Behavior of subharmonic functions of slow growth outside exclusive sets
UDC517.53 Let $v$ be a slowly growing function unbounded on $[0,\,+\infty),$ $u$ be subharmonic (in plane) function of zero order, $\mu$~be its Riesz measure, $n(t,u)=\mu(\{x\colon |x|\le t\}),$ $N(t,u)=\int_{1}^{t}n(\tau,u)/\tau d\tau,$ and $n(r,u)=O(v(r)),$ $r\to+\infty.$ &nbs...
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| Date: | 2024 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8157 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC517.53
Let $v$ be a slowly growing function unbounded on $[0,\,+\infty),$ $u$ be subharmonic (in plane) function of zero order, $\mu$~be its Riesz measure, $n(t,u)=\mu(\{x\colon |x|\le t\}),$ $N(t,u)=\int_{1}^{t}n(\tau,u)/\tau d\tau,$ and $n(r,u)=O(v(r)),$ $r\to+\infty.$  A  set $E \in \mathbb{C}$ is called a $C_0^\beta$-set, $0 < \beta \le 1,$ if $E$ can be covered by a system of disks $K(a_n,r_n)=\{z\colon |z-a_n| < r_n\}$ such that $\sum_{|a_n| \le r} r_n^\beta = o(r^\beta),$ $r\to+\infty.$ Then, for every nondecreasing function  $\phi$ unbounded on $[0,\,+\infty),$  there exists a $C_0^\beta$-set $E$ such that \begin{equation*}u(z)=N(r,u)+o(\phi(r)v(r)),\qquad r=|z|\to+\infty,\quad z \notin E.\end{equation*} It is shown that, in this asymptotic formula, the remainder term $o(\phi(r)v(r))$ cannot be changed by $O(v(r)).$ |
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| DOI: | 10.3842/umzh.v76i7.8157 |