Relative growth of Hadamard compositions of Dirichlet series absolutely convergent in a half-plane - O.M. Mulyava; M. M. Sheremeta; Yu.S. Trukhan (Математичні Студії, 2025)
Let $Λ=(λ_n)$ be a positive sequence increasing to $+∞$ and $S(Λ,A)$ be a class of Dirichlet series $F(s)=\sum\limits_{n=1}^{∞}a_n\exp \{sλ_n\}$ with the abscissa of absolute convergence $A\in (-∞,\,+∞]$. The function $F$ is called Hadamard composition of the genus $m\ge 1$ of the functions $F_j(s)=\sum\limits_{n=0}^{∞}a_{n,j} \exp \{sλ_n\}$ $(j=1,2,…,p)$, if $a_n=\sum\limits_{k_1+…+k_p=m}c_{k_1...k_p}a_{n,1}^{k_1}·...· a_{n,p}^{k_p}$ for all $n$. The growth of the function $F\in S(Λ,0)$ with respect to a function $G(s)=\sum\limits_{n=1}^{∞}g_n\exp\{sλ_n\}\in S(Λ,+∞)$ is identified with the growth of the function $M^{-1}_G(M_F(σ))$ as $σ\uparrow 0$, where $M_F(σ)=\sup\{|F(σ+it)|:\,t\in{\Bbb R}\}$. The dependence of the growth of a function $M^{-1}_G(M_F(σ))$ on the growth of functions $M^{-1}_G(M_{F_j}(σ))$ is studied in terms of generalized orders and generalized convergence classes.