Tur\'an-Type Inequalities for Gaussian Hypergeometric Functions, and Baricz's Conjecture

TL;DR

This paper disproves Baricz's conjecture on Turán-type inequalities for Gaussian hypergeometric functions and introduces new inequalities and bounds for related elliptic integrals.

Contribution

The authors disprove Baricz's conjecture and establish new Turán-type inequalities and sharp bounds for elliptic integrals of the first kind.

Findings

Disproved Baricz's conjecture on hypergeometric functions.

Established new Turán-type inequalities for Gaussian hypergeometric functions.

Provided sharp bounds for complete and generalized elliptic integrals.

Abstract

In 2007, \'A. Baricz put forward a conjecture concerning Tur\'an-type inequalities for Gaussian hypergeometric functions (see Conjecture \ref{ConjA} in Section \ref{Sec1}). In this paper, the authors disprove this conjecture with several methods, and present Tur\'an-type double inequalities for Gaussian hypergeometric functions, and sharp bounds for complete and generalized elliptic integrals of the first kind.