On quasiinvariance of harmonic measure and Hayman-Wu theorem
https://doi.org/10.26907/0021-3446-2024-2-22-36
Abstract
The article is devoted to the definition and properties of the class of diffeomorphisms of
the unit disk D = { z : | z| < 1} on the complex plane C for which the harmonic measure of the
boundary arcs of the slit disk has a limited distortion, i.e. is quasiinvariant. Estimates for derivative
mappings of this class are obtained. We prove that such mappings are quasiconformal and are also
quasiisometries with respect to the pseudohyperbolic metric. An example of a mapping with the
specified property is given. As an application, a generalization of the Hayman–Wu theorem to this
class of mappings is proved.
About the Author
S. Yu. Graf
Tver State University; Petrozavodsk State University
Russian Federation
Russian Federation
Sergey Yur’evich Graf
33 Zheliabova str., Tver, 170100
33 Lenina Ave., Petrozavodsk, 185910
References
1. Nevanlinna R. Das harmonische mass von punktmengen und seine anwendung in der funktionentheorie, in : Comptes rendus du huit´eme Congr´es des math´ematiciens Scandinaves, 116–133 (Stockholm, 1934).
2. Голузин Г.М. Геометрическая теория функций комплексного переменного (Наука, М., 1966).
3. Garnett J.B., Marshall D.E. Harmonic measure (Cambridge Univ. Press, 2005).
4. Ahlfors L.V. Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill, New York, 1973).
5. Beurling A. The Collected Works of Arne Beurling. Complex analysis (L. Carleson, et al., eds.) V. 1 (Birkh¨auser, 1989).
6. Betsakos D., Solynin A.Yu. Extensions of Beurling’s shove theorem for harmonic measure, Complex Variables and Elliptic Equat. 42 (1), 57–65 (2000).
7. Kelingos J. A. Characterizations of quasiconformal mappings in terms of harmonic and hyperbolic measure, Ann. Acad. Sci. Fenn. Ser. A I 368, 1–16 (1965).
8. Graf S.Yu. On distortion of harmonic measure under locally quasiconformal mappings, Complex Variables and Elliptic Equat. 66 (9), 1550–1564 (2020).
9. Ahlfors L.V. Lectures on Quasiconformal Mappings: Second Edition (AMS, University Lecture Ser., V. 38,2006).
10. Anderson G.D., Vamanamurthy M.K., Vuorinen M. Conformal Invariants, Inequalities an QuasiconformalMaps (Wiley, New York, 1997).
11. Vasil’ev A. Moduli of Families of Curves for Conformal and Quasiconformal Mappings (Springer, Berlin;N. Y., 2002).
12. Граф С.Ю. Аналог леммы Шварца для локально-квазиконформных автоморфизмов круга, Изв. вузов.Матем. (11), 87–92 (2014).
13. Hayman W.K., Wu J.M.G. Level sets of univalent functions, Comment. Math. Helv. 56 (3), 366–403 (1981).
14. Rohde S. On the theorem of Hayman and Wu, Proc. Amer. Math. Soc. 130, 387–394 (2002).
15. {O}yma K. Harmonic measure and conformal length, Proc. Amer. Math. Soc. 115 (3), 687–689 (1992).
16. {O}yma K. The Hayman–Wu Constant, Proc. Amer. Math. Soc. 119 (1), 337–338 (1993).
Review
For citations:
Graf S.Yu. On quasiinvariance of harmonic measure and Hayman-Wu theorem. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2024;(2):22-36. (In Russ.) https://doi.org/10.26907/0021-3446-2024-2-22-36
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