Entire Dirichlet series with monotonous coefficients and logarithmic h-measure

Let $F$ be an entire function represented by absolutely convergent for all $z\in \mathbb{C}$ Dirichlet series of the form $ F(z) = \sum\nolimits_{n=0}^{+\infty} a_{n}e^{z\lambda_{n}},$whereasequence$(\lambda_n)$suchthat$\lambda_n\in$\mathbb{R}$\ \ (n\geq0)$,$\lambda_n\not=\lambda_k$forany$n\not=k$and$(\forall n\geq 0):
0\leq\lambda_n<\beta:=\sup{\lambda_j:\ j\geq0}\leq +\infty.{Leth$ be non-decrease positive continuous function on $[0,+\mathrm{\infty })$ and $\mathrm{\Phi }$ increase positive continuous on $[0,+\mathrm{\infty })$ function.} In this paper we {find} the condition {on} $({\mu }_n)$ and $({\lambda }_n)$ {such that} the relation $F(x+iy)=(1+o(1))a_{\nu (x,F)}{e}^{(x+iy){\lambda }_{\nu (x,F)}}$ holds as $x\to +\mathrm{\infty }$\ outside some set $E$ of finite logarithmic $h$-measure uniformly in $y\in \mathbb{R}$.