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Volume 2008, Article ID 134932, 12 pagesdoi:10.1155/2008/134932 Research Article A New Subclass of Analytic Functions Defined by Generalized Ruscheweyh Differential Operator Serap Bulut

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 ˙Izmit-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 diskU 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

Copyrightq 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},

R0

−1: R−1\ {0}

1.1

LetA denote the class of all functions of the form

f z  z ∞

n2

which are analytic in the open unit diskU : {z ∈ C : |z| < 1}.

For f j∈ A given by

fj z  z ∞

n2

the Hadamard productor convolution f1∗f2of f1and f2is defined by

f1∗f2z  z ∞

n2

Using the convolution1.4, Shaqsi and Darus 1 introduced the generalization of the Ruscheweyh derivative as follows

For f ∈ A, λ ≥ 0, and u ∈ R−1, we consider

R u λ f z  z

where R λf z  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

R u λ f z  z ∞

n2

where

C u, n  1  u n−1

and where a n is the Pochhammer symbol or shifted factorial defined in terms of the Gamma function by

a n: Γa  n Γa 



a a  1 · · · a  n − 1, if n ∈ N, a ∈ C. 1.8

In the case m∈ N0, we have

R m λ f z  z z m−1f z

m

and for λ  0, we obtain uth Ruscheweyh derivative introduced in 2 , R m

0  R m

Using the generalized Ruscheweyh derivative operator R u λ, we define the following classes

Definition 1.1 LetSλ u, v; α be the class of functions f ∈ A satisfying

Re

R u

λ f z

R v

λ f z



for some 0≤ α < 1, u ∈ R0

−1, v∈ R−1, λ ≥ 0, and all z ∈ U.

In this paper, basic properties of the classSλ 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

∞



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 Let2.1 be true for 0 ≤ α < 1, u ∈ R0

−1, v ∈ R−1, and λ ≥ 0 For f ∈ A, define the function F by

F z : R u λ f z

It is sufficient to show that



for z∈ U

So, we have



F F z − 1 z  1 R u λ f z − 1  αR v

λ f z

R u

λ f z  1 − αR v

λ f z







 α−∞n21  n − 1λ Cu, n − 1  αCv, n a nz n−1

2 − α ∞n21  n − 1λ Cu, n  1 − αCv, n a n z n−1





∞

n21  n − 1λ |Cu, n − 1  αCv, n||a n ||z| n−1

2 − α −∞n21  n − 1λ Cu, n  1 − αCv, n |a n ||z| n−1

< α∞n21  n − 1λ |Cu, n − 1  αCv, n||a n|

2 − α −∞n21  n − 1λ Cu, n  1 − αCv, n |a n|

< 1 by 2.1.

2.5

Therefore, f ∈ Sλ u, v; α.

Theorem 2.2 If f ∈ S λ u, v; α, then

1  n − 1λ |Cu, n − Cv, n|

n−1



u1

1  n − u − 1λ Cv, n − u|a n −u| 2.6

for n ≥ 2, with a1 1.

Proof Define the function

G z : 1

1− α

R u

λ f z

R v λ f z − α

: 1 ∞

n1

Since Re{Gz} > 0, we get

for n  1, 2,

From the definition of Gz, we obtain

R u

λ f z − αR v

λ f z

1∞

n1

nz n

So, by1.6, we have

z 1 λ

1− α Cu, 2 − αCv, 2 a2z2 1 2λ

1− α Cu, 3 − αCv, 3 a3z3 · · ·

1z2

2z3

3z4 · · ·

 1  λCv, 2a2z2 1  λCv, 2a2 1z3 1  λCv, 2a2 2z4 · · ·

 1  2λCv, 3a3z3 1  2λCv, 3a3 1z4 · · ·

2.10

or

z 1 λ

1− α Cu, 2 − Cv, 2 a2z21 2λ

1− α Cu, 3 − Cv, 3 a3z3 · · ·

1z2 1  λCv, 2a2 1 2 z3

 1  2λCv, 3a3 1 1  λCv, 2a2 2 3 z4 · · ·

2.11

or, equivalently,

z∞

n2

1 n − 1λ

1− α Cu, j − Cv, j a n z n

 z ∞

n2

n−1



u1

1  n − u − 1λ Cv, n − ua n −u u



z n

2.12

When we consider the coefficients of znof both series in the above equality, we have

