Monotonicity Properties of Gaussian Hypergeometric Functions with Respect to the Parameter

TL;DR

This paper investigates the conditions under which Gaussian hypergeometric functions and related elliptic integrals exhibit monotonicity with respect to parameters, extending recent results and providing new theorems.

Contribution

It provides necessary and sufficient conditions for monotonicity of hypergeometric functions and elliptic integrals, generalizing prior findings and introducing new monotonicity theorems.

Findings

Established conditions for monotonicity of hypergeometric functions.

Generalized results for elliptic integrals of the first and second kinds.

Proved new monotonicity theorems from alternative perspectives.

Abstract

The authors establish the necessary and sufficient conditions under which certain combinations of Gaussian hypergeometric function and elementary function are monotone in the parameter, which generalize the recent results of generalized elliptic integrals of the first and second kinds obtained by Qiu et al. Moreover, the authors also prove two monotonicity theorems of generalized elliptic integrals from another point of view.