Subclasses of Multivalent Analytic Functions Associated with a q-Difference Operator

Ekram Elsayed Ali,Teodor Bulboacă

doi:10.3390/math8122184

Abstract

In this article we introduced and studied some inclusion properties for new subclasses of multivalent analytic functions defined by using the q-derivative operator. With the aid of the Jackson q-derivative we defined two new operators that generalize many other previously studied operators, and help us to define two new subclasses of functions with several interesting properties studied in this paper. The methods used for the proof of our results are special tools of the differential subordination theory of one-variable functions.

Highlights

Introduction and PreliminariesLet A( p) denote the class of functions of the form f (z) = z p + ∞∑ ak+ p zk+ p, z ∈ D, p ∈ N := {1, 2, . . . }, (1)k =1 that are analytic multivalent in the open unit disc D := {z ∈ C : |z| < 1}
We denote by f ∗ h the Hadamard product of the functions f and h analytic in D, that is, if f is given by (1) and h(z) = z p +
∑ bk + p z k + p, z ∈ D, k =1 ( f ∗ h)(z) := z p +

Summary

Introduction

Let A( p) denote the class of functions of the form f (z) = z p + For 0 < q < 1, in [1,2] Jackson defined the q-derivative operator Dq of a function f by 3. By specializing the parameters η, m, n and p we obtain the following operators studied by various authors: m (i) S0,m η,p = : Dη,p (see Aouf et al [4]); m (ii) S0,m

Results

Conclusion