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Matematicheskie Zametki, 1996, Volume 60, Issue 1, Pages 58–65
DOI: https://doi.org/10.4213/mzm1803
(Mi mzm1803)
 

This article is cited in 1 scientific paper (total in 1 paper)

Borel transformations on Dirichlet spaces

V. V. Napalkov

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
Full-text PDF (172 kB) Citations (1)
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Abstract: We study the growth of an entire function $f$, whose Borel transform $\gamma_f$ is holomorphic outside a bounded convex region $D_f$ with boundary curvature bounded away from 0 and $\infty$. The function $\gamma_f$ is assumed to belong to the Dirichlet space, i.e., it satisfies
$$ \int_{\mathbb C\setminus D_f}|\gamma_f'(\xi)|^2dv(\xi)<\infty, $$
where $dv(\xi)$ is the area element. It is shown that for $\gamma_f$ to satisfy the above conditions, it is necessary and sufficient to have
$$ \int_0^{2\pi}\int_0^\infty|f(re^{i\varphi})|^2 e^{-2rh_f(\varphi)}r^{3/2}drd\varphi<\infty, $$
where $h_f(\varphi)\overset{\operatorname{def}}=\varlimsup_{r\to \infty}\bigl(\ln|f(re^{i\varphi})|\bigr)/r$, $\varphi\in [0,2\pi]$ is the growth indicatrix of $f$ satisfies the relation $0<m\le h''(\varphi)+h(\varphi)\le M<\infty$.
Received: 25.05.1994
English version:
Mathematical Notes, 1996, Volume 60, Issue 1, Pages 42–48
DOI: https://doi.org/10.1007/BF02308878
Bibliographic databases:
UDC: 517.5
Language: Russian
Citation: V. V. Napalkov, “Borel transformations on Dirichlet spaces”, Mat. Zametki, 60:1 (1996), 58–65; Math. Notes, 60:1 (1996), 42–48
Citation in format AMSBIB
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\by V.~V.~Napalkov
\paper Borel transformations on Dirichlet spaces
\jour Mat. Zametki
\yr 1996
\vol 60
\issue 1
\pages 58--65
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\crossref{https://doi.org/10.4213/mzm1803}
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\zmath{https://zbmath.org/?q=an:0897.30011}
\elib{https://elibrary.ru/item.asp?id=13241326}
\transl
\jour Math. Notes
\yr 1996
\vol 60
\issue 1
\pages 42--48
\crossref{https://doi.org/10.1007/BF02308878}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1996WE97100006}
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  • https://www.mathnet.ru/eng/mzm1803
  • https://doi.org/10.4213/mzm1803
  • https://www.mathnet.ru/eng/mzm/v60/i1/p58
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математические заметки Mathematical Notes
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