Certain Families of Multivalent Analytic Functions Associated with Iterations of the Owa-Srivastava Fractional Differintegral Operator

A. K. Mishra and S. N. Kund 1 National Institute of Science and Technology, Palur Hills, Berhampur 761008, India 2Department of Mathematics, Khallikote Autonomous College, Ganjam District, Berhampur, Odisha 760001, India Correspondence should be addressed to A. K. Mishra; akshayam2001@yahoo.co.in Received 19 May 2014; Accepted 18 September 2014; Published 14 October 2014 Academic Editor: Haakan Hedenmalm Copyright © 2014 A. K. Mishra and S. N. Kund.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.


Introduction
Let A denote the class of analytic functions in the open unit disk and let A  be the subclass of A consisting of functions represented by the following Taylor-Maclaurin's series: In a recent paper Patel and Mishra [1] studied several interesting mapping properties of the fractional differintegral operator: where  ∈ A  is given by (2).In the particular case  = 1 and −∞ <  < 1 the fractional-differintegral operator Ω (,1)  := Ω   (5) was earlier introduced by Owa and Srivastava [2] (see also [3]) and this is popularly known as the Owa-Srivastava operator [4][5][6].Moreover, for 0 ≤  < 1 and  ∈ N, Ω (,)  was investigated by Srivastava and Aouf [7] which was further extended to the range −∞ <  < 1,  ∈ N by Srivastava and Mishra [8].The following are some of the interesting particular cases of Ω (,)  : and, for  ∈ N, Similarly, for  ∈ A  , represented by (2), let the operator be defined by the following: and, for  ∈ N \ {1}, Very recently Srivastava et al. [6] considered the composition of the operators C   and Ω (,) , and introduced the following operator: That is, for  ∈ A  , given by (2), we know that The transformation D (,)  (, ) includes, among many, the following two previously studied interesting operators as particular cases.
We next recall the definition of subordination.Suppose that  ∈ A and  in A is univalent in U. We say that () is subordinate to () in U if (0) = (0) and (U) ⊆ (U).Considering the function () =  −1 (()), it is readily checked that () satisfies the conditions of the Schwarz lemma and In a broader sense the function  ∈ A is said to be subordinate to the function  ∈ A ( need not be univalent in U), written as if condition (15) holds for some Schwarz function () (see [12] for details).We also need the following definition of Hadamard product (or convolution).For the functions  and  in A  , given by the following Taylor-Maclaurin's series their Hadamard product (or convolution)  *  is defined by It is easy to see that  *  ∈ A  .
The study of iterations of entire and meromorphic functions, as the number of iterations tends to infinity, is a popular topic in complex analysis.However, investigations have been initiated only recently regarding iterations of certain transforms defined on classes of analytic and meromorphic functions.For example, Al-Oboudi and Al-Amoudi [9,10] investigated properties of certain classes of analytic functions associated with conical domains, by making use of the operator D ,  .Their work generalized several earlier results of Srivastava and Mishra [13].This theme has been further pursued in our more recent papers [6,14,15].In the sequel to these current investigations, in the present paper, we define the following subclass of A  associated with the iterated operator D (,)  (, ) and investigate its several interesting properties.Our work is also motivated by earlier works in [16][17][18][19][20], connecting subordination and Hadamard product.Definition 1.The function  ∈ A  is said to be in the class H ,  (, ; ℎ) if the following subordination condition is satisfied: where  is a complex number and ℎ is an analytic convex univalent function in U.
In the present paper we primarily focus on a variety of convolution theorems for the class H ,  (, ; ℎ).We also find inclusion theorems and study behavior of the Libera-Livingston integral operator.

Some More Definitions and Preliminary Lemmas
We need the following definitions and results for the presentation of our results.Let CV() and S * () (0 ≤  < 1) denote, respectively, the classes of univalent convex functions of order  and starlike functions of order  (see [12] for details).The function  ∈ A 1 is said to be in the class PS * () consisting of prestarlike functions of order  [25] if It is readily seen that Furthermore, it is well known [25] that We will also need the following lemmas in order to derive our main results.
Then, for any analytic function where (F(U)) denotes the closed convex hull of F(U).
Lemma 4 (see [28]).Let  and  be univalent convex functions in U, and let ℎ and  be functions in A. Suppose that ℎ ≺  and  ≺  in U. Then ℎ *  ≺  *  in U.
The following well known result is a consequence of the principle of subordination and can be found, for example, in [12,29].

Convolution Results
We state and prove the following convolution results.

Journal of Complex Analysis
Proof.For every  and  in A  , we have where Now, if  ∈ H ,  (, ; ℎ), then Furthermore, condition ( 27) is equivalent to Therefore, an application of Lemma 4 in (29) yields This shows that  *  ∈ H ,  (, ; ℎ).The proof of Theorem 6 is completed.
Then the function is in the class H ,  (, ; ℎ).
Proof.Let  ∈ H ,  (, ; ℎ).We note that where Also, for  ∈ N \ {1}, it is well known [18] that In view of ( 36) and ( 38), an application of Theorem 6 gives The proof of Corollary 8 is completed.
Taking  = 1/2 in Theorem 9 we get the following.
In the following theorem we discuss convolution properties of the function class H(, ; ℎ) when ℎ is a right half plane mapping.
Theorem 11.Let  ≥ 0 and suppose that each of the functions   ( = 1, 2) is a member of the class H ,  (, ; ℎ  ), where If  ∈ A  is defined by the following then  ∈ H ,  (, ; ℎ), where and  is given by The bound on  is the best possible.
In order to show that the value of  is the least possible, we take the functions   () ∈ A  ( = 1, 2) defined by for which we have Hence, for  ∈ A  , given by (45), we obtain →  (as  → −1) .
Finally, for the case  = 0, the proof of Theorem 11 is simple, so we choose to omit the details involved.

Properties of the Libera-Livingston Transform
For the function  ∈ A  , the function F defined by is popularly known as the Libera-Livingston transform of .
We state and prove the following.