a n 1− α

1  n − 1λ Cu, n − Cv, n

n−1



u1

1  n − u − 1λ Cv, n − ua n −u u 2.13

Therefore,

|a n| ≤ 1  n − 1λ |Cu, n − Cv, n|1− α n−1

u1

1  n − u − 1λ Cv, n − u|a n −u u|

≤ 1  n − 1λ |Cu, n − Cv, n|21 − α n−1

u1

1  n − u − 1λ Cv, n − u|a n −u|,

2.14

since u | ≤ 2, u  1, 2, .

3 Extreme points

Definition 3.1 Let Sλ u, v; α be the subclass of S λ u, v; α which consists of function

f z  z ∞

n2

whose coefficients satisfy inequality 2.1

Theorem 3.2 Let f1z  z and

f k z  z B21 − α

whereBk u, v, α is given by 2.2.

Then f ∈ Sλ u, v; α if and only if it can be expressed in the form

f z ∞

k1

where δk ≥ 0 and∞k1δk  1.

Proof Assume that

f z ∞

k1

Then

f z  δ1f1z ∞

k2

δkfk z

 δ1z∞

k2

δ k



z B21 − α

k u, v, α z k

k1

δ k



z∞

k2

δ k 21 − α

Bk u, v, α z k

 z ∞

k2

δk 21 − α

Bk u, v, α z k .

3.5

Thus

∞



k2

δ k 21 − α

Bk u, v, αBk u, v, α  21 − α∞

k2

δ k  21 − α1 − δ1 ≤ 21 − α. 3.6

Therefore, we have f∈ Sλ u, v; α.

Conversely, suppose that f∈ Sλ u, v; α Since

ak≤ B21 − α

we can set

δk: Bk u, v, α

21 − α ak k  2, 3, ,

δ1 : 1 −∞

k2

δ k

3.8

f z  z ∞

k2

akz k

k1

δ k



z∞

k2

δ k 21 − α

Bk u, v, α z k

 δ1z∞

k2

δ k



z B21 − α

k u, v, α z k

 δ1f1z ∞

k2

δkfk z

∞

k1

δkfk z.

3.9

This completes the proof ofTheorem 3.2

Corollary 3.3 The extreme points of Sλ u, v; α are given by

f1z  z, fk z  z B21 − α

k u, v, α z k k  2, 3, , 3.10

whereBk 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 Owa3 also by Srivastava and Owa4  will be required in our investigation

Definition 4.1 The fractional derivative of order η is defined, for a function f, by

D η z f z  Γ1 − η1 d

dz

z 0

f ξ

where the function f is analytic in a simply connected region of the complex z-plane

containing the origin, and the multiplicity ofz − ξ −η is removed by requiring logz − ξ

to be real when z − ξ > 0.

Definition 4.2 Under the hypothesis ofDefinition 4.1, the fractional derivative of order p η

is defined, for a function f, by

D p z η f z  d p

where 0≤ η < 1 and p ∈ N0

It readily follows from4.1 inDefinition 4.1that

D z η z k Γk  1 − η Γk  1 z k −η 0 ≤ η < 1, k ∈ N. 4.3

We will also need the concept of subordination between analytic functions and a subordination theorem of Littlewood5 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

such that

We denote this subordination by

Lemma 4.4 If the functions f and g are analytic in U with

then, for μ > 0 and z  re iθ 0 < r < 1,

2π 0

|fz| μ dθ≤

2π 0

Our main theorem is contained in the following

Theorem 4.5 Let f ∈ Sλ u, v; α and suppose that

∞



n2

n − p p1an≤ B 21 − αΓk  1Γ3 − η − p

for 0 ≤ p ≤ n, k ≥ p, 0 ≤ η < 1, where n − p p1denotes the Pochhammer symbol defined by

