Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator

By using a method based upon the Briot-Bouquet differential subordination, we investigate some subordination properties of the generalized fractional integral operator , , 0,z       p    which was defined by Owa, Saigo and Srivastava [1]. Some interesting further consequences are also considered.


Introduction
Also let f and g be analytic in with .Then we say that f is subordinate to g in , written , if there exists the Schwarz function w, analytic in such that , . We also observe that Let a, b and c be complex numbers with .Then the Gaussian/classical hypergeometric function where k  is the Pochhammer symbol defined, in terms of the Gamma function, by 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

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 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]).
where 2 1 F is the Gaussian hypergeometric function defined by (1.2) and   f z   is taken to be an analytic function in a simply-connected region of the z-plane containing the origin with the order , and the multiplicity of 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 , , 0, z also maps onto itself as follows: 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 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][8][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 0, z     by using the technique of differential subordination.

Main Results
In order to establish our results, we shall need the following lemma due to Miller and Mocanu [10].
be analytic and convex univalent in with   0 1 h  , and let We begin by proving the following theorem.
, and let .
The result is sharp.
For and , it follows from (2.3) that (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 where is analytic in with   and .
To prove sharpness, we take For this function we find that Hence the proof of Theorem 1 is evidently completed.

 
2.4 and satisfies the condition where k is given by .
Then, from (1.7) we observe that 2.4) and (2.13) that and 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.

20)
The result is sharp. .
Hence, by applying the same argument as in the proof of Theorem 1 2), we obtain (2.20), which evidently proves Theorem 3.

Proof. 1 )
If we put using same techniques as in the proof of Theorem 1 1), we obtain the desired result.
15) We prove the bound in(2.14).The bound in (2.15) is immediately obtained from (2.14) and will be omitted.Let Proof.