Notes on Generalized Gr\"otzsch Ring Function and Generalized Hersch-Pfluger Distortion Function

For $a\in(0,1)$, $r\in(0,1)$ and $K\in(1,\infty)$, let $\mu_{a}(r)$ and $\varphi_{K}^{a}(r)$ be the generalized Gr\"{o}tzsch ring function and generalized Hersch-Pfluger distortion function. In the past few years, the functions $\mu_{a}(r)$ and $\varphi_{K}^{a}(r)$, and their special cases $\mu_{1/2}(r)$ and $\varphi_{K}^{1/2}(r)$ 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 $\mu_{a}(r)$, and thus prove that the function $r\mapsto -[\mu_{a}(r)-\log{(e^{R(a)/2})/r}]$ 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 $\varphi_{K}^{a}(r)$, and establish some new inequalities for $\varphi_{K}^{a}(r)$ in terms of elementary functions.