The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrig...The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.展开更多
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 doubl...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.展开更多
基金Supported by the National Natural Science Foundation of China(10671174, 10401036)a Foundation for the Author of National Excellent Doctoral Dissertation of China(200518)
文摘The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.
文摘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.
