On New Subclass of Harmonic Univalent Functions Associated with Modified q-Operator

Shujaat Ali Shah,Asghar Ali Maitlo,Muhammad Afzal Soomro,Khalida Inayat Noor

doi:10.28924/2291-8639-19-2021-826

Abstract

In this article, we introduce new subclasses of harmonic univalent functions associated with the q-difference operator. The modified q-Srivastava-Attiya operator is defined and certain applications of this operator are discussed. We investigate the sufficient condition, distortion result, extreme points and invariance of convex combination of the elements of the subclasses.

Highlights

A real-valued function u (x, y) is said to be harmonic in a domain D if it has continuous second order partial derivatives in D and satisfies uxx + uyy = 0
We introduce new subclasses of harmonic univalent functions associated with the q-difference operator
It is observed that every harmonic function f in any connected domain Ω can be written as f (z) = h(z) + g(z), where h and g are analytic in Ω, and are called, respectively, the analytic and co-analytic parts of f

Summary

Introduction

We denote by H the class of complex-valued harmonic functions f = h + g defined in the open unit disc. We denote by SH the subclass of H consisting of all sense-preserving univalent harmonic functions f. In motivation of above said literature, first we modify the q-Srivastava-Attiya operator and we define some new subclasses of SH. For f = h + g ∈ SH, we define a new class HSq (γ, λ, β) as follows: Definition 1.1. We further define HSq (γ, λ, β) = HSq (γ, λ, β) ∩ SH, where SH denote the subclass of SH consisting of functions of the type f (z) = h(z) + g (z), where (1.5). By using modified q-Srivastava-Attiya operator given by (1.4), we define the following.

The sufficient condition is obvious from the

If f

We take