Borel transformations on Dirichlet spaces

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 $0<m\le h''(\varphi)+h(\varphi)\le M<\infty$.