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This article is cited in 1 scientific paper (total in 1 paper)
Borel transformations on Dirichlet spaces
V. V. Napalkov Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
Abstract:
We study the growth of an entire function $f$, whose Borel transform $\gamma_f$ is holomorphic outside a bounded convex region $D_f$ with boundary curvature bounded away from 0 and $\infty$. The function $\gamma_f$ is assumed to belong to the Dirichlet space, i.e., it satisfies
$$
\int_{\mathbb C\setminus D_f}|\gamma_f'(\xi)|^2dv(\xi)<\infty,
$$
where $dv(\xi)$ is the area element. It is shown that for $\gamma_f$ to satisfy the above conditions, it is necessary and sufficient to have
$$
\int_0^{2\pi}\int_0^\infty|f(re^{i\varphi})|^2
e^{-2rh_f(\varphi)}r^{3/2}drd\varphi<\infty,
$$
where $h_f(\varphi)\overset{\operatorname{def}}=\varlimsup_{r\to \infty}\bigl(\ln|f(re^{i\varphi})|\bigr)/r$, $\varphi\in [0,2\pi]$ is the growth indicatrix of $f$ satisfies the relation $0<m\le h''(\varphi)+h(\varphi)\le M<\infty$.
Received: 25.05.1994
Citation:
V. V. Napalkov, “Borel transformations on Dirichlet spaces”, Mat. Zametki, 60:1 (1996), 58–65; Math. Notes, 60:1 (1996), 42–48
Linking options:
https://www.mathnet.ru/eng/mzm1803https://doi.org/10.4213/mzm1803 https://www.mathnet.ru/eng/mzm/v60/i1/p58
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