Summary: For zero-balanced Gaussian hypergeometric functions
, we determine maximal regions of
plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each
. Thereby an exhausting answer is given to the open problem from the work by G. D. Anderson, M. K. Vamanamurthy
, and M. Vuorinen
[Conformal invariants, inequalities, and quasiconformal maps. Wiley (1997; Zbl 0885.30012
), p. 79].