Notes on Generalized Grötzsch Ring Function and Generalized Hersch-Pfluger Distortion Function

Abstract

For $a\in (0,1)$, $r\in (0,1)$ and $K\in (1,\infty )$, let $\hat{I}{¼}_a(r)$ and $\ddot{I}{†}_K^a(r)$ be the generalized Grötzsch ring function and generalized Hersch-Pfluger distortion function. In the past few years, the functions $\hat{I}{¼}_a(r)$ and $\ddot{I}{†}_K^a(r)$, and their special cases $\hat{I}{¼}_{1/2}(r)$ and $\ddot{I}{†}_K^{1/2}(r)$ have been playing the very important role on the theory of quasiconformal mappings and (generalized) Ramanujan's modular equations. In this paper, we present a series expansion of $\hat{I}{¼}_a(r)$, and thus prove that the function $r\mapsto -[\hat{I}{¼}_a(r)-\log (e^{R(a)/2})/r]$ is absolutely monotonic on $(0,1)$. Here $R(a)$ is the Ramanujan constant. In addition, we also investigate the submultiplicative and power submultiplicative properties of $\ddot{I}{†}_K^a(r)$, and establish some new inequalities for $\ddot{I}{†}_K^a(r)$ in terms of elementary functions.