On entire Dirichlet series similar to Hadamard compositions

Abstract

A function $F(s)=\sum_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with $0\le\lambda_n\uparrow+\infty$ is called the Hadamard composition of the genus $m\ge 1$ of functions $F_j(s)=\sum_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where
$P(x_1,...,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}x_1^{k_1}\cdot...\cdot x_p^{k_p}$ is a homogeneous polynomial of degree $m\ge 1$. Let $M(\sigma,F)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$ and functions $\alpha,\,\beta$ be positive continuous and increasing to $+\infty$ on
$[x_0, +\infty)$. To characterize the growth of the function $M(\sigma,F)$, we use generalized order $\varrho_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\alpha(\ln\,M(\sigma,F))}{\beta(\sigma)}$, generalized type
$T_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\ln\,M(\sigma,F)}{\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}$
and membership in the convergence class defined by the condition
$\displaystyle \int_{\sigma_0}^{\infty}\frac{\ln\,M(\sigma,F)}{\sigma\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}d\sigma<+\infty.$

Assuming the functions $\alpha, \beta$ and $\alpha^{-1}(c\beta(\ln\,x))$ are slowly increasing for each $c\in (0,+\infty)$ and $\ln\,n=O(\lambda_n)$ as $n\to \infty$, it is proved, for example, that if the functions $F_j$ have the same generalized order $\varrho_{\alpha,\beta}[F_j]=\varrho\in (0,+\infty)$ and the types $T_{\alpha,\beta}[F_j]=T_j\in [0,+\infty)$, $c_{m0...0}=c\not=0$, $|a_{n,1}|>0$ and $|a_{n,j}|= o(|a_{n,1}|)$ as $n\to\infty$ for $2\le j\le p$, and $F$ is the Hadamard composition of genus
$m\ge 1$ of the functions $F_j$ then $\varrho_{\alpha,\beta}[F]=\varrho$ and
$\displaystyle T_{\alpha,\beta}[F]\le \sum_{k_1+\dots+k_p=m}(k_1T_1+...+k_pT_p).$
It is proved also that $F$ belongs to the generalized convergence class if and only if
all functions $F_j$ belong to the same convergence class.