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Ufimsk. Mat. Zh., 2010, Volume 2, Issue 3, Pages 46–53 (Mi ufa62)  

This article is cited in 2 scientific papers (total in 2 papers)

On the accuracy of asymptotic approximation of subharmonic functions by the logarithm of the modulus of an entire function

V. I. Lutsenkoa, R. S. Yulmukhametovb

a Bashkir State University, Ufa, Russia
b Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia

Abstract: We study the degree of possible accuracy of the asymptotic approximation of subharmonic functions by the logarithm of the modulus of an entire function. It is proved that if a subharmonic function $u$ is twice differentiable and satisfies the condition
$$ m\le|z|\Delta u(z)\le M,\qquad|z|>0, $$
where $M,m>0$, then approximation with the accuracy $q\ln|z|+O(1)$ with the constant $q\in(0,\frac14)$ is possible only outside sets of non-$C_0$-set. On the other hand, it is shown that approximation with the accuracy to $q\ln|z|+O(1)$ with the constant $q\ge\frac14$ is possible outside sets, that can be covered by circles $B(z_k,r_k)$ so that
$$ \sum_{|z_k|\le R}r_k=O(R^{\frac34-q}) $$
when $q\in[\frac14,\frac34]$ and
$$ \sum_{|z_k|\ge R}r_k=O(R^{\frac34-q}) $$
when $q>\frac34$. In particular, these sets are $C_0$-sets when $q>\frac14$. In the second case, the approximating function is the same for all $q\ge\frac14$, and this function is only a small modification of sine type functions, constructed by Yu. Lubarsky and M. Sodin.

Keywords: subharmonic functions, entire functions.

Full text: PDF file (359 kB)
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Bibliographic databases:
UDC: 517.574
Received: 03.07.2010

Citation: V. I. Lutsenko, R. S. Yulmukhametov, “On the accuracy of asymptotic approximation of subharmonic functions by the logarithm of the modulus of an entire function”, Ufimsk. Mat. Zh., 2:3 (2010), 46–53

Citation in format AMSBIB
\Bibitem{LutYul10}
\by V.~I.~Lutsenko, R.~S.~Yulmukhametov
\paper On the accuracy of asymptotic approximation of subharmonic functions by the logarithm of the modulus of an entire function
\jour Ufimsk. Mat. Zh.
\yr 2010
\vol 2
\issue 3
\pages 46--53
\mathnet{http://mi.mathnet.ru/ufa62}
\zmath{https://zbmath.org/?q=an:1240.30185}
\elib{http://elibrary.ru/item.asp?id=15240755}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. R. Bagautdinova, A. V. Lutsenko, V. I. Lutsenko, E. D. Shaimuratova, “Integralnye otsenki proizvodnykh analiticheskikh funktsii vne vypuklykh oblastei”, Ufimsk. matem. zhurn., 4:4 (2012), 13–21  mathnet  mathscinet
    2. K. P. Isaev, “On entire functions with given asymptotic behavior”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 12–30  mathnet  crossref  elib
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