Landen inequalities for zero-balanced hypergeometric functions

TL;DR

This paper investigates Landen inequalities for zero-balanced hypergeometric functions, identifying maximal parameter regions where these inequalities hold, thus providing a comprehensive solution to an open problem in the field.

Contribution

It determines the maximal regions in the parameter space where Landen inequalities are valid for zero-balanced hypergeometric functions, solving an open problem.

Findings

Maximal regions for inequalities are explicitly characterized.

Landen inequalities are extended to zero-balanced hypergeometric functions.

The results provide a complete answer to the open problem.

Abstract

For zero-balanced Gaussian hypergeometric functions $ F(a,b;a+b;x),$ $a,b>0,$ we determine maximal regions of $ab$ plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each $x\in (0,1)$. Thereby an exhausting answer is given to an open problem.