This article is cited in 9 scientific papers (total in 9 papers)
Evolution families of conformal mappings with fixed points and the Löwner-Kufarev equation
V. V. Goryainovab
a Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
b The Volzhsky Institute of Humanities
The paper is concerned with evolution families of conformal mappings of the unit disc to itself that fix an interior point and a boundary point. Conditions are obtained for the evolution families to be differentiable, and an existence and uniqueness theorem for an evolution equation is proved. A convergence theorem is established which describes the topology of locally uniform convergence of evolution families in terms of infinitesimal generating functions. The main result in this paper is the embedding theorem which shows that any conformal mapping of the unit disc to itself with two fixed points can be embedded into a differentiable evolution family of such mappings. This result extends the range of the parametric method in the theory of univalent functions. In this way the problem of the mutual change of the derivative at an interior point and the angular derivative at a fixed point on the boundary is solved for a class of mappings of the unit disc to itself. In particular, the rotation theorem is established for this class of mappings.
Bibliography: 27 titles.
conformal mapping, fixed point, evolution family, angular derivative, rotation theorem.
|Russian Foundation for Basic Research
|This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 12-01-00434-a).
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Sbornik: Mathematics, 2015, 206:1, 33–60
MSC: Primary 30C55, 30C75; Secondary 30C35, 30D05, 39B12, 39B32
Received: 12.08.2013 and 20.10.2013
V. V. Goryainov, “Evolution families of conformal mappings with fixed points and the Löwner-Kufarev equation”, Mat. Sb., 206:1 (2015), 39–68; Sb. Math., 206:1 (2015), 33–60
Citation in format AMSBIB
\paper Evolution families of conformal mappings with fixed points and the L\"owner-Kufarev equation
\jour Mat. Sb.
\jour Sb. Math.
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V. V. Goryainov, “Golomorfnye otobrazheniya edinichnogo kruga s konechnymi uglovymi proizvodnymi”, Geometricheskii analiz i ego prilozheniya, Materialy III Mezhdunarodnoi shkoly-konferentsii, Volgogradskii gosudarstvennyi universitet, Institut matematiki im. S.L. Soboleva Sibirskogo otdeleniya RAN, 2016, 62–65
V. V. Goryainov, “Holomorphic mappings of the unit disc into itself with two fixed points”, Sb. Math., 208:3 (2017), 360–376
J. Koch, S. Schleißinger, “Three value ranges for symmetric self-mappings of the unit disc”, Proc. Amer. Math. Soc., 145:4 (2017), 1747–1761
O. S. Kudryavtseva, “Analog of the Löwner–Kufarev Equation for the Semigroup of Conformal Mappings of the Disk into Itself with Fixed Points and Invariant Diameter”, Math. Notes, 102:2 (2017), 289–293
P. Gumenyuk, “Parametric representation of univalent functions with boundary regular fixed points”, Constr. Approx., 46:3 (2017), 435–458
V. V. Goryainov, “Some inequalities for holomorphic self-maps of the unit disc with two fixed points”, Complex analysis and dynamical systems VII, Contemp. Math., 699, Amer. Math. Soc., Providence, RI, 2017, 129–136
P. Gumenyuk, D. Prokhorov, “Value regions of univalent self-maps with two boundary fixed points”, Ann. Acad. Sci. Fenn. Math., 43:1 (2018), 451–462
D. V. Prokhorov, “Value regions in classes of conformal mappings”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 19:3 (2019), 258–279
O. S. Kudryavtseva, A. P. Solodov, “Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points”, Sb. Math., 210:7 (2019), 1019–1042
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