Inverse functions of Grötzsch’s and Teichmüller’s modulus functions

Abstract

Let $\chi$ be the inverse of the Grötzsch modulus function and let $\sigma _{n}$ be the $n$-th iteration of the function $\sigma (r) = 2\sqrt{r}/(1 + r)$, $r > 0$. For a real constant $\beta \neq 0$ with $\beta >-2$, the difference $\chi (x)^{\beta}-\sigma _{n}(4e^{-2^{n}x})^{\beta}$ is estimated. In the particular case where $\beta =-2$ one has an approximation of the inverse $S$ of the Teichmüller modulus function, which is applied to improving the known upper and lower estimates concerning the error term of $\lambda (K) = \chi (\pi K/2)^{-2} -1$ from $16^{-1}e^{\pi K}-2^{-1}$ for the variable $K \geqslant 1$. Expressions of $\chi$ and $S$ in terms of theta functions are studied. Lipschitz continuity of $f$ or log $f$ for $f = \chi , S$, as well as other functions are proved.