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Israel Journal of Mathematics

, Volume 91, Issue 1–3, pp 157–171 | Cite as

Behavior of domain constants under conformal mappings

  • W. Ma
  • D. Minda
Article

Abstract

Domain constants are numbers attached to regions in the complex plane ℂ. For a region Ω in ℂ, letd(Ω) denote a generic domain constant. If there is an absolute constantM such thatM −1d(Ω)/d(Δ)≤M whenever Ω and Δ are conformally equivalent, then the domain constant is called quasiinvariant under conformal mappings. IfM=1, the domain constant is conformally invariant. There are several standard problems to consider for domain constants. One is to obtain relationships among different domain constants. Another is to determine whether a given domain constant is conformally invariant or quasi-invariant. In the latter case one would like to determine the best bound for quasi-invariance. We also consider a third type of result. For certain domain constants we show there is an absolute constantN such that |d(Ω)−d(Δ)|≤N whenever Ω and Δ and conformally equivalent, sometimes determing the best possible constantN. This distortion inequality is often stronger than quasi-invariance. We establish results of this type for six domain constants.

Keywords

Conformal Mapping Equivalent Region Covering Projection Convex Region Schwarzian Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Magnes Press 1995

Authors and Affiliations

  • W. Ma
    • 1
  • D. Minda
    • 1
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

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