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Properties of $q$-gamma and $q$-beta functions derived from the $q$-Gauss-P\' olya inequalities

We consider log-convexity and other properties of several functions related to $q$-gamma and $q$-beta functions. These properties are consequences of the general inequality, so-called $q$- analogue of the Gauss-P\' olya inequality. Various inequalities involving these special functions are also given.

the Jackson integral ; log-convexity ; $q$-beta function ; $q$-gamma function ; $q$-Gauss-P\' olya inequalities

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