Behavior of coefficients of series of exponents of finite order near the boundary

Let $G$ be a bounded convex domain with a smooth boundary in which a given system of exponents is not complete. For a class of analytic functions in $G$ that can be represented in $G$ by a series of exponents, we examine the behavior of coefficients of the series expansion in terms of the growth order near the boundary $\partial G$. We establish two-sided estimates for the order through characteristics depending only on the indices of the series of exponents and the supporting function of the domain (these estimates are strong). As a consequence, we obtain a formula for calculating the growth order through the coefficients.