**Title:** Generalized quantum Airy differential operator in a complex domain

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-21-00065; **Volume 5 / Issue 3 / Year 2023**, Pages 37-48

**Document Type:** Research Paper

**Author(s):** Rabha Waell Ibrahim ^{a}

^{a}IEEE: 94086547, USA

Received: 14 November 2021, Accepted: 18 March 2022, Published: 22 June 2022.

**Corresponding Author:** Rabha Waell Ibrahim (Email address: rabhaibrahim@yahoo.com)

**Full Text:** PDF

**Abstract**

In this work, we express a generalization of Airy functions in virtue of the *q*–calculus, in a complex domain. Based on this generalization, we formulate the Airy equation and study its behavior in view of the geometric function theory. This formula will be considered in some classes of analytic functions. In this investigation, we act the suggested *q*–Airy operator on the subclass of normalized analytic functions. Our method is given by the theory of subordination and superordination. Some examples will be illustrated in the sequel. Moreover, an application is formulated for finding the upper solution of a complex wave diffusive equation using the suggested *q*-operator.

**Keywords:** Fractional calculus, quantum calculus, analytic function, univalent function, subordination, open unit disk

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