To the Sheremeta theorem concerning relations between the Maximal Term and the Maximum Modulus of an entire Dirichlet series

For an entire Dirichlet series $F(s)=\sum_{n=0}^{\infty} a_ne^{s\lambda_n}$, $s=\sigma+it$, $0\le\lambda_n\uparrow \infty$, necessary and sufficient condition on $a_n$ is established in order that $M(\sigma,F) < \mu (\sigma,F) h(\ln\mu (\sigma, F))$, $\sigma\ge\sigma_0$, where $M(\sigma,F) =\sup\{|F(\sigma +it)|:t\in{\Bbb R}\}$, $\mu (\sigma,F) =\max\{|a_n|\exp\{\sigma\lambda_n\}: n\ge 0\}$ and $h$ is a continuous function on $[0;+\infty)$ such that $h(r)\to +\infty$ $(r\to +\infty)$.