Landen inequalities for zero-balanced hypergeometric functions (Q417225)

Summary: 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 the open problem from the work by \textit{G. D. Anderson, M. K. Vamanamurthy}, and \textit{M. Vuorinen} [Conformal invariants, inequalities, and quasiconformal maps. Wiley (1997; Zbl 0885.30012), p. 79].