Borel transformations on Dirichlet spaces

V. V. Napalkov

Matematičeskie zametki, Tome 60 (1996) no. 1, pp. 58-65

Résumé

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 $0$.