Landen Inequalities for Zero-Balanced Hypergeometric Functions

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 $x\in \left(\mathrm{0,1}\right)$. Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.