This article is cited in 3 papers

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

V. I. Lutsenko^a, R. S. Yulmukhametov<sup>b

a</sup> 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 $q\in\bigl[\frac14,\frac34\bigr]$ 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.

UDC: 517.574

Received: 03.07.2010

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