On subclasses of analytic functions based on a quantum symmetric conformable differential operator with application

Rabha W Ibrahim,Rafida M Elobaid,Suzan J Obaiys

doi:10.1186/s13662-020-02788-6

Abstract

Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric function theory to extend differential operators, integral operators, and classes of analytic functions, especially the classes that are generated by convolution product (Hadamard product). In this effort, we aim to introduce a quantum symmetric conformable differential operator (Q-SCDO). This operator generalized some well-know differential operators such as Sàlàgean differential operator. By employing the Q-SCDO, we present subclasses of analytic functions to study some of its geometric solutions of q-Painlevé differential equation (type III).

Highlights

1 Introduction The conception of q-calculus model is a creative method for designs of the q-special functions
The procedure of q-calculus improves various kinds of orthogonal polynomials, operators, and special functions, which realize the form of their typical complements
Numerous integral and derivative features were formulated by using the convolution concept; for example, the Sàlàgean derivative [6], Al-Oboudi derivative [7], and the symmetric Sàlàgean derivative [8]

Summary

Introduction

The conception of q-calculus model is a creative method for designs of the q-special functions. The procedure of q-calculus improves various kinds of orthogonal polynomials, operators, and special functions, which realize the form of their typical complements. The idea of q-calculus was principally realized by Carmichael [1], Jackson [2], Mason [3], and Trjitzinsky [4]. An analysis of this calculus for the early mechanism was offered by Ismail et al [5]. It is significant to notify that the procedure of convolution finds its uses in different research, analysis, and study of the geometric properties of regular functions (see [9,10,11]). The consequences are generalized classes in the open unit disk

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Methods