Computational Methods and Function Theory 8 (2008), No. 2, 303--325
Copyright Heldermann Verlag 2008
Aimo Hinkkanen
aimo@uiuc.edu , University of Illinois at Urbana-Champaign, Department of Mathematics,
Let $G$ be a bounded domain in the complex plane, let $f$ be analytic in $G$ and continuous in $\overline{G}$, and let $\mu$ be a majorant, that is, a non-negative non-decreasing function defined for $t\geq 0$ such that $\mu(2t)\leq 2\mu(t)$ for all $t\geq 0$. Suppose that $z_1\in\partial G$ and that $|f(z_1)-f(z_2)|\leq \mu(|z_1-z_2|)$ for all $z_2\in\partial G$. We show that then $|f(z_1)-f(z_2)|\leq C\mu(|z_1-z_2|)$ for all $z_2\in G$ where $C=3456$. If the assumption is made for all $z_1,z_2\in\partial G$, then the conclusion holds for all $z_1,z_2\in\overline{G}$. Earlier such a result, with an absolute constant $C$, had only been known when $G$ is simply or doubly connected. The same result holds when $G$ is an open set with only bounded components. We also give a survey of results on this type of problems, and explain the reductions that can be made.
Keywords: Analytic functions, modulus of continuity, majorization, maximum principle.
MSC 2000: Primary 30C80.
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