Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions

TL;DR

This paper proves longstanding conjectures on the monotonicity of ratios of Kummer, Gauss, and generalized hypergeometric functions, extending previous results and connecting to Turan-type inequalities.

Contribution

It establishes the monotonicity conjectures for a broad class of hypergeometric functions, generalizing earlier partial results from 1993.

Findings

Proved conjectures on monotonicity of ratios for Kummer hypergeometric functions.

Extended results to Gauss and generalized hypergeometric functions.

Connected monotonicity properties with Turan-type inequalities.

Abstract

In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from 1993. In this paper we prove these conjectures in the more general setting for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turan--type inequalities.