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Sbornik: Mathematics, 2007, Volume 198, Issue 12, Pages 1747–1761
DOI: https://doi.org/10.1070/SM2007v198n12ABEH003904
(Mi sm3774)
 

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

Representation of subharmonic functions in a half-plane

K. G. Malyutina, N. Sadikb

a V. N. Karazin Kharkiv National University
b İstanbul University
References:
Abstract: The theory of subharmonic functions of finite order is based to a considerable extent on integral formulae. In the present paper representations are obtained for subharmonic functions in the upper half-plane with more general growth $\gamma(r)$ than finite order. The main result can be stated as follows. Let $\gamma(r)$ be a growth function such that either $\ln\gamma(r)$ is a convex function of $\ln r$ or the lower order of $\gamma(r)$ is infinite. Then for each proper subharmonic function $v$ of growth $\gamma(r)$ there exist an unbounded set $\mathbf R$ of positive numbers and a family $\{u_R:R\in\mathbf R\}$ of proper subharmonic functions in the upper half-plane $\mathbb{C}_+$ such that
1) the full measures of the $u_R$ in the discs $|z|\leqslant R$ are equal to the full measure of the function $v$;
2) $v-u_R\rightrightarrows0$ uniformly on compact subsets of $\mathbb{C}_+$ as $R\to\infty$, $R\in\mathbf R$;
3) the function family $\{u_R:R\in\mathbf R\}$ satisfies the growth constraints uniformly in $R$, that is, $T(r,u_R)\leqslant A\gamma(Br)/r$, where $A$ and $B$ are constants and $T(r,\,\cdot\,)$ is the growth characteristic.
Bibliography: 16 titles.
Received: 13.10.2006 and 06.04.2007
Russian version:
Matematicheskii Sbornik, 2007, Volume 198, Number 12, Pages 47–62
DOI: https://doi.org/10.4213/sm3774
Bibliographic databases:
UDC: 517.574
MSC: 31A05
Language: English
Original paper language: Russian
Citation: K. G. Malyutin, N. Sadik, “Representation of subharmonic functions in a half-plane”, Mat. Sb., 198:12 (2007), 47–62; Sb. Math., 198:12 (2007), 1747–1761
Citation in format AMSBIB
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\by K.~G.~Malyutin, N.~Sadik
\paper Representation of subharmonic functions in a~half-plane
\jour Mat. Sb.
\yr 2007
\vol 198
\issue 12
\pages 47--62
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\crossref{https://doi.org/10.4213/sm3774}
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\transl
\jour Sb. Math.
\yr 2007
\vol 198
\issue 12
\pages 1747--1761
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  • https://doi.org/10.1070/SM2007v198n12ABEH003904
  • https://www.mathnet.ru/eng/sm/v198/i12/p47
  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математический сборник Sbornik: Mathematics
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    Russian version PDF:227
    English version PDF:26
    References:55
    First page:5
     
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