Some inequalities for the $q$-Extension of the Gamma Function

TL;DR

This paper derives inequalities and bounds for the $q$-extension of the Gamma function using geometric convexity and monotonicity properties, contributing to the mathematical understanding of $q$-special functions.

Contribution

It introduces new inequalities for the $q$-Gamma function, utilizing geometric convexity and monotonicity, which were not previously established.

Findings

Established bounds for ratios of the $q$-Gamma function

Derived inequalities based on convexity properties

Enhanced understanding of $q$-Gamma function behavior

Abstract

In this paper, the authors establish some inequalities involving the $q$-extension of the classical Gamma function. These inequalities provide bounds for certain ratios of the $q$-extended Gamma function. The procedure makes use of geometric convexity and monotonicy properties of certain functions associated with the $q$-extended Gamma function.