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

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