Majorization of the Modulus of Continuity of Analytic Functions

Let G be a bounded domain in the complex plane, let f be analytic in G and continuous in G, and let μ be a majorant, that is, a non-negative non-decreasing function defined for t ≥ 0 such that μ(2t) ≤ 2μ(t) for all t ≥ 0. Suppose that z1 ∈ ∂G and that |f(z1) − f(z2)| ≤ μ(|z1 − z2|) for all z2 ∈ ∂G. We show that then |f(z1)−f(z2)| ≤ Cμ(|z1−z2|) for all z2 ∈ G where C = 3456. If the assumption is made for all z1, z2 ∈ ∂G, then the conclusion holds for all z1, z2 ∈ 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.