A Class of Quantum Briot–Bouquet Differential Equations with Complex Coefficients

Rafida M. Elobaid,Suzan J. Obaiys,Rabha W. Ibrahim

doi:10.3390/math8050794

Rafida M. Elobaid, Suzan J. Obaiys + Show 1 more

Open Access

https://doi.org/10.3390/math8050794

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Abstract

Quantum inequalities (QI) are local restraints on the magnitude and range of formulas. Quantum inequalities have been established to have a different range of applications. In this paper, we aim to introduce a study of QI in a complex domain. The idea basically, comes from employing the notion of subordination. We shall formulate a new q-differential operator (generalized of Dunkl operator of the first type) and employ it to define the classes of QI. Moreover, we employ the q-Dunkl operator to extend the class of Briot–Bouquet differential equations. We investigate the upper solution and exam the oscillation solution under some analytic functions.

Highlights

Quantum calculus exchanges the traditional derivative by a difference operator, which permits dealing with sets of non-differentiable curves and admits several formulas
The most common formula of quantum calculus is constructed by the q-operator (q-indicates for the quantum), which is created by the Jackson q-difference operator [1] as follows: let δq be the q-calculus which is formulated by δq (g(ξ )) = g(qξ ) − g(ξ ), the derivatives of functions are presented as fractions by the q-derivative
The suggested q-differential operator indicates a generalization of well-known differential operators in the open unit disk, such as the Dunkl operator and the Sàlàgean operator

Summary

Introduction

Quantum calculus exchanges the traditional derivative by a difference operator, which permits dealing with sets of non-differentiable curves and admits several formulas. Mao et al formulated a new quantum key distribution based on quantum inequalities [8] In this investigation, we formulate a novel q-differential operator of complex coefficients and discuss its behavior in view of the theory of geometric functions. The suggested q-differential operator indicates a generalization of well-known differential operators in the open unit disk, such as the Dunkl operator and the Sàlàgean operator. It will be considered in some subclasses of starlike functions. Quantum inequalities involve the q-differential operator and some special functions are studied.

Related Works

Q-Differential Equations

Conclusions