Inverse functions of Grötzsch's and Teichmüller's modulus functions
The Grötzsch ring domain is the planar doubly-connected domain \(\{z:| z| <1\}\setminus [0,r]\), \(0<r<1\). It can be mapped conformally onto the annulus \(\{z:1<| z| <e^{\mu(r)}\}\), where \(\mu\) is the Grötzsch modulus function which can be expressed in terms of elliptic integrals. Let \(\chi:(0,\infty)\to (0,1)\) be the inverse function of \(\mu\). We also consider the Teichmüller modulus function \(T(t)= 2\mu(1/\sqrt{1+t})\), \(t>0\) and its inverse \(S(x)=\chi(x/2)^{-2}-1,\;x>0\). Let also \(\sigma_n\) denote the \(n\)th iteration of the function \(\sigma(r)=2\sqrt{r}/(1+r)\), \(r>0\). The author proves error bounds for the approximation \(\chi(x)^\beta\approx \sigma_n(4e^{-2^nx})^\beta\) for \(\beta\geq -2,\,\beta\neq 0\). Let \(\lambda(K)=S(KT(1))\), \(K\geq 1\). This function appears in the theory of \(K\)-quasiconformal maps. The author proves that the above mentioned error bounds lead to the inequalities \[ 1.2425\dots<(\lambda(K)-16^{-1}e^{\pi K}+2^{-1})e^{\pi K}<1.25,\quad K\geq 1. \] The right hand inequality is sharp and improves earlier results that appear in the book ``Conformal invariants, inequalities, and quasiconformal maps'' (1997; Zbl 0885.30012) by \textit{G. D. Anderson, M. K. Vamanamurthy}, and \textit{M. K. Vuorinen}. In this book one can find more details about the Grötzsch and Teichmüller functions, their expressions in terms of elliptic functions, and their use in quasiconformal maps. Yamashita also studies the Lipschitz continuity of \(\chi\) and \(S\) and their expressions in terms of elliptic theta functions.
scientific article; zbMATH DE number 2124478