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Mat. Sb., 2015, Volume 206, Number 1, Pages 39–68 (Mi msb8276)  

This article is cited in 7 scientific papers (total in 7 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

Abstract: 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.

Keywords: conformal mapping, fixed point, evolution family, angular derivative, rotation theorem.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00434-а
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 12-01-00434-a).


DOI: https://doi.org/10.4213/sm8276

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English version:
Sbornik: Mathematics, 2015, 206:1, 33–60

Bibliographic databases:

UDC: 517.54
MSC: Primary 30C55, 30C75; Secondary 30C35, 30D05, 39B12, 39B32
Received: 12.08.2013 and 20.10.2013

Citation: 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
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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  elib
    2. V. V. Goryainov, “Holomorphic mappings of the unit disc into itself with two fixed points”, Sb. Math., 208:3 (2017), 360–376  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. J. Koch, S. Schleißinger, “Three value ranges for symmetric self-mappings of the unit disc”, Proc. Amer. Math. Soc., 145:4 (2017), 1747–1761  crossref  mathscinet  zmath  isi  scopus
    4. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    5. P. Gumenyuk, “Parametric representation of univalent functions with boundary regular fixed points”, Constr. Approx., 46:3 (2017), 435–458  crossref  mathscinet  zmath  isi  scopus
    6. 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  crossref  mathscinet  isi  scopus
    7. 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  crossref  mathscinet  zmath  isi  scopus
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