Growth conditions for analytic functions in unbounded open sets

Abstract

Let G be a non-empty open set in the complex plane ℂ with at least two finite boundary points. Let be a continuous function that is analytic in G. Let μ be a non-decreasing non-negative function defined for t ≥ 0 such that μ(2t) ≤ 2μ(t) for all t ≥ 0. Suppose that |f(z 1) − f(z 2)| ≤ μ(|z 1 − z 2|) for a fixed z 1 ∈ ∂G and for all z 2 ∈ ∂G. Suppose that for each unbounded component D of G, if any, there is a positive number q such that f(z) = O(|z| q ) as z → ∞ in D. Then at least one of the following holds: i. For all z 2 ∈ G we have |f(z 1) − f(z 2)| ≤ Cμ(|z 1 − z 2|) where C = 3456. ii. The set G contains a neighbourhood of infinity, so that G has exactly one unbounded component, and f has a pole at infinity. Furthermore, if |f(z 1) − f(z 2)| ≤ μ(|z 1 − z 2|) for all z 1, z 2 ∈ ∂G, then |f(z 1) − f(z 2)| ≤ Cμ(|z 1 − z 2|) for all with C = 3456 except perhaps when G contains a neighbourhood of infinity, so that G has exactly one unbounded component, and f has a pole at infinity.