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Majorization of the Modulus of Continuity of Analytic Functions

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 ${\overline G}$ , 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Majorization of the Modulus of Continuity of Analytic Functions

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Publisher
Springer Journals
Copyright
Copyright © 2008 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321690
Publisher site
See Article on Publisher Site

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.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 20, 2007

References