Certain Inequalities Involving the $q$-Deformed Gamma Function

TL;DR

This paper derives new inequalities involving the $q$-deformed Gamma function using $q$-integral techniques, extending classical results and analyzing their sharpness.

Contribution

It introduces novel double inequalities for the $q$-Gamma function ratio and extends classical asymptotic relations to the $q$-analogue setting.

Findings

Established double inequalities for the $q$-Gamma ratio

Derived $q$-analogues of Wendel's asymptotic relation

Investigated the sharpness of the inequalities

Abstract

This paper is inspired by the work of J. S\'{a}ndor in 2006. In the paper, the authors establish some double inequalities involving the ratio $ \frac{\Gamma_{q}(x+1)}{ \Gamma_{q} \left( x+\frac{1}{2}\right)}$, where $\Gamma_{q}(x)$ is the $q$-deformation of the classical Gamma function denoted by $\Gamma(x)$. The method employed in presenting the results makes use of Jackson's $q$-integral representation of the $q$-deformed Gamma function. In addition, H\"{o}lder's inequality for the $q$-integral, as well as some basic analytical techniques involving the $q$-analogue of the psi function are used. As a consequence, $q$-analogues of the classical Wendel's asymptotic relation are obtained. At the end, sharpness of the inequalities established in this paper is investigated.