Journal of Mathematics of Kyoto University

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

Shinji Yamashita

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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.

Article information

J. Math. Kyoto Univ., Volume 43, Number 4 (2003), 771-805.

First available in Project Euclid: 14 August 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C20: Conformal mappings of special domains
Secondary: 30C62: Quasiconformal mappings in the plane 33E05: Elliptic functions and integrals


Yamashita, Shinji. Inverse functions of Grötzsch’s and Teichmüller’s modulus functions. J. Math. Kyoto Univ. 43 (2003), no. 4, 771--805. doi:10.1215/kjm/1250281735.

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