Compositions of Dirichlet series similar to the Hadamard
compositions, and convergence classes

O. M. Mulyava; M. M. Sheremeta

doi:10.15330/ms.51.1.25-34

Let $(\lambda_n)$ be a positive sequence increasing to $+\infty$, $m\ge2$ and Dirichlet series $F_j(s)=\sum\nolimits_{n=0}^{\infty}a_{n,j} \exp \{s\lambda_n\}$ $(j=1,2,\dots,m)$ have the abscissa $A\in (-\infty,\,+\infty]$ of absolute convergence. We say that Dirichlet series $F(s)=\sum\nolimits_{n=0}^{\infty}a_{n} \exp \{s\lambda_n\}$ is similar to Hadamard compositions of of Dirichlet series $F_j$ if $a_n=w(a_{n,1},\,a_{n,2})$ for all $n$, where $w\colon\mathbb{C}^2\to\mathbb{C}$ is some function. Clearly, if $w(a_{n,1},\,a_{n,2})=a_{n,1}a_{n,2}$ then $F$ is the Hadamard composition of the functions $F_1$ and $F_2$. In the case $|a_n|\asymp\prod\nolimits_{j=1}^m |a_{n,j}|^{\omega_j}$ as $n\to +\infty$, where $\omega_j>0$ and $\sum\nolimits_{j=1}^m \omega_j=1$, it is investigated the belonging of $F$ to some convergence class with respect of the belonging to this class of functions $F_j$.

1. Calys E.G. A note on the order and type of integral functions// Riv. Mat. Univer. Parma (2). – 1964. – V.5. – P. 133–137.

2. Kulyavets’ L.V., Mulyava O.M. On the growth of a class of entire Dirichlet series// Carpathian Math. Publ. – 2014. – V.6, №2. – P. 300–309. (in Ukrainian)

3. Sheremeta M.N. Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion// Izv. Vyssh. Uchebn. Zaved. Mat. – 1967, – №2 – P. 100–108. (in Russian)

4. Kulyavets’ L.V., Mulyava O.M. On the growth of a class of Dirichlet series absolutely convergent in half–plane// Carpathian Math. Publ. – 2017. – V.9, №1. – P. 63–71.

5. Mulyava O.M. On convergence classes of Dirichlet series// Ukr. Math. Journ. – 1999. – V.51, №1. – P. 1485–1494. (in Ukrainian)

6. Leont’ev A.F. Series of exponents. – M.: Nauka, 1956. – 536 p. (in Russian)

7. Mulyava O.M. On belonging of entire Dirichlet series to a modified generalized convergence class// Mat. Stud. – 2018. – V.50, №2. – P. 135–142.

8. Sheremeta M.M. On two classes of positive functions and the belonging to them of main characteristics of entire functions// Mat. Stud. – 2003. – V.19, №1. – P. 75–82.

9. Seneta E. Regularly varying functions, Lecture Notes in Mathematics, V.508. – Berlin: Springer-Verlag – 1976.

10. Gal’ Yu.M. On the growth of analytic functions given by Dirichlet series absolutely convergent in half– plane. – Drogobych, 1980. – 39 p. Manuscript dep. in VINITI 16.09.1980, no. 4080–80 Dep. (in Russian)

11. Mulyava O.M. Convergence classes in the theory of Dirichlet series// Dopov. Nats. Akad. Nauk. Ukr. – 1999. – №3. – P. 35–39. (in Ukrainian)

	Compositions of Dirichlet series similar to the Hadamard compositions, and convergence classes
Author	O. M. Mulyava¹, M. M. Sheremeta² info@nuft.edu.ua¹, m_m_sheremeta@gmail.com² 1) Kyiv National University of Food Technologies; 2) Ivan Franko National University of Lviv
Abstract	Let $(\lambda_n)$ be a positive sequence increasing to $+\infty$, $m\ge2$ and Dirichlet series $F_j(s)=\sum\nolimits_{n=0}^{\infty}a_{n,j} \exp \{s\lambda_n\}$ $(j=1,2,\dots,m)$ have the abscissa $A\in (-\infty,\,+\infty]$ of absolute convergence. We say that Dirichlet series $F(s)=\sum\nolimits_{n=0}^{\infty}a_{n} \exp \{s\lambda_n\}$ is similar to Hadamard compositions of of Dirichlet series $F_j$ if $a_n=w(a_{n,1},\,a_{n,2})$ for all $n$, where $w\colon\mathbb{C}^2\to\mathbb{C}$ is some function. Clearly, if $w(a_{n,1},\,a_{n,2})=a_{n,1}a_{n,2}$ then $F$ is the Hadamard composition of the functions $F_1$ and $F_2$. In the case $\|a_n\|\asymp\prod\nolimits_{j=1}^m \|a_{n,j}\|^{\omega_j}$ as $n\to +\infty$, where $\omega_j>0$ and $\sum\nolimits_{j=1}^m \omega_j=1$, it is investigated the belonging of $F$ to some convergence class with respect of the belonging to this class of functions $F_j$.
Keywords	Dirichlet series; convergence class; Hadamard composition
DOI	doi:10.15330/ms.51.1.25-34
Reference	1. Calys E.G. A note on the order and type of integral functions// Riv. Mat. Univer. Parma (2). – 1964. – V.5. – P. 133–137. 2. Kulyavets’ L.V., Mulyava O.M. On the growth of a class of entire Dirichlet series// Carpathian Math. Publ. – 2014. – V.6, №2. – P. 300–309. (in Ukrainian) 3. Sheremeta M.N. Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion// Izv. Vyssh. Uchebn. Zaved. Mat. – 1967, – №2 – P. 100–108. (in Russian) 4. Kulyavets’ L.V., Mulyava O.M. On the growth of a class of Dirichlet series absolutely convergent in half–plane// Carpathian Math. Publ. – 2017. – V.9, №1. – P. 63–71. 5. Mulyava O.M. On convergence classes of Dirichlet series// Ukr. Math. Journ. – 1999. – V.51, №1. – P. 1485–1494. (in Ukrainian) 6. Leont’ev A.F. Series of exponents. – M.: Nauka, 1956. – 536 p. (in Russian) 7. Mulyava O.M. On belonging of entire Dirichlet series to a modified generalized convergence class// Mat. Stud. – 2018. – V.50, №2. – P. 135–142. 8. Sheremeta M.M. On two classes of positive functions and the belonging to them of main characteristics of entire functions// Mat. Stud. – 2003. – V.19, №1. – P. 75–82. 9. Seneta E. Regularly varying functions, Lecture Notes in Mathematics, V.508. – Berlin: Springer-Verlag – 1976. 10. Gal’ Yu.M. On the growth of analytic functions given by Dirichlet series absolutely convergent in half– plane. – Drogobych, 1980. – 39 p. Manuscript dep. in VINITI 16.09.1980, no. 4080–80 Dep. (in Russian) 11. Mulyava O.M. Convergence classes in the theory of Dirichlet series// Dopov. Nats. Akad. Nauk. Ukr. – 1999. – №3. – P. 35–39. (in Ukrainian)
Pages	25-34
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Compositions of Dirichlet series similar to the Hadamard compositions, and convergence classes