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 a b plane where well‐known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each x ∈ (0,1). Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.