RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

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

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-à 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

Full text: PDF file (630 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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
\Bibitem{Gor15} \by V.~V.~Goryainov \paper Evolution families of conformal mappings with fixed points and the L\"owner-Kufarev equation \jour Mat. Sb. \yr 2015 \vol 206 \issue 1 \pages 39--68 \mathnet{http://mi.mathnet.ru/msb8276} \crossref{https://doi.org/10.4213/sm8276} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3354961} \zmath{https://zbmath.org/?q=an:06439407} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015SbMat.206...33G} \elib{http://elibrary.ru/item.asp?id=23421597} \transl \jour Sb. Math. \yr 2015 \vol 206 \issue 1 \pages 33--60 \crossref{https://doi.org/10.1070/SM2015v206n01ABEH004445} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000351527000004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84925238085} 

• http://mi.mathnet.ru/eng/msb8276
• https://doi.org/10.4213/sm8276
• http://mi.mathnet.ru/eng/msb/v206/i1/p39

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. V. V. Goryainov, “Holomorphic mappings of the unit disc into itself with two fixed points”, Sb. Math., 208:3 (2017), 360–376
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
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
5. P. Gumenyuk, “Parametric representation of univalent functions with boundary regular fixed points”, Constr. Approx., 46:3 (2017), 435–458
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
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
•  Number of views: This page: 2018 Full text: 525 References: 45 First page: 998