Convolution Properties of p-Valent Functions Associated with a Generalization of the Srivastava-Attiya Operator

LetAp denote the class of functions analytic in the open unit disc U and given by the series f(z) = z p + ∑ ∞ n=p+1 anz . For f ∈ Ap, the transformationI p,δ : Ap → Ap defined byI λ p,δ f(z) = z + ∑ ∞ n=p+1 ((p + δ)/(n + δ))λanz , (δ + p ∈ C\Z 0 , λ ∈ C; z ∈ U), has been recently studied as fractional differintegral operator by Mishra and Gochhayat (2010) . In the present paper, we observed thatI p,δ can also be viewed as a generalization of the Srivastava-Attiya operator. Convolution preserving properties for a class of multivalent analytic functions involving an adaptation of the popular Srivastava-Attiya transform are investigated.


Introduction and Preliminaries
Let A be the class of functions analytic in the open unit disk U := { :  ∈ C, || < 1} . ( Suppose that  and  are in A. We say that  is subordinate to  (or  is superordinate to ), written as if there exists a function  ∈ A, satisfying the conditions of the Schwarz lemma (i.e., (0) = 0 and |()| < 1) such that  () =  ( ()) ( ∈ U) .
By making use of the following normalized function: Srivastava-Attiya [2] introduced the linear operator L , : A 1 → A 1 by the following series: where the function  ∈ A 1 is, respectively, by The operator L , is now popularly known in the literature as the Srivastava-Attiya operator.Various basic properties of L , are systematically investigated in [6][7][8][9][10][11].
For a function  ∈ A  and represented by the series (8), the transformation defined by has been recently studied as fractional differintegral operator by the authors [12].We observed that I  , can also be viewed as a generalization of the Srivastava-Attiya operator (take  = 1,  = ,  =  in ( 14)), suitable for the study of multivalent functions.(Also see [13] for a variant.)Furthermore, transformation I  , generalizes several previously studied familiar operators.For example taking  = 0 we get the identity transformation; the choices  = −1,  = 0 yield the Alexander transformation and  = 1,  a negative integer,  = 0 give the S ȃ l ȃ gean operator.Some more interesting particular cases are also pointed out by the authors in [12] (also see [14]).
Using (14) it can be verified that For the functions   () ∈ A  given by their Hadamard product (or convolution) is defined by Observe that when  =  ∈ N, the operator I  , given by ( 14) can be represented in terms of convolution as follows: where In the sequel to earlier investigations, in the present paper we find a convolution result involving I  , is also presented.With a view to state a well-known result, we denote by ℘() the class of functions as follows: where The result is the best possible for  1 =  2 = −1.
Proof.Suppose that each of the functions   () ∈ A  ( = 1, 2) satisfies the condition (22).Set Then, by making use of the identity (15) in (26) we get Therefore, a simple computation, by using ( 24) and ( 27), shows that where The proof will be completed by finding the best possible lower bound for ℜ( 0 ()).A change of variable also gives Since   () ∈ ℘(  ) ( = 1, 2), where   = ((1 −   )/(1 −   )) ( = 1, 2), it follows from a result in [15] that and the bound  3 is the best possible.An application of Lemma 1, in (30), yields In order to show that  is the best possible in the assertion (23) when  1 =  2 = −1, we consider the function   () ∈ A  given by It is readily checked that   satisfies (22) with   = −1.Since it follows from (30) and Lemma 1 that This completes the proof of Theorem 2.