Also let the function fk be defined by

f k z  z B21 − α

If there exists an analytic function w defined by

wz k−1: Bk u, v, αΓk  1 − η − p21 − αΓk  1 ∞

n2

n − p p1Ψna nz n−1 4.12

with

Ψn  Γn  1 − η − p Γn − p , 0 ≤ η < 1, n  2, 3, , 4.13

then, for μ > 0 and z  re iθ 0 < r < 1,

2π 0

D p η

z f zμ

dθ≤

2π 0

D p η

z fk zμ

Proof By means of4.3 andDefinition 4.2, we find from3.1 that

D p z η f z  Γ2 − η − p z1−η−p

1∞

n2

Γ2 − η − pΓn  1

Γn  1 − η − p anz n−1

 Γ2 − η − p z1−η−p

1∞

n2

Γ2 − η − pn − p p1Ψna nz n−1

,

4.15

where

Ψn  Γn  1 − η − p Γn − p , 0 ≤ η < 1, n  2, 3, . 4.16

SinceΨ is a decreasing function of n, we get

0 < Ψn ≤ Ψ2  Γ3 − η − p Γ2 − p 4.17

Similarly, from4.11, 4.3, andDefinition 4.2, we have

D p z η f k z  Γ2 − η − p z1−η−p



1B21 − α

k u, v, α

Γ2 − η − pΓk  1

Γk  1 − η − p z k−1



For μ > 0 and z  re iθ 0 < r < 1, we want to show that

2π 0





1

∞



n2

Γ2 − η − pn − p p1Ψna nz n−1





μ dθ

≤

2π 0



1 B21 − αk u, v, α Γ2 − η − pΓk  1 Γk  1 − η − p z k−1

μ dθ.

4.19

So, by applyingLemma 4.4, it is enough to show that

1∞

n2

Γ2 − η − pn − p p1Ψna n z n−1≺ 1 B21 − α

k u, v, α

Γ2 − η − pΓk  1

Γk  1 − η − p z k−1.

4.20

If the above subordination holds true, then we have an analytic function w with w0  0 and

|wz| < 1 such that

1∞

n2

Γ2 − η − pn − p p1Ψna nz n−1 1 B21 − α

k u, v, α

Γ2 − η − pΓk  1

Γk  1 − η − p wz k−1.

4.21

By the condition of the theorem, we define the function w by

wz k−1 Bk u, v, αΓk  1 − η − p

21 − αΓk  1

∞



n2

n − p p1Ψna n z n−1, 4.22

which readily yields w0  0 For such a function w, we have

|wz| k−1≤ Bk u, v, αΓk  1 − η − p

21 − αΓk  1

∞



n2

n − p p1Ψna n |z| n−1

≤ |z|Bk u, v, αΓk  1 − η − p21 − αΓk  1 Ψ2∞

n2

n − p p1an

 |z|Bk u, v, αΓk  1 − η − p

21 − αΓk  1

Γ2 − p

Γ3 − η − p

∞



n2

n − p p1a n

≤ |z| < 1

4.23

by means of the hypothesis of the theorem

Thus the theorem is proved

As a special case p 0, we have the following result fromTheorem 4.5

Corollary 4.6 Let f ∈ Sλ u, v; α and suppose that

∞



n2

nan≤ 21 − αΓk  1Γ3 − ηB

k u, v, αΓk  1 − η k  2, 3, . 4.24

If there exists an analytic function w defined by

wz k−1 Bk u, v, αΓk  1 − η

21 − αΓk  1

∞



n2

with

Ψn  Γn  1 − η Γn , 0 ≤ η < 1, n  2, 3, , 4.26

then, for μ > 0 and z  re iθ 0 < r < 1,

2π 0

D η

z f zμ

dθ≤

2π 0

D η

z fk zμ

Letting p 1 inTheorem 4.5, we have the following

Corollary 4.7 Let f ∈ Sλ u, v; α and suppose that

∞



n2

n n − 1a n≤ 21 − αΓk  1Γ2 − ηB

k u, v, αΓk − η k  2, 3, . 4.28

If there exists an analytic function w defined by

wz k−1 B21 − αΓk  1k u, v, αΓk − η∞

n2

with

Ψn  Γn − η Γn − 1 , 0 ≤ η < 1, n  2, 3, , 4.30

then, for μ > 0 and z  re iθ 0 < r < 1,

2π 0

D1η

z f zμ

dθ≤

2π 0

D1η

z f k zμ

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.

whereBk u, v, α is given by 2.2.

Then f ∈ Sλ...