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Mathematical Notes

, Volume 60, Issue 2, pp 130–136 | Cite as

Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

  • E. P. Dolzhenko
Article

Abstract

Letd(Γ;z, t) be the smallest diameter of the arcs of a Jordan curve Γ with endsz andt. Consider the rapidity of decreasing ofd(Γ;ρ)=sup{d(Γ;z, t):z, t ∈ Γ, ¦z−t¦≤ρ} (asρ ↘ 0,ρ≥0) as a measure of “nicety” of Γ. Letg(x) (x≥0) be a continuous and nondecreasing function such thatg(x)≥x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(√x))2. LetH(x) be an arbitrary primitive of 1/h−1(x). Note that the functionH−1x is positive and increasing on (−∞, +∞),H−1 →0 asx→−∞ andH−1→+∞ asx → +∞. The following statement is proved in the paper.

Key words

modulus of continuity conformal mapping Jordan domain holomorphic function Hölder class Carathéodory boundary element 

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Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • E. P. Dolzhenko
    • 1
  1. 1.Moscow State UniversityUSSR

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