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Journal of Mathematical Analysis and Applications

The authors present sharp Landen transformation inequalities for the hypergeometric functions F12(a,b;a+b;x) and F12(a,b;(a+b+1)/2;x), by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear (or rational) functions, thus giving complete solutions of the problem on extending the well-known Landen transformation identities for the complete elliptic integrals of the first kind to these two hypergeometric functions, and substantially improving the related known results. As applications of these results, sharp Landen transformation inequalities are obtained for the generalized Grötzsch ring functions and the modular functions, which appear in Ramanujan's modular equations. Some other properties of hypergeometric functions and several properties of the Ramanujan constant are obtained, too.

Sharp Landen transformation inequalities for hypergeometric functions, with applications