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


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.


Conformal Mapping Equivalent Region Covering Projection Convex Region Schwarzian Derivative 
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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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