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Journal d'Analyse Mathématique

, Volume 134, Issue 1, pp 107–126 | Cite as

On the existence of strips inside domains convex in one direction

  • Dimitrios Betsakos
Article
  • 48 Downloads

Abstract

A plane domain Ω is convex in the positive direction if for every ωΩ, the entire half-line {ω + t: t ≥ 0} is contained in Ω. Suppose that h maps the unit disk onto such a domain Ω with the normalization h(0) = 0 and limt→∞h−1(h(z) + t) = 1. We show that if ∠limz→−1 Re h(z) = −∞ and ∠limz→−1(1 + z)h′(z) = ν ∈ (0, +∞), then Ω contains a maximal horizontal strip of width πν. We also prove a converse statement. These results provide a solution to a problem posed by Elin and Shoikhet in connection with semigroups of holomorphic functions.

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Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

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