Some remarks on the modulus of continuity of a conformal mapping of the disk onto a~Jordan domain

E. P. Dolzhenko

Matematičeskie zametki, Tome 60 (1996) no. 2, pp. 176-184

Résumé

Let $d(\Gamma;z,t)$ be the smallest diameter of the arcs of a Jordan curve $\Gamma$ with ends $z$ and $t$. Consider the rapidity of decreasing of $d(\Gamma;\rho)=\sup\bigl\{d(\Gamma;z,t): z,t\in \Gamma, |z-t|\le\rho\bigr\}$ (as $\rho\searrow0$, $\rho\ge0$) as a measure of “nicety” of $\Gamma$. Let $g(x)$ ($x\ge0$) be a continuous and nondecreasing function such that $g(x)\ge x$, $g(0)=0$. Put $\overline g(x):=g(x)+x$, $h(x):=\bigl(\overline g(x^{1/2})\bigr)^2$. Let $H(x)$ be an arbitrary primitive of $1/h^{-1}(x)$. Note that the function $H^{-1}(x)$ is positive and increasing on $(-\infty,+\infty)$, $H^{-1}(x)\to0$ as $x\to-\infty$ and $H^{-1}(x)\to+\infty$ as $x\to+\infty$. The following statement is proved in the paper.