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

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.

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Bibliographic databases:
UDC: 517.574

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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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
2. K. P. Isaev, “On entire functions with given asymptotic behavior”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 12–30
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