Inequalities for some basic hypergeometric functions

Abstract: We establish conditions for the discrete versions of logarithmic concavity and convexity of the higher order regularized basic hypergeometric function with respect simultaneous shift of all its parameters. For a particular case of Heine's basic hypergeometric function we prove logarithmic concavity and convexity with respect to the bottom parameter. We further establish a linearization identity for the generalized Tur\'{a}nian formed by a particular case of Heine's basic hypergeometric function. Its $q=1$ case also appears to be new.