Monotonicity theorems and inequalities for the Hübner function with applications

In this paper, the authors present sharp bounds of the Hubner 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.