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\nolimits_{n=0}^{+\infty} a_{n}e^{z\lambda_{n}},$\ where a sequence $(\lambda_n)$ such that $\lambda_n\in\mathbb{R}\ \ (n\geq0)$, $\lambda_n\not=\lambda_k$ for any $n\not=k$ and $(\forall n\geq 0):\ 0\leq\lambda_n<\beta:=\sup\{\lambda_j:\ j\geq0\}\leq +\infty.$ {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 $y\in\mathbb{R}$.