A proof of Sethares' conjecture  

作　　者：YAO Guowu 

机构地区：[1]School of Mathematical Sciences, Peking University, Beijing 100871, China

出　　处：《Science China Mathematics》2004年第2期236-244,共9页中国科学：数学（英文版）

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.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.

关 键 词：Teichmulier mapping EXTREMALITY Hamilton sequence substantial boundary point. 

分 类 号：O1[理学—数学]