On belonging of entire Dirichlet series to convergence class

Let $\Lambda=\{\lambda_n\}_{n=0}^{\infty}$ be a sequence of nonnegative numbers increasing to $+\infty$ and $S(\Lambda)$ be the class of entire Dirichlet series $F(s)=\sum_{n=0}^{\infty} a_ne^{s\lambda_n}$, $s=\sigma+it$. Put $M(\sigma,F)=\max\{|F(s)|:\Re s=\sigma\}$ and let $\mu(\sigma,F)=\max\{|a_n|\exp\{\sigma\lambda_n\}:n\ge 0\}$ be the maximal term of the function $F\in S(\Lambda)$. We prove that in order that $\int_0^{+\infty}e^{-\sigma\rho}\ln \mu(\sigma,F) d\sigma < \infty$ imply $\int_0^{+\infty}e^{-\sigma\rho}\ln M(\sigma,F) d\sigma < \infty$ for every $F\in S(\Lambda)$, it is necessary and sufficient that $\ln n=O(\lambda_n)$, $n\to\infty$.