Valiron-type and Valiron–Titchmarsh-type theorems for subharmonic functions of slow growth
UDC 517.53 Let $u$ be a function of zero order subharmonic in $\mathbb{R}^m,$ $m\geq 2,$ with Riesz measure $\mu$ on the negative semiaxis $Ox_1,$  n(r,u)=\mu\big(\big\{x\in\mathbb{R}^m\colon |x|\leq r\big\}\big),$ $d_m=m-2$ for $m\geq 3,$ $d_2=1,$ and $N(r,u)=d_m\displaystyle\int\nolim...
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| Datum: | 2026 |
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| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7251 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | UDC 517.53
Let $u$ be a function of zero order subharmonic in $\mathbb{R}^m,$ $m\geq 2,$ with Riesz measure $\mu$ on the negative semiaxis $Ox_1,$  n(r,u)=\mu\big(\big\{x\in\mathbb{R}^m\colon |x|\leq r\big\}\big),$ $d_m=m-2$ for $m\geq 3,$ $d_2=1,$ and $N(r,u)=d_m\displaystyle\int\nolimits_1^r \dfrac{n(t,u)}{t^{m-1}}\,dt.$Under the condition of slow growth of $N(r,u),$ we determine the asymptotics of $u(x)$ as $|x|=r\to+\infty.$ We also study the inverse relationship between the regular growth of $u$ and the behavior of $N(r,u)$ for $r\to+\infty.$ |
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| DOI: | 10.37863/umzh.v74i11.7251 |