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.