Geodesic
Exponential series for functions with specified growth near the boundary
Izvestiya. Mathematics , Tome 17 (1981) no. 3, pp. 505-521.

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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(\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. Bibliography: 2 titles. 
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     author = {A. F. Leont'ev},
     title = {Exponential series for functions with specified growth near the boundary},
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A. F. Leont'ev. Exponential series for functions with specified growth near the boundary. Izvestiya. Mathematics , Tome 17 (1981) no. 3, pp. 505-521. http://geodesic.mathdoc.fr/item/IM2_1981_17_3_a3/

[1] Leontev A. F., Ryady eksponent, Nauka, M., 1976 | MR

[2] Aronszajn N., “Sur les décompositions des fonctions analytiques uniformes et sur leurs applications”, Acta Math., 65:1–2 (1935), 1–156 | DOI | MR | Zbl