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

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.

Submitted 23 January 2019, pages 17