Entire Dirichlet series with monotonous coefficients and logarithmic h-measure

Abstract

Let $F$ be an entire function represented by absolutely convergent for all $z\in \mathbb{C}$ Dirichlet series of the form $F(z)={\sum }_{n=0}^{+\infty }{a}_n{e}^{z{\lambda }_n},$\ where a sequence $({\lambda }_n)$ such that ${\lambda }_n\in \mathbb{R}(n\ge 0)$, ${\lambda }_n\ne {\lambda }_k$ for any $n\ne k$ and $(\forall n\ge 0):0\le {\lambda }_n<\beta :=\sup \{{\lambda }_j:j\ge 0\}\le +\infty .$ {Let $h$ be non-decrease positive continuous function on $[0,+\infty )$ and $\Phi$ increase positive continuous on $[0,+\infty )$ 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 $x\to +\infty$\ outside some set $E$ of finite logarithmic $h$-measure uniformly in $y\in \mathbb{R}$.