Majorization of the Modulus of Continuity of Analytic Functions

Abstract

Let G be a bounded domain in the complex plane, let f be analytic in G and continuous in ${\overline G}$, 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 z ₁ ∈ ∂G and that ¦ f(z₁) − f(z ₂)¦ ≤ μ(¦z ₁ − z ₂¦) for all z ₂ ∈ ∂G. We show that then ¦f(z ₁) − f(z ₂)¦ ≤ Cμ(¦z ₁ − z ₂¦t) for all z ₂ ∈ G where C = 3456. If the assumption is made for all z ₁,z ₂ ∈ ∂G, then the conclusion holds for all z ₁, z ₂ ∈ ${\overline G}$. 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.