Hadamard product of convex functions and Jackson operator

TL;DR

This paper explores properties of Jackson's difference operator for convex univalent functions in the unit disk, focusing on its Hadamard product representation and implications in q-theory.

Contribution

It introduces new properties of Jackson's difference operator for convex functions using Hadamard products, extending the understanding of q-theory applications.

Findings

Properties of Jackson's difference operator analyzed for convex functions.

Representation of the operator as a Hadamard product of power series.

Connections established between the operator and applications in hypergeometric series and physics.

Abstract

In this paper we consider some properties of Jackson's difference operator for convex univalent functions in $|z|<1$ with complex parameter $q$ as a Hadamard product of two power series. Jackson in 1908 introduced for a real $q$, $q\in[0,1)$, the difference operator \mbox{${\rm d}_qf(z)$} for an analytic function $f$ in the unit disc $|z|<1$ in the complex plane. Thanks to this operator, many mathematicians have extended the theory of functions in $q$-theory. The $q$-theory has found many applications in theory of hypergeometric series, special functions, combinatorics, number theory, fluid mechanics, quantum mechanics and physics.