Abstract
In this paper we study the relations among various constants of quasiextremal distance (QED) domains in the plane. In particular, we give an affirmative answer to Shen’s conjecture about the relation between the QED constant \(M(\Omega )\) and the boundary dilatation \(H(\Omega )\). This leads to the conclusion that, for a large class of domains, the equality \(M(\Omega )=1+R(\Omega )\) conjectured by Garnett and Yang does not hold, where \(R(\Omega )\) is the quasiconformal reflection constant.
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Acknowledgments
We wish to express our sincere gratitude to the referee, whose extremely careful reading of the manuscript led to many clarifications and improvements in the text. This work was partially done when the first author was visiting the Department of Mathematics and Computer Sciences at Emory University. He wishes to express his gratitude to the department and university for their hospitality and support.
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The first author is partially supported by National Natural Science Foundation of China (Nos. 11001081 and 11371268).
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Cheng, T., Yang, S. Decomposition of extremal length and a proof of Shen’s conjecture on QED constant and boundary dilatation. Math. Z. 278, 1195–1211 (2014). https://doi.org/10.1007/s00209-014-1353-z
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DOI: https://doi.org/10.1007/s00209-014-1353-z