{"title":"On entire Dirichlet series similar to Hadamard compositions","authors":"O. Mulyava, M. Sheremeta","doi":"10.30970/ms.59.2.132-140","DOIUrl":null,"url":null,"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 ifall functions $F_j$ belong to the same convergence class.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.59.2.132-140","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
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 ifall functions $F_j$ belong to the same convergence class.