Inequalities for some basic hypergeometric functions

TL;DR

This paper investigates inequalities related to basic hypergeometric functions, establishing conditions for their logarithmic concavity and convexity, and introduces new linearization identities, including a novel case at q=1.

Contribution

It provides new conditions for the concavity and convexity of hypergeometric functions and introduces a new linearization identity, including a novel case at q=1.

Findings

Conditions for discrete logarithmic concavity and convexity established.

Logarithmic properties proved for Heine's basic hypergeometric function.

New linearization identity for generalized Turánian, including a novel q=1 case.

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.