Exponential series for functions with specified growth near the boundary

Let $D$ be a bounded convex region and $F(z)$ a function analytic in $D$ and satisfying
\begin{equation} |F(z)|<\exp\biggl[\biggl(\frac1r\biggr)^{\rho+\varepsilon}\biggr],\qquad r=\rho(z,\partial D),\qquad r<r_0(\varepsilon),\quad\forall\varepsilon>0. \end{equation}
This paper considers the question of expanding $F(z)$ in $D$ in an exponential series for which the sum of the series of moduli of the terms satisfies an inequality of the form (1). It is shown that such an expansion is always possible if $D$ is a convex polygon.
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