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Ufa Mathematical Journal, 2020, Volume 12, Issue 2, Pages 35–49
DOI: https://doi.org/10.13108/2020-12-2-35
(Mi ufa515)
 

This article is cited in 6 scientific papers (total in 6 papers)

Growth of subharmonic functions along line and distribution of their Riesz measures

A. E. Salimova, B. N. Khabibullin

Bashkir State University, Zaki Validi str. 32, 450076, Ufa, Russia
References:
Abstract: Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ be two subharmonic functions on a complex plane $\mathbb C$ with Riesz measures $\nu_u$ and $\mu_M$, respectively, such that $u(z)\leq O(|z|)$ and $M(z)\leq O(|z|)$ as $z\to \infty$, and $q$ is some positive continuous function on a real axis $\mathbb{R}$, and ${\rm mes}$ is a linear Lebesgue measure on $\mathbb{R}$. We assume that the following condition for the growth of function $u$ along the imaginary axis $i\mathbb{R}$ of the form
$$ u(iy)\leq \frac{1}{2\pi}\int\limits_0^{2\pi}M\bigl(iy+q(y)e^{i\theta}\bigr)\,{\rm d}\theta +q(y) \quad\text{for all } y\in \mathbb{R}\setminus E, $$
where $E\subset \mathbb{R}$ is some small set, for instance, ${\rm mes}\bigl(E\cap [-r,r]\bigr)\leq q(r)$ as $r\geq 0$. Under such restrictions for the function $u$ it is natural to expect that the Riesz measure $\nu_u$ is in some sense majorized by the Riesz measure $\mu_M$ of the function $M$ or by integral characteristics of the function $M$. We provide a rigorous quantitative form of such majorizing. The need in such estimates arises naturally in the theory of entire functions in its applications to the completeness issues of exponential systems, analytic continuation, etc. Our results are formulated in terms of special logarithmic characteristics of measures $\nu_u$ and $\mu_M$ arisen earlier in classical works by P. Malliavin, L.A. Rubel and other for sequences of points and also in terms of special logarithmic characteristics of the behavior of the function $M$ along the imaginary axis and of the function $q$ along the real axis. The obtained results are new also for distribution of the zeroes of entire functions of exponential type under restrictions for the growth of such function along a line. The latter is demonstrated by a new uniqueness theorem for entire functions of exponential type employing so-called logarithmic block-densities of the distribution of the points on the complex plane.
Keywords: subharmonic function of a finite type, Riesz measure, entire function of exponential type, distribution of zeroes, uniqueness theorem.
Funding agency Grant number
Russian Science Foundation 18-11-00002
The research is financially supported by the grant of Russian Science Foundation (project no. 18-11-00002).
Received: 26.11.2019
Bibliographic databases:
Document Type: Article
UDC: 517.574 : 517.547.22
MSC: 31A05, 30D20, 30D15
Language: English
Original paper language: Russian
Citation: A. E. Salimova, B. N. Khabibullin, “Growth of subharmonic functions along line and distribution of their Riesz measures”, Ufa Math. J., 12:2 (2020), 35–49
Citation in format AMSBIB
\Bibitem{SalKha20}
\by A.~E.~Salimova, B.~N.~Khabibullin
\paper Growth of subharmonic functions along line and distribution of their Riesz measures
\jour Ufa Math. J.
\yr 2020
\vol 12
\issue 2
\pages 35--49
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\crossref{https://doi.org/10.13108/2020-12-2-35}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85097425077}
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  • https://doi.org/10.13108/2020-12-2-35
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  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Уфимский математический журнал
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    References:43
     
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