The Hayman--Wu constant is $π^2$

Abstract

We show that for every conformal map $φ:\mathbb{D}\toΩ\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $Ω$, the length of $φ^{-1}(Ω\cap L)$ is at most $π^2$ for every line $L$.

Disclosure

“Acknowledgments The author acknowledges partial support from the NSF CAREER grant DMS-2152401, NSF grant DMS-2554183, a Simons Fellowship, and a Humboldt Research Fellowship for Experienced Researchers. The author acknowledges the use of AI tools. References [1] B. Brown Flinn, Hyperbolic convexity and level sets of analytic functions, Indiana Univ. Math. J. 32 (1983), no. 6, 831–841. [2] E. Crane, A note on the Hayman–Wu theorem, Comput. Methods Funct. Theory 8 (2008), no.”

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2026-08-23T05:12:40.723114Z

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