Applications for Certain Classes of Spirallike Functions Defined by the Srivastava-Attiya Operator

Let the function f be analytic in { } : and 1 z z z = ∈ <   and be given by ( ) 2 k k k f z z a z ∞ = = +∑ . In this paper, making use of the Srivastava-Attiya operator , s b  , we introduce two classes of analytic functions and investigate some convolution properties and coefficient estimates for these classes. Furthermore, several inclusion properties involving these and other families of integral operators are also considered.


Introduction and Definitions
Let  denote the class of functions .Also let f and g be analytic in  with ( ) ( ) . Then we say that f is subordinate to g in  , written f g  or ( ) ( ) , if there exists the Schwarz function w, analytic in  such that ( ) ( ) 1 w z < and ( ) ( ) ( ) We also observe that  ( ) .
Making use of the principle of subordination between analytic functions, Bhoosnurmath and Devadas [1] considered the subclasses [ ] ≤ as following (see also [2] and [3]): : e cos sin .
With a view to define the Srivastava-Attiya transform, we recall here a general Hurwitz-Lerch zeta function, which is defined in [8] by the following series: For further interesting properties and characteristics of the Hurwitz-Lerch Zeta and other related functions ( ) , , z s a Φ see [9] [10] and [11].
We note that: It is easily verified from (1.6) that Next, by using the linear operator , : : , , : : , .
It follows from the definitions (1.8) and (1.9) that
In this article, we investigate some convolution properties and coefficient . Furthermore, several inclusion properties and relevant connections of the results presented here with those obtained in earlier works are also discussed.

Convolution Properties and Coefficient Estimates
Unless otherwise mentioned, we will assume in the reminder of this paper that In order to establish our convolution properties, we shall need the following lemmas due to Bhoosnurnath and where We begin by proving the following theorem.
Theorem 2.3 The function ( ) where M is given by (2.2).Then, by applying (1.6), the left hand side of (2.5) becomes 1 which completes the proof of Theorem 2.3.
Theorem 2. 4 The function ( ) where N is given by (2.4).Then, by using (1.6), the left hand side of (2.6) may be written as which evidently proves Theorem 2.4.
Next, we determine coefficients estimates for a function of the form if its coefficients satisfy the condition ( )

Inclusion Properties and Applications
To prove the inclusion properties for the classes , we shall require the following lemma due to Eenigenburg et al. [20].Lemma 3.1 ([20]).Let ( ) By applying Lemma 3.1, we prove Theorem 3.2 Let , where ( ) p z is analytic in  with ( ) 0 e i p α = .By applying the identity (1.7), we , , , ,


is now popularly known in the literature as the Srivastava-Attiya operator.Various class-mapping properties of the operator , s b  (and its variants) are discussed in the recent works of Srivastava and Attiya J. H. Choi DOI: 10.4236/apm.2018.86035617 Advances in Pure Mathematics


, we introduce the following new classes of analytic functions for 0

3 ) 4 ) 1 b > − and 0 s
Making use of the logarithmic differentiation on both side in (3.3), we haveSince the function ( )h z is convex univalent in  with ( ) using Lemma 3.1 and (3.4), we observe that ( ) ( ) ≥ .Suppose that (3.1) holds for all z ∈  .Then (1.10) and Theorem 3.2, we observe that in Theorem 3.2 and 3.3, we have the following corollary.


defined by (cf.[21] [22] and[23]) By using same argument as in the proof of Theorem 3.2 with (3.6)which completes the proof of Theorem 3.6.