|
This article is cited in 11 scientific papers (total in 11 papers)
Representation of subharmonic functions in a half-plane
K. G. Malyutina, N. Sadikb a V. N. Karazin Kharkiv National University
b İstanbul University
Abstract:
The theory of subharmonic functions of finite order is based to a considerable
extent on integral formulae. In the present paper representations are obtained for subharmonic
functions in the upper half-plane with more general growth $\gamma(r)$ than finite order.
The main result can be stated as follows. Let $\gamma(r)$ be a growth function such that
either $\ln\gamma(r)$ is a convex function of $\ln r$ or the lower order of $\gamma(r)$ is infinite. Then for each proper subharmonic function $v$ of growth $\gamma(r)$ there exist an unbounded set $\mathbf R$ of positive numbers and a family
$\{u_R:R\in\mathbf R\}$ of proper subharmonic functions in the upper
half-plane $\mathbb{C}_+$ such that
1) the full measures of the $u_R$ in the discs $|z|\leqslant R$
are equal to the full measure of the function $v$;
2) $v-u_R\rightrightarrows0$ uniformly on compact subsets of $\mathbb{C}_+$
as $R\to\infty$, $R\in\mathbf R$;
3) the function family
$\{u_R:R\in\mathbf R\}$ satisfies the growth constraints uniformly in $R$,
that is, $T(r,u_R)\leqslant A\gamma(Br)/r$, where $A$ and $B$ are constants
and $T(r,\,\cdot\,)$ is the growth characteristic.
Bibliography: 16 titles.
Received: 13.10.2006 and 06.04.2007
Citation:
K. G. Malyutin, N. Sadik, “Representation of subharmonic functions in a half-plane”, Mat. Sb., 198:12 (2007), 47–62; Sb. Math., 198:12 (2007), 1747–1761
Linking options:
https://www.mathnet.ru/eng/sm3774https://doi.org/10.1070/SM2007v198n12ABEH003904 https://www.mathnet.ru/eng/sm/v198/i12/p47
|
Statistics & downloads: |
Abstract page: | 538 | Russian version PDF: | 227 | English version PDF: | 26 | References: | 55 | First page: | 5 |
|