Monotonicity theorems and inequalities for the Hübner function with applications

Abstract

In this paper, the authors present sharp bounds of the Hübner function M, which is defined by (1.11) and has important applications in the theories of quasiconformal maps and Ramanujan's modular equations, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of M and elementary functions. By these results, several well-known results for M including its bounds and logarithmic inequalities, the explicit quasiconformal Schwarz lemma and the estimates of the solutions to Ramanujan's classical modular equations are remarkably improved. A simpler and more concise proof of the series expansion of M(r) is given, too.