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






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


Mat. Sb. (N.S.), 1984, Volume 124(166), Number 2(6), Pages 272–279 (Mi msb2051)  

This article is cited in 12 scientific papers (total in 12 papers)

On the change in harmonic measure under symmetrization

V. N. Dubinin


Abstract: Let $D_\alpha$ be the unit disc cut along the segments $l_k=ż:\arg z=\alpha_k, r\leqslant|z|\leqslant1\}$, $k=0,1,…,n-1$ ($\alpha=(\alpha_0,\alpha_1,…,\alpha_{n-1})$, $0<r<1$), and let $\omega_\alpha$ be the harmonic measure of the set $\bigcup\limits_{k=0}^{n-1}l_k$ relative to the region $D_\alpha$ at the point $z=0$.
An affirmative solution is given of a problem of A. A. Gonchar:
$$ \omega_\alpha\leqslant\omega_{\alpha^*} $$
where $\alpha^*=(0,\frac{2\pi}n,…,\frac{2\pi}n(n-1))$. Equality holds only when $D_\alpha$ coincides with $D_{\alpha^*}$ to within a rotation about the origin. The proof is based on a property of certain condensers under dissymmetrization, i.e. under a transformation of symmetric condensers into nonsymmetric ones.
Bibliography: 4 titles.

Full text: PDF file (446 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1985, 52:1, 267–273

Bibliographic databases:

UDC: 517.54
MSC: 30C85, 31A15
Received: 13.05.1983

Citation: V. N. Dubinin, “On the change in harmonic measure under symmetrization”, Mat. Sb. (N.S.), 124(166):2(6) (1984), 272–279; Math. USSR-Sb., 52:1 (1985), 267–273

Citation in format AMSBIB
\Bibitem{Dub84}
\by V.~N.~Dubinin
\paper On the change in harmonic measure under symmetrization
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 124(166)
\issue 2(6)
\pages 272--279
\mathnet{http://mi.mathnet.ru/msb2051}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=746071}
\zmath{https://zbmath.org/?q=an:0571.30025|0548.30017}
\transl
\jour Math. USSR-Sb.
\yr 1985
\vol 52
\issue 1
\pages 267--273
\crossref{https://doi.org/10.1070/SM1985v052n01ABEH002888}


Linking options:
  • http://mi.mathnet.ru/eng/msb2051
  • http://mi.mathnet.ru/eng/msb/v166/i2/p272

    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

    This publication is cited in the following articles:
    1. Baernstein A., “Dubinins Symmetrization Theorem”, Lect. Notes Math., 1275 (1987), 23–30  crossref  mathscinet  zmath  isi
    2. Baernstein A., “Ahlfors and Conformal Invariants”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 13:3 (1988), 289–312  crossref  mathscinet  isi
    3. Haliste K., “On an Extremal Configuration for Capacity”, Ark. Mat., 27:1 (1989), 97–104  crossref  mathscinet  zmath  isi
    4. V. N. Dubinin, “Symmetrization in the geometric theory of functions of a complex variable”, Russian Math. Surveys, 49:1 (1994), 1–79  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. A. Yu. Solynin, “Harmonic measure of radial line segments and symmetrization”, Sb. Math., 189:11 (1998), 1701–1718  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Betsakos, D, “Geometric theorems and problems for harmonic measure”, Rocky Mountain Journal of Mathematics, 31:3 (2001), 773  crossref  isi  elib
    7. [Anonymous], “Moduli on Teichmüller Spaces”, Moduli of Families of Curves for Conformal and Quasconformal Mapping, Lecture Notes in Mathematics, 1788, Springer-Verlag Berlin, 2002, 175–206  isi
    8. Bucur D., Buttazzo G., Varchon N., “On the Problem of Optimal Cutting”, SIAM J. Optim., 13:1 (2002), 157–167  crossref  mathscinet  zmath  isi
    9. G. V. Kuz'mina, “Gennadii Mikhailovich Goluzin and geometric function theory”, St. Petersburg Math. J., 18:3 (2007), 347–372  mathnet  crossref  mathscinet  zmath  elib
    10. Dixit, A, “Monotonicity of quotients of theta functions related to an extremal problem on harmonic measure”, Journal of Mathematical Analysis and Applications, 336:2 (2007), 1042  crossref  isi  elib
    11. Arcozzi N., “Capacity of Shrinking Condensers in the Plane”, J. Funct. Anal., 263:10 (2012), 3102–3116  crossref  mathscinet  zmath  isi
    12. J. Math. Sci. (N. Y.), 222:5 (2017), 645–689  mathnet  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:325
    Full text:119
    References:44
    First page:2

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