On the theorem of Hayman and Wu

Rohde, S.

doi:10.1090/S0002-9939-01-06209-8

On the theorem of Hayman and Wu
HTML articles powered by AMS MathViewer

by S. Rohde PDF

Proc. Amer. Math. Soc. 130 (2002), 387-394 Request permission

Abstract:

We show that the Hayman-Wu constant Ø is strictly smaller than $4\pi .$ Previously it has been shown that $\pi ^2\leq$ Ø$\ \leq 4\pi .$ A main tool in our proof is an analysis of the hyperbolic geodesic curvature of straight lines in simply connected domains.

References

Barbara Brown Flinn, Hyperbolic convexity and level sets of analytic functions, Indiana Univ. Math. J. 32 (1983), no. 6, 831–841. MR 721566, DOI 10.1512/iumj.1983.32.32056
José L. Fernández, Juha Heinonen, and Olli Martio, Quasilines and conformal mappings, J. Analyse Math. 52 (1989), 117–132. MR 981499, DOI 10.1007/BF02820475
J. B. Garnett, F. W. Gehring, and P. W. Jones, Conformally invariant length sums, Indiana Univ. Math. J. 32 (1983), no. 6, 809–829. MR 721565, DOI 10.1512/iumj.1983.32.32055
J. L. Fernández and A. Granados, On geodesic curvature and conformal mapping, Algebra i Analiz 9 (1997), no. 3, 220–244; English transl., St. Petersburg Math. J. 9 (1998), no. 3, 615–637. MR 1466803
W. K. Hayman and J. M. G. Wu, Level sets of univalent functions, Comment. Math. Helv. 56 (1981), no. 3, 366–403. MR 639358, DOI 10.1007/BF02566219
Brad G. Osgood, Some properties of $f^{\prime \prime }/f^{\prime }$ and the Poincaré metric, Indiana Univ. Math. J. 31 (1982), no. 4, 449–461. MR 662912, DOI 10.1512/iumj.1982.31.31037
Knut Øyma, Harmonic measure and conformal length, Proc. Amer. Math. Soc. 115 (1992), no. 3, 687–689. MR 1101991, DOI 10.1090/S0002-9939-1992-1101991-1
Knut Øyma, The Hayman-Wu constant, Proc. Amer. Math. Soc. 119 (1993), no. 1, 337–338. MR 1149976, DOI 10.1090/S0002-9939-1993-1149976-4
Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768