Some new inequalities for the $q$-gamma and related functions

TL;DR

This paper explores convexity and monotonicity properties of functions related to the $q$-gamma function, deriving new inequalities and improving existing results on zeros of the $q$-digamma function using $q$-analogues.

Contribution

It introduces new inequalities for the $q$-gamma and related functions, utilizing $q$-analogues of classical formulas to enhance understanding of their properties.

Findings

Derived new inequalities for $q$-gamma and $q$-digamma functions.

Improved results on the zeros of the $q$-digamma function.

Utilized $q$-analogues of Gauss multiplication formula for closed-form inequalities.

Abstract

We consider convexity and monotonicity properties for some functions related to the $q$-gamma function. As applications, we give a variety of inequalities for the $q$-gamma function, the $q$-digamma function $\psi_q(x)$, and the $q$-series. Among other consequences, we improve a result of Azler~and~Grinshpan about the zeros of the function $\psi_q(x)$. We use $q$-analogues for the Gauss multiplication formula to put in closed form members of some of our inequalities.