RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2019, Volume 19, Issue 3, Pages 258–279 (Mi isu806)  

Scientific Part
Mathematics

Value regions in classes of conformal mappings

D. V. Prokhorovab

a Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia
b Petrozavodsk State University, 33 Lenin St., Petrozavodsk 185910, Republic of Karelia, Russia

Abstract: The survey is devoted to most recent results in the value region problem over different classes of holomorphic univalent functions represented by solutions to the Loewner differential equations both in the radial and chordal versions. It is important also to present classical and modern solution methods and to compare their efficiency. More details are concerned with optimization methods and the Pontryagin maximum principle, in particular. A value region is the set $\{f(z_0)\}$ of all possible values for the functional $f\mapsto f(z_0)$ where $z_0$ is a fixed point either in the upper half-plane for the chordal case or in the unit disk for the radial case, and $f$ runs through a class of conformal mappings. Solutions to the Loewner differential equations form dense subclasses of function families under consideration. The coefficient value regions $\{(a_2,…,a_n):f(z)=z+\sum_{n=2}^{\infty}a_nz^n\}$, $|z|<1$, are the part of the field closely linked with extremal problems and the Bombieri conjecture about the structure of the coefficient region for the class $S$ in a neighborhood of the point $(2,…,n)$ corresponding to the Koebe function.

Key words: value region, Loewner equation, reachable set, boundary curve.

Funding Agency Grant Number
Russian Science Foundation 17-11-01229
This work was supported by the Russian Science Foundation (project No. 17-11-01229).


DOI: https://doi.org/10.18500/1816-9791-2019-19-3-258-279

Full text: PDF file (350 kB)

Bibliographic databases:

UDC: 517.54
Received: 07.04.2018
Accepted:12.05.2019
Language:

Citation: D. V. Prokhorov, “Value regions in classes of conformal mappings”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 19:3 (2019), 258–279

Citation in format AMSBIB
\Bibitem{Pro19}
\by D.~V.~Prokhorov
\paper Value regions in classes of conformal mappings
\jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.
\yr 2019
\vol 19
\issue 3
\pages 258--279
\mathnet{http://mi.mathnet.ru/isu806}
\crossref{https://doi.org/10.18500/1816-9791-2019-19-3-258-279}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000487123700002}
\elib{http://elibrary.ru/item.asp?id=39542327}


Linking options:
  • http://mi.mathnet.ru/eng/isu806
  • http://mi.mathnet.ru/eng/isu/v19/i3/p258

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
    Number of views:
    This page:11
    Full text:3

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019