Relative growth of Hadamard compositions of Dirichlet series absolutely convergent in a half-plane

Abstract

Let $\Lambda=(\lambda_n)$ be a positive sequence increasing to $+\infty$ and $S(\Lambda,A)$ be a class of Dirichlet series $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp \{s\lambda_n\}$ with the abscissa of absolute  convergence $A\in (-\infty,\,+\infty]$. The function $F$ is called Hadamard composition of the genus $m\ge 1$ of the functions $F_j(s)=\sum\limits_{n=0}^{\infty}a_{n,j} \exp \{s\lambda_n\}$ $(j=1,2,\dots,p)$, if $a_n=\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}a_{n,1}^{k_1}\cdot...\cdot a_{n,p}^{k_p}$ for all $n$. The growth of the function $F\in S(\Lambda,0)$ with respect to a function $G(s)=\sum\limits_{n=1}^{\infty}g_n\exp\{s\lambda_n\}\in S(\Lambda,+\infty)$ is identified with the growth of the function $M^{-1}_G(M_F(\sigma))$ as  $\sigma\uparrow 0$, where $M_F(\sigma)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$. The dependence of the growth of a function $M^{-1}_G(M_F(\sigma))$ on the growth of functions $M^{-1}_G(M_{F_j}(\sigma))$ is studied in terms of generalized orders and generalized convergence classes.