| A proof of Sethares' conjecture |
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| 作者姓名： | YAO Guowu
School of Mathematical Sciences Peking University Beijing China |
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| 作者单位： | YAO Guowu
School of Mathematical Sciences,Peking University,Beijing 100871,China |
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| 基金项目： | 国家重点基础研究发展计划(973计划) |
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| 摘 要： | Let (?)(z) be holomorphic in the unit disk △ and meromorphic on △. Suppose / is a Teichmuller mapping with complex dilatation In 1968, Sethares conjectured that f is extremal if and only if either (i)(?) has a double pole or (ii)(?) has no pole of order exceeding two on (?)△. The "if" part of the conjecture had been solved by himself. We will give the affirmative answer to the "only if" part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper, which generalizes the "if" part of Sethares' conjecture and improves the result by Reich and Shapiro in 1990.
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A proof of Sethares’ conjecture |
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| YAO Guowu
School of Mathematical Sciences,Peking University,Beijing ,China.A proof of Sethares' conjecture[J].Science in China(Mathematics),2004,47(2):236-244. |
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| Authors: | Guowu Yao |
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| Institution: | School of Mathematical Sciences, Peking University, Beijing 100871, China |
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| Abstract: | Let ψ(z) be holomorphic in the unit disk △ and meromorphic on -△. Suppose f is a Teichmuller mapping with complex dilatation k-ψ/|ψ|. In 1968, Sethares conjectured that f is extremal if and only if either (i) ψ has a double pole or (ii) ψ has no pole of order exceeding two on △. The "if" part of the conjecture had been solved by himself. We will give the affirmative answer to the "only if" part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper,which generalizes the "if" part of Sethares' conjecture and improves the result by Reich and Shapiro in 1990. |
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| Keywords: | Teichmulier mapping extremality Hamilton sequence substantial boundary point |
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