Monotonicity theorems and inequalities for the Hubner function with applications

摘要

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. (C) 2021 Elsevier Inc. All rights reserved.