Integrals and indicators of subharmonic functions.~II

K. G. Malyutin ; M. V. Kabanko ; T. I. Malyutina

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 236-269.

Résumé

This article is a direct continuation of the article [18]. At the beginning of the paper, we present those known results of the general theory of subharmonic functions, which are used in what follows. In the definition of a semi-formal order, it is required the existence of the real numbers $\delta>0$, $q\in(0,1)$ and $N$ such that an arbitrary domain $D(R,q,\delta)$ contains a point $z$ which satisfies $v(z)>NV(|z|)$ condition. This condition we call the Levin's condition. We weak this condition and require only that the necessary point $z$ be contained not in an arbitrary domain $D(R,q,\delta)$, but only for $R=R_n$, where $R_n$ is a sequence that converges to infinity. Functions that satisfy this weakened condition are called functions that locally satisfy Levin's condition. Our result related to this class of functions is that on the set $E=\left\{z: \arg z\in (0,\pi), |z|\in\bigcup\limits_{n=1}^{\infty}\biggl[qR_n,R_n/q\biggr]\right\}$ the function $v(z)$ behaves like a function of the semi-formal order $\rho(r)$. We also note assertions 1 and 3 of the theorem 2 associated with estimates of the full measure of sets that are not subsets of the set $E$. The main result is the theorem 7. In assertion 3 of this theorem we fix a new property of subharmonic functions of finite order, which, together with the property formulated in theorem 3, can be considered as one of the most important properties that distinguish subharmonic functions in the class of all functions. If shift the Riesz measures of a subharmonic function $v(z)$, that are located inside some angle $S$ of size $2\Delta$, to the boundary of this angle and to denote by $v_{\Delta}(z)$ a subharmonic function with a shifted Riesz measure, then the resulting function can be regarded as an approximation of the function $v(z)$. This approximation is a harmonic function inside $S$. We obtain an integral estimate for the modulus of the difference $|v(z)-v_{\Delta}(z)|$, which is qualitatively better than the estimate of the corresponding integral for $|v(z)| $. A special case when the lower indicator of the function $v$ is finite on the bisector of the angle $S$ is investigated.

Détail

Keywords: subharmonic function, semi-formal order, local Levin's condition, Riesz measure, function indicator.

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