On the representation of analytic functions in a closed convex region by a Dirichlet series

For a closed bounded convex region $\overline D$ we give conditions on $\{\lambda_k\}$ for which any function $f(z)$, analytic in the open region $D$ and continuous together with its first two derivatives on $\overline D$, can be expanded in the closed region $\overline D$ in an absolutely convergent Dirichlet series with exponents $\{\lambda_k\}$ ($k\geqslant1$).