Solvability of a New q-Differential Equation Related to q-Differential Inequality of a Special Type of Analytic Functions

Abstract

The current study acts on the notion of quantum calculus together with a symmetric differential operator joining a special class of meromorphic multivalent functions in the puncher unit disk. We formulate a quantum symmetric differential operator and employ it to investigate the geometric properties of a class of meromorphic multivalent functions. We illustrate a set of differential inequalities based on the theory of subordination and superordination. In this real case study, we found the analytic solutions of q-differential equations. We indicate that the solutions are given in terms of confluent hypergeometric function of the second type and Laguerre polynomial.

Q‐Differential Equations of Higher Order and Structural Properties of the Approximate Roots for q‐Modified Derangements’ Polynomials

“…Several mathematicians discovered q-diferential equations by using special polynomials as a solution and studying their properties and identities, see [3][4][5][6][7]. Te q-diferential equation based on q-Hermit polynomials were studied by Hermoso, Huertas, and Lastra in [8].…”

On Starlike Functions of Negative Order Defined by q-Fractional Derivative

“…The q-derivative and q-integral are the main tools introduced by Jackson [1,2] in a systematic way. The linear q-difference equation, and q-differential equations, are studied in [3,4]. Mansour [5] investigated linear sequential q-differential equation of fractional order.…”

Multivalent Functions and Differential Operator Extended by the Quantum Calculus

“…This calculus proved its efficiency and accuracy to generalize the families of differential and integral operators in a complex domain. In addition, special functions (see [7,8]) have associated with this calculus, especially the queen of special functions: Mittag-Leffler function (see [9][10][11][12]). The quantum calculus (q-calculus) has tremendous applications in different fields, for example, integral inequalities [13], summability [14], approximation and polynomials [15], and sequence spaces [16].…”