Notes on Generalized Grötzsch Ring Function and Generalized Hersch-Pfluger Distortion Function
Qi Bao, MiaoKun Wang
TL;DR
This paper addresses the generalized Grötzsch ring function $\mu_a(r)$ and generalized Hersch-Pfluger distortion function $\varphi_K^a(r)$, which play central roles in quasiconformal mapping theory and Ramanujan modular equations. It derives a new series expansion for $\mu_a(r)$ and proves that the function $r \mapsto -[\mu_a(r)-\log(e^{R(a)/2}/r)]$ is absolutely monotonic on $(0,1)$, via a representation $\mu_a(r)+\log r=\frac{R(a)}{2}-\sum_{n\ge1} \theta_n r^{2n}$ with $\theta_n$ given by entires $\lambda_n$. It also extends sharp submultiplicative and power submultiplicative inequalities to $\varphi_K^a(r)$ (and $\varphi_{1/K}^a(r)$) using $m_a(r)$ and $\mu_a(r)$, proved through monotonicity of Borwein-type elliptic integrals. The results yield practical bounds and algorithms for estimating these functions and strengthen the connections between quasiconformal distortion, elliptic integrals, and Ramanujan modular equations.
Abstract
For $a\in(0,1)$, $r\in(0,1)$ and $K\in(1,\infty)$, let $μ_{a}(r)$ and $\varphi_{K}^{a}(r)$ be the generalized Grötzsch ring function and generalized Hersch-Pfluger distortion function. In the past few years, the functions $μ_{a}(r)$ and $\varphi_{K}^{a}(r)$, and their special cases $μ_{1/2}(r)$ 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 $μ_{a}(r)$, and thus prove that the function $r\mapsto -[μ_{a}(r)-\log{(e^{R(a)/2})/r}]$ 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.