On Dirichlet series similar to Hadamard compositions in half-plane

Abstract

Let $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ and $F_j(s)=\sum\limits_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\},$ $j=\overline{1,p},$ be Dirichlet series with exponents $0\le\lambda_n\uparrow+\infty,$ $n\to\infty,$ and the abscissas of absolutely convergence equal to $0$. The function $F$ is called Hadamard composition of the genus $m\ge 1$ of the functions $F_j$ if $a_n=P(a_{n,1},\dots ,a_{n,p})$, where $$P(x_1,\dots ,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1\dots\, k_p}x_1^{k_1}\cdots x_p^{k_p}$$ is a homogeneous polynomial of degree $m$. In terms of generalized orders and convergence classes the connection between the growth of the functions $F_j$ and the growth of the Hadamard composition $F$ of the genus $m\ge 1$ of $F_j$ is investigated. The pseudostarlikeness and pseudoconvexity of the Hadamard composition of the genus $m\ge 1$ are studied.