Applications of a q-Salagean type operator on multivalent functions

Saqib Hussain,Maslina Darus,Shahid Khan,Muhammad Asad Zaighum

doi:10.1186/s13660-018-1888-3

Abstract

In this paper, we introduce a new class k-mathcal{US}(q,gamma ,m,p), gamma inmathbb{C}backslash {0}, of multivalent functions using a newly defined q-analogue of a Salagean type differential operator. We investigate the coefficient problem, Fekete–Szego inequality, and some other properties related to subordination. Relevant connections of the results presented here with those obtained in the earlier work are also pointed out.

Highlights

1 Introduction For a positive integer p, let Ap denote the set of all functions f (z) which are analytic and p-valent in the open unit disk E = {z ∈ C : |z| < 1} and have series expansion of the form
It is worth mentioning that convolution theory helps many researchers to investigate a number of properties of analytic univalent and multivalent functions
Due to growing applications of q-calculus, investigators are interested in studying properties of functions using q-operators instead of ordinary differential operators; for comprehensive study, we refer to Kanas and Reducanu [15], Mahmood and Darus [19], and Hussain et al Journal of Inequalities and Applications

Summary

Introduction

Let f ∗ g denote the convolution (or Hadamard product) of f , g ∈ Ap defined as follows:. It is worth mentioning that convolution theory helps many researchers to investigate a number of properties of analytic univalent and multivalent functions. Several differential and integral operators were defined using ordinary derivative; for details, see [29]. Due to growing applications of q-calculus, investigators are interested in studying properties of functions using q-operators instead of ordinary differential operators; for comprehensive study, we refer to Kanas and Reducanu [15], Mahmood and Darus [19], and Hussain et al Journal of Inequalities and Applications. For any non-negative integer n, the q-integer number n denoted by [n]q is defined by. For a non-negative integer n, the q-number shift factorial is defined as [n]q! For f ∈ A, in [5], the q-derivative operator or q-difference operator is defined as follows:

It can easily be seen that

Proof Let

Hence we have

Methods