Certain Integral Operator Related to the Hurwitz–Lerch Zeta Function

Zhi-Ren Wang,Lei Shi,Xiao-Yuan Wang

doi:10.1155/2018/5915864

Abstract

The aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operator Ws,bf(z) involving the Hurwitz–Lerch Zeta function. We make some applications of the operator Ws,bf(z) for meromorphic functions.

Highlights

Denote by H(U) the class of functions analytic in the unite diskU = {z : z ∈ C, |z| < 1} (1) of the form ∞H [a, n] = {f : f ∈ H (U), f (z) = a + ∑akzk} k=n (2)(a ∈ C; n ∈ N = {1, 2, . . .})and let H = H[1, 1]
The aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operator Ws,bf(z) involving the Hurwitz–Lerch Zeta function
As the second-order differential superordinations were introduced and investigated by Miller and Mocanu [16], Tang et al [17] introduced the following third-order differential superordinations

Summary

Introduction

For two functions f(z) and g(z) to be analytic in U, f(z) is said to be subordinate to g(z) in U and written by f (z) ≺ g (z) (z ∈ U) ,. Denote by Q the set of functions q(z) that are analytic and univalent on U \ E(q), where. We recall the general Hurwitz–Lerch Zeta function Φ(z, s, a) The general Hurwitz–Lerch Zeta function Φ(z, s, a) was investigated by many researchers. In 2007, by involving the general Hurwitz–Lerch Zeta function Φ(z, s, a), Srivastava and Attiya [7] ( see [8,9,10,11]) introduced the integral operator. The main purpose of this paper is to derive some thirdorder differential subordination, differential superordination properties, and sandwich-type theorems of the integral operator Ws,bf(z)

Preliminary Results

Main Results

Conclusions