## 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(z_{1})-f(z_{2}){pipe} ≤ μ({pipe}z_{1}-z_{2}{pipe}) 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({pipe}z{pipe}^{q}) as z→∞ in D. Then at least one of the following holds: (i) For all z_{2} ∈ G we have {pipe}f(z_{1})-f(z_{2}){pipe} ≤ Cμ({pipe}z_{1}-z_{2}{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(z_{1})-f(z_{2}){pipe} ≤ μ({pipe}z_{1}-z_{2}{pipe}) for all z_{1}, z_{2} ∈ ∂G, then {pipe}f(z_{1})-f(z_{2}){pipe} ≤ Cμ({pipe}z_{1}-z_{2}{pipe}) for all z_{1}, z_{2} ∈ 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 language | English (US) |
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Pages (from-to) | 59-80 |

Number of pages | 22 |

Journal | Complex Variables and Elliptic Equations |

Volume | 56 |

Issue number | 1-4 |

DOIs | |

State | Published - Jan 2011 |

## Keywords

- Analytic functions
- Modulus of continuity
- Subharmonic functions

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics