﻿Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator

Vol.3 No.1(2013), Article ID:26506,5 pages DOI:10.4236/apm.2013.31001

Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator

Jae Ho Choi

Department of Mathematics Education, Daegu National University of Education, Daegu, South Korea

Email: choijh@dnue.ac.kr

Received August 29, 2012; revised October 29, 2012; accepted November 10, 2012

Keywords: Multivalent Functions; Subordination; Gaussian Hypergeometric Function; Fractional Integral Operator

ABSTRACT

By using a method based upon the Briot-Bouquet differential subordination, we investigate some subordination properties of the generalized fractional integral operator which was defined by Owa, Saigo and Srivastava [1]. Some interesting further consequences are also considered.

1. Introduction

Let denote the class of functions of the form

(1.1)

which are analytic in the open unit disk Also let f and g be analytic in with. Then we say that f is subordinate to g in, written or, if there exists the Schwarz function w, analytic in such that, and. We also observe that

if and only if

whenever is univalent in.

Let a, b and c be complex numbers with. Then the Gaussian/classical hypergeometric function is defined by

(1.2)

where is the Pochhammer symbol defined, in terms of the Gamma function, by

(1.3)

The hypergeometric function is analytic in and if a or b is a negative integer, then it reduces to a polynomial.

For each A and B such that, let us define the function

(1.4)

It is well known that, for, is the conformal map of the unit disk onto the disk symmetrical respect to the real axis having the center and the radius. The boundary circle cuts the real axis at the points and.

Many essentially equivalent definitions of fractional calculus have been given in the literature (cf., e.g. [2,3]). We state here the following definition due to Saigo [4] (see also [1,5]).

Definition 1. For, , the fractional integral operator is defined by

(1.5)

where is the Gaussian hypergeometric function defined by (1.2) and is taken to be an analytic function in a simply-connected region of the z-plane containing the origin with the order

for, and the multiplicity of is removed by requiring that to be real when.

The definition (1.5) is an interesting extension of both the Riemann-Liouville and Erdélyi-Kober fractional operators in terms of Gauss’s hypergeometric functions.

With the aid of the above definition, Owa, Saigo and Srivastava [1] defined a modification of the fractional integral operator by

(1.6)

for and. Then it is observed that also maps onto itself as follows:

(1.7)

We note that, where the operator was introduced and studied by Jung, Kim and Srivastava [6] (see also [7]).

It is easily verified from (1.7) that

(1.8)

The identity (1.8) plays an important and significant role in obtaining our results.

Recently, by using the general theory of differential subordination, several authors (see, e.g. [7-9]) considered some interesting properties of multivalent functions associated with various integral operators. In this manuscript, we shall derive some subordination properties of the fractional integral operator by using the technique of differential subordination.

2. Main Results

In order to establish our results, we shall need the following lemma due to Miller and Mocanu [10].

Lemma 1. Let be analytic and convex univalent in with, and let be analytic in. If

(2.1)

then for and,

(2.2)

We begin by proving the following theorem.

Theorem 1. Let, , , , and, and let

. Suppose that

(2.3)

where

(2.4)

and is given by (1.3).

1) If, then

(2.5)

2) If and, then

(2.6)

The result is sharp.

Proof. 1) If we set

then, from (1.7) we see that

(2.7)

For and, it follows from (2.3) that

(2.8)

which implies that

2) Let

(2.9)

Then the function is analytic in. Using (1.8) and (2.9), we have

(2.10)

From (2.5), (2.9) and (2.10) we obtain

Thus, by applying Lemma 1, we observe that

or

(2.11)

where is analytic in with and. In view of and , we conclude from (2.11) that

(2.12)

Since for and, from (2.12) we see that the inequality (2.6) holds.

To prove sharpness, we take defined by

For this function we find that

and

Hence the proof of Theorem 1 is evidently completed.

Theorem 2. Let, , , , and. Suppose that

, and . If the sequence is nondecreasing with

(2.13)

where is given by and satisfies the condition, then

(2.14)

and

(2.15)

Each of the bounds in (2.14) and (2.15) is best possible for.

Proof. We prove the bound in (2.14). The bound in (2.15) is immediately obtained from (2.14) and will be omitted. Let

Then, from (1.7) we observe that

where, for convenience,

It is easily seen from (2.4) and (2.13) that and

(2.16)

Hence, by applying (2.3) and (2.16), we have

which readily yields the inequality (2.14).

If we take, then

This show that the bound in (2.14) is best possible for each m, which proves Theorem 2.

Finally, we consider the generalized Bernardi-LiveraLivingston integral operator defined by (cf. [11-13])

(2.17)

Theorem 3. Let, , , , , andand let. Suppose that

(2.18)

where

and is given by (1.3).

1) If, then

(2.19)

2) If and, then

(2.20)

The result is sharp.

Proof. 1) If we put

then, from (1.7) and (2.17) we have

Therefore, by using same techniques as in the proof of Theorem 1 1), we obtain the desired result.

2) From (2.17) we have

(2.21)

Let

(2.22)

Then, by virtue of (2.21), (2.22) and (2.19), we observe that

Hence, by applying the same argument as in the proof of Theorem 1 2), we obtain (2.20), which evidently proves Theorem 3.

3. Acknowledgements

This work was supported by Daegu National University of Education Research grant in 2011.

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