Geodesic
On quasiinvariance of harmonic measure and Hayman–Wu theorem
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2024), pp. 22-36 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article is devoted to the definition and properties of the class of diffeomorphisms of the unit disk $\mathbb{D}=\{z: |z|<1\}$ on the complex plane $\mathbb{C}$ 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.
Keywords: harmonic measure, quasiconformal mapping, pseudohyperbolic metric, quasiisometry, Hayman–Wu theorem. 
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     title = {On quasiinvariance of harmonic measure and {Hayman{\textendash}Wu} theorem},
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S. Yu. Graf. On quasiinvariance of harmonic measure and Hayman–Wu theorem. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2024), pp. 22-36. http://geodesic.mathdoc.fr/item/IVM_2024_2_a1/

[1] Nevanlinna R., “Das harmonische mass von punktmengen und seine anwendung in der funktionentheorie”, Comptes rendus du huitéme Congrés des mathématiciens Scandinaves (Stockholm, 1934), 116–133 | Zbl

[2] Goluzin G.M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[3] Garnett J.B., Marshall D.E., Harmonic measure, Cambridge Univ. Press, 2005 | MR | Zbl

[4] Ahlfors L.V., Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York, 1973 | MR | Zbl

[5] Beurling A., The Collected Works of Arne Beurling. Complex analysis, v. 1, eds. L. Carleson, et al., Birkhäuser, 1989 | MR | Zbl

[6] Betsakos D., Solynin A.Yu., “Extensions of Beurling's shove theorem for harmonic measure”, Complex Variables and Elliptic Equat., 42:1 (2000), 57–65 | MR

[7] Kelingos J. A., “Characterizations of quasiconformal mappings in terms of harmonic and hyperbolic measure”, Ann. Acad. Sci. Fenn. Ser. A I, 368 (1965), 1–16 | MR

[8] Graf S.Yu., “On distortion of harmonic measure under locally quasiconformal mappings”, Complex Variables and Elliptic Equat., 66:9 (2020), 1550–1564 | DOI | MR

[9] Ahlfors L.V., Lectures on Quasiconformal Mappings, University Lecture Ser., 38, Second Edition, AMS, 2006 | DOI | MR | Zbl

[10] Anderson G.D., Vamanamurthy M.K., Vuorinen M., Conformal Invariants, Inequalities an Quasiconformal Maps, Wiley, New York, 1997 | MR

[11] Vasil'ev A., Moduli of Families of Curves for Conformal and Quasiconformal Mappings, Springer, Berlin–N. Y., 2002 | MR | Zbl

[12] Graf S.Yu., “Analog lemmy Shvartsa dlya lokalno-kvazikonformnykh avtomorfizmov kruga”, Izv. vuzov. Matem., 2014, no. 11, 87–92 | Zbl

[13] Hayman W.K., Wu J.M.G., “Level sets of univalent functions”, Comment. Math. Helv., 56:3 (1981), 366–403 | DOI | MR | Zbl

[14] Rohde S., “On the theorem of Hayman and Wu”, Proc. Amer. Math. Soc., 130 (2002), 387–394 | DOI | MR | Zbl

[15] $\O$yma K., “Harmonic measure and conformal length”, Proc. Amer. Math. Soc., 115:3 (1992), 687–689 | DOI | MR

[16] $\O$yma K., “The Hayman-Wu Constant”, Proc. Amer. Math. Soc., 119:1 (1993), 337–338 | DOI | MR