Relative growth of Hadamard compositions of Dirichlet series absolutely convergent in a half-plane
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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2025-03-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/541 |
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| Summary: | 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. |
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| ISSN: | 1027-4634 2411-0620 |