Log-convexity and log-concavity of hypergeometric-like functions

TL;DR

This paper establishes new conditions for log-convexity and log-concavity of hypergeometric-like functions, generalizing known inequalities and providing new insights into their behavior, with implications for classical hypergeometric functions.

Contribution

It introduces sufficient conditions for log-convexity and log-concavity of hypergeometric-like functions, generalizing recent inequalities and establishing new reverse inequalities.

Findings

Generalized Turán inequalities for hypergeometric functions

Log-convexity results for confluent hypergeometric functions

A conjecture on monotonicity of hypergeometric function quotients

Abstract

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms $a\mapsto\sum{f_k}(a)_kx^k$, $a\mapsto\sum{f_k}\Gamma(a+k)x^k$ and $a\mapsto\sum{f_k}x^k/(a)_k$. The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Tur\'{a}n inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.