# On entire Dirichlet series similar to Hadamard compositions

• O.M. Mulyava Kyiv National University of Food Technologies Kyiv, Ukraine
• M. M. Sheremeta Ivan Franko National University of Lviv, Lviv
Keywords: Dirichlet series, Hadamard composition, generalized order, convergence class.

### Abstract

A function $F(s)=\sum_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with $0\le\lambda_n\uparrow+\infty$ is called the Hadamard composition of the genus $m\ge 1$ of functions $F_j(s)=\sum_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where
$P(x_1,...,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}x_1^{k_1}\cdot...\cdot x_p^{k_p}$ is a homogeneous polynomial of degree $m\ge 1$. Let $M(\sigma,F)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$ and functions $\alpha,\,\beta$ be positive continuous and increasing to $+\infty$ on
$[x_0, +\infty)$. To characterize the growth of the function $M(\sigma,F)$, we use generalized order $\varrho_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\alpha(\ln\,M(\sigma,F))}{\beta(\sigma)}$, generalized type
$T_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\ln\,M(\sigma,F)}{\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}$
and membership in the convergence class defined by the condition
$\displaystyle \int_{\sigma_0}^{\infty}\frac{\ln\,M(\sigma,F)}{\sigma\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}d\sigma<+\infty.$

Assuming the functions $\alpha, \beta$ and $\alpha^{-1}(c\beta(\ln\,x))$ are slowly increasing for each $c\in (0,+\infty)$ and $\ln\,n=O(\lambda_n)$ as $n\to \infty$, it is proved, for example, that if the functions $F_j$ have the same generalized order $\varrho_{\alpha,\beta}[F_j]=\varrho\in (0,+\infty)$ and the types $T_{\alpha,\beta}[F_j]=T_j\in [0,+\infty)$, $c_{m0...0}=c\not=0$, $|a_{n,1}|>0$ and $|a_{n,j}|= o(|a_{n,1}|)$ as $n\to\infty$ for $2\le j\le p$, and $F$ is the Hadamard composition of genus
$m\ge 1$ of the functions $F_j$ then $\varrho_{\alpha,\beta}[F]=\varrho$ and
$\displaystyle T_{\alpha,\beta}[F]\le \sum_{k_1+\dots+k_p=m}(k_1T_1+...+k_pT_p).$
It is proved also that $F$ belongs to the generalized convergence class if and only if
all functions $F_j$ belong to the same convergence class.

### Author Biographies

O.M. Mulyava, Kyiv National University of Food Technologies Kyiv, Ukraine

Kyiv National University of Food Technologies Kyiv, Ukraine

M. M. Sheremeta, Ivan Franko National University of Lviv, Lviv

Department of Mechanics and Mathematics, Professor

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Published
2023-06-23
How to Cite
Mulyava, O., & Sheremeta, M. M. (2023). On entire Dirichlet series similar to Hadamard compositions. Matematychni Studii, 59(2), 132-140. https://doi.org/10.30970/ms.59.2.132-140
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