# Inequalities for series in q-shifted factorials and q-gamma functions

## TL;DR

This paper investigates the convexity and concavity properties of power series involving q-gamma functions and q-shifted factorials, providing conditions for the sign of their Turánian and applying results to hypergeometric functions.

## Contribution

It extends previous work by establishing new inequalities and sign conditions for series with q-analogues of gamma functions and factorials, including applications to hypergeometric functions.

## Key findings

- Conditions for constant sign of Turánian coefficients
- Seven examples of hypergeometric functions satisfying the theorems
- Extension of earlier results on gamma function-based series

## Abstract

The paper studies logarithmic convexity and concavity of power series with coefficients involving q-gamma functions or q-shifted factorials with respect to a parameter contained in their arguments. The principal motivating examples of such series are basic hypergeometric functions. We consider four types of series. For each type we establish conditions sufficient for the power series coefficients of the generalized Tur\'anian formed by these series to have constant sign. Finally, we furnish seven examples of basic hypergeometric functions satisfying our general theorems. This investigation extends our previous results on power series with coefficient involving the ordinary gamma functions and the shifted factorials.