Abstract
For a function ω, we establish a condition sufficient for the sum ∑i, ω(diam φ(L i )) to be finite for any quasiconformal curve L i , simply connected domain Ω, and a function φ which conformally and univalently maps this domain onto the unit disk. Here, L i denote the components of Ω∩L.
This is a preview of subscription content, log in via an institution to check access.
References
J. Fernandez, J. Heinonen, and O. Martio, “Quasilines and conformal mappings” J. D’Anal. Math., 52, 117–132 (1989).
W. Hayman and G. Wu, “Level sets of univalent function,” Comm. Math. Helv., 56, 366–403 (1981).
J. Fernandez and D. Hamilton, “Lengths of curves under conformal mappings,” Comm. Math. Helv., 62, 122–134 (1987).
Ch. Bishop and P. Jones, “Harmonic measure and arc length,” Ann. Math., 132, 511–547 (1990).
K. Astala, J. Fernandez, and S. Ronde, Quasilines and the Hayman-Wu Theorem, Preprint, Berlin 1991.
L. Ahlfors, Lectures in Quasiconformal Mappings, Van-Nostrand, Princeton 1966.
J. Vaisala, “Bounded turning and quasiconformal mappings,” Monastshefte Math., 11, 233–244 (1991).
O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer, Berlin 1973.
L. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory, McGraw-Hill, New York 1973.
Rights and permissions
About this article
Cite this article
Maimeskul, V.V. On the Hayman-Wu theorem for quasilines. Ukr Math J 48, 959–964 (1996). https://doi.org/10.1007/BF02384181
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02384181