关于类似于Hadamard合成的整个Dirichlet级数

Q3 Mathematics Matematychni Studii Pub Date : 2023-06-23 DOI:10.30970/ms.59.2.132-140
O. Mulyava, M. Sheremeta
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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":"{\"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 $[x_0, +\\\\infty)$. 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引用次数: 0

摘要

函数$F(s)=\sum_{n=1}^{\infty}A_n\exp\{s\lambda_n\}$与$0\le\lambda\uparrow+\infty$被称为函数$F_j(s)=\sum_{n=1}^{\ infty}A_{n,j}\exp\}$的亏格$m\ge 1$的Hadamard合成,如果$A_n=P(A_,1},…,A_,P})$,其中$P(x_ 1,…,x_P)=\sum\limits_{k_1+\dots+k_P=m}c_{k_1...k_p}x_1^{k_1}\cdot。。。\cdotx_p^{k_p}$是一个次为$m\ge1$的齐次多项式。设$M(\sigma,F)=\sup\{|F(\sigma+it)|:\,t\ in{\Bbb R}\}$和函数$\alpha,\,\beta$是正连续的,并在$[x_0,+\infty)$上增加到$+\infty$。为了刻画函数$M(\ sigma,F)$的增长,我们使用广义阶$\varrho_{\beta(\sigma)}$,广义类型$T_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\ln\,M(\sigma,F)}{\alpha^{-1}(\varrho_{\elpha,\beta}[F]\r\n\beta(\sigm))}$和条件$\displaystyle\int_{\sigma_0}^{\infty}\fracβ}[F]\beta(\sigma))}d\sigma0$和$|a_{n,j}|=o(|a_{n,1}|)$作为$n\to\infty$$2\le j\le p$,$F$是函数$F_j$然后$\varrho_{\alpha,\beta}[F]=\varrho$和$\displaystyle T_{\elpha,\peta}[F]\le \sum_{k_1+\dots+k_p=m}(k_1_1+…+k_pT_p)的亏格$m\ge 1$的Hadamard合成。$还证明了$F$属于广义收敛类,当且仅当所有函数$F_j-$属于同一收敛类。
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On entire Dirichlet series similar to Hadamard compositions
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.
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Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
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期刊介绍: Journal is devoted to research in all fields of mathematics.
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On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series Almost periodic distributions and crystalline measures Reflectionless Schrodinger operators and Marchenko parametrization Existence of basic solutions of first order linear homogeneous set-valued differential equations Real univariate polynomials with given signs of coefficients and simple real roots
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