Growth conditions for analytic functions in unbounded open sets

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Abstract

Let G be a non-empty open set in the complex plane ℂ with at least two finite boundary points. Let f: G → ℂ 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 {pipe}f(z1)-f(z2){pipe} ≤ μ({pipe}z1-z2{pipe}) for a fixed z1 ∈ ∂G and for all z2 ∈ ∂G. Suppose that for each unbounded component D of G, if any, there is a positive number q such that f(z) = O({pipe}z{pipe}q) as z→∞ in D. Then at least one of the following holds: (i) For all z2 ∈ G we have {pipe}f(z1)-f(z2){pipe} ≤ Cμ({pipe}z1-z2{pipe}) 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 {pipe}f(z1)-f(z2){pipe} ≤ μ({pipe}z1-z2{pipe}) for all z1, z2 ∈ ∂G, then {pipe}f(z1)-f(z2){pipe} ≤ Cμ({pipe}z1-z2{pipe}) for all z1, z2 ∈ G 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.

Original languageEnglish (US)
Pages (from-to)59-80
Number of pages22
JournalComplex Variables and Elliptic Equations
Volume56
Issue number1-4
DOIs
StatePublished - Jan 2011

Keywords

  • Analytic functions
  • Modulus of continuity
  • Subharmonic functions

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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