Dz fk zμ dθ,
0 ≤ η < 1.
4.27
k 2, 3, . . ..
4.28
0
Letting p 1 in Theorem 4.5, we have the following.
Corollary 4.7. Let f ∈ Sλ u, v; α and suppose that
∞
21 − αΓk 1Γ2 − η
nn − 1an ≤
Bk u, v, αΓk − η
n2
If there exists an analytic function w defined by
wzk−1 ∞
Bk u, v, αΓk − η nn − 1Ψnan zn−1
21 − αΓk 1 n2
4.29
with
Ψn Γn − 1
,
Γn − η
0 ≤ η < 1, n 2, 3, . . .,
4.30
μ
1η
Dz fk z dθ,
4.31
then, for μ > 0 and z reiθ 0 < r < 1,
2π
0
μ
1η
Dz fz dθ ≤
2π
0
0 ≤ η < 1.
12
Journal of Inequalities and Applications
References
1 K. A. Shaqsi and M. Darus, “On univalent functions with respect to K-symmetric points given by a
generalised Ruscheweyh derivatives operator,” submitted.
2 S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society,
vol. 49, no. 1, pp. 109–115, 1975.
3 S. Owa, “On the distortion theorems. I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978.
4 H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis
Horwood Series in Mathematics and Its Applications, Ellis Horwood, Chichester, UK; John Wiley &
Sons, New York, NY, USA, 1989.
5 J. E. Littlewood, “On inequalities in the theory of functions,” Proceedings of the London Mathematical
Society, vol. 23, no. 1, pp. 481–519, 1925.