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 125(167), Number 2(10), Pages 181–198 (Mi msb2078)  

This article is cited in 11 scientific papers (total in 11 papers)

On separation of singularities of meromorphic functions

V. I. Danchenko


Abstract: Let $E$ be an arbitrary bounded proper continuum on $\overline{\mathbf C}$, $\lambda$ a finite collection of pairwise distinct domains that are components of $\overline{\mathbf C}\setminus E$, $f$ a function meromorphic in each domain $G\in\lambda$ and continuous in some neighborhood of $E$, $f_\lambda$ the sum of the principal parts of the Laurent expansions of $f$ with respect to its poles in the union of the domains in $\lambda$, and $n_\lambda$ the degree of the rational function $f_\lambda$. If all the domains $G\in\lambda$ are bounded, then $\|f_\lambda\|_{C(E)}\leqslant\mathrm{const}\cdot n_\lambda\|f\|_{C(E)}$. If $E$ is a rectifiable curve $\Gamma$, then the total variation $\operatorname{Var}(f_\lambda,\Gamma)=\int_\Gamma|f_\lambda'(\zeta)|\cdot|d\zeta|$ of $f_\lambda$ along $\Gamma$ satisfies $\operatorname{Var}(f_\lambda,\Gamma)\leqslant\mathrm{const}\cdot n_\lambda\ln^3(en_\lambda)\|f\|_{C(\Gamma)}V(\Gamma)$, where $V(\Gamma)$ is the supremum of the set $\{\operatorname{Var}(r,\Gamma)\}$ of total variations along $\Gamma$ of all the partial fractions $r(z)=a/(bz+c)$ with $\|r\|_{C(\Gamma)}=1$.
Bibliography: 11 titles.

Full text: PDF file (930 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1986, 53:1, 183–201

Bibliographic databases:

UDC: 517.53
MSC: 30A10, 30C99, 30D30
Received: 19.09.1983

Citation: V. I. Danchenko, “On separation of singularities of meromorphic functions”, Mat. Sb. (N.S.), 125(167):2(10) (1984), 181–198; Math. USSR-Sb., 53:1 (1986), 183–201

Citation in format AMSBIB
\Bibitem{Dan84}
\by V.~I.~Danchenko
\paper On separation of singularities of meromorphic functions
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 125(167)
\issue 2(10)
\pages 181--198
\mathnet{http://mi.mathnet.ru/msb2078}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=764477}
\zmath{https://zbmath.org/?q=an:0611.30032}
\transl
\jour Math. USSR-Sb.
\yr 1986
\vol 53
\issue 1
\pages 183--201
\crossref{https://doi.org/10.1070/SM1986v053n01ABEH002916}


Linking options:
  • http://mi.mathnet.ru/eng/msb2078
  • http://mi.mathnet.ru/eng/msb/v167/i2/p181

    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. E. P. Dolzhenko, V. I. Danchenko, “Mapping of sets of finite $\alpha$-measure by rational functions”, Math. USSR-Izv., 31:3 (1988), 621–633  mathnet  crossref  mathscinet  zmath
    2. A. A. Gonchar, L. D. Grigoryan, “On an estimate of the components of bounded analytic functions”, Math. USSR-Sb., 60:2 (1988), 291–295  mathnet  crossref  mathscinet  zmath  isi
    3. E. P. Dolzhenko, V. I. Danchenko, “Mapping sets of locally finite length by a rational function”, Proc. Steklov Inst. Math., 180 (1989), 120–123  mathnet  zmath
    4. V. I. Danchenko, “Several integral estimates of the derivatives of rational functions on sets of finite density”, Sb. Math., 187:10 (1996), 1443–1463  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. D. Ya. Danchenko, “On interpolation in the classes $E^p$”, Math. Notes, 66:3 (1999), 388–392  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. L. Lukashov, “A Bernstein-type inequality for derivatives of rational functions on two segments”, Math. Notes, 66:4 (1999), 415–420  mathnet  crossref  crossref  mathscinet  isi
    7. V. I. Danchenko, “Estimates of Green potentials. Applications”, Sb. Math., 194:1 (2003), 63–88  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. V. I. Danchenko, “Estimates of derivatives of simplest fractions and other questions”, Sb. Math., 197:4 (2006), 505–524  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. V. I. Danchenko, A. E. Dodonov, “Estimates for exponential sums. Applications”, J Math Sci, 2012  crossref
    10. V. I. Danchenko, “Cauchy and Poisson formulas for polyanalytic functions and applications”, Russian Math. (Iz. VUZ), 60:1 (2016), 11–21  mathnet  crossref  isi
    11. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49  mathnet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:274
    Full text:59
    References:20

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