Log-convexity and log-concavity for series in gamma ratios and applications

TL;DR

This paper investigates the log-convexity and log-concavity properties of series involving gamma function ratios, establishing new theoretical results and inequalities for hypergeometric functions and related special functions.

Contribution

It introduces a novel notion of q-logarithmic concavity for power series with gamma ratio parameters and proves multiple theorems on their convexity and concavity properties.

Findings

Proved six theorems on q-logarithmic convexity and concavity.

Derived new inequalities for hypergeometric functions.

Applied theoretical results to specific functions like Kummer and Gauss functions.

Abstract

Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of parameter. Four types of such power series are considered where parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of $q$-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions. In the last section of the paper we present new inequalities for the Kummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems.