The order of growth of an exponential series near the boundary of the convergence domain

For a class of analytic functions in a bounded convex domain G G that admit an exponential series expansion in D D , the behavior of the coefficients of this expansion is studied in terms of the growth order near the boundary ∂ G \partial G . In the case where G G has a smooth boundary, unimprovable two-sided estimates are established for the order via characteristics depending only on the exponents of the exponential series and the support function of G G . As a consequence, a formula is obtained for the growth of the exponential series via the coefficients and the support function of the convergence domain  G G .