伪星形和伪凸方向上的多重狄利克雷级数

Q3 Mathematics Matematychni Studii Pub Date : 2023-01-23 DOI:10.30970/ms.58.2.182-200
M. Sheremeta, O. Skaskiv
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Let ${\\bf b}=(b_1,...,b_p)$ and $\\partial_{{\\bf b}}F( {s})=\\sum\\limits_{j=1}^p b_j\\dfrac{\\partial F( {s})}{\\partial {s}_j}$ be the derivative of $F$ in the direction ${\\bf b}$. In this paper, in particular, the following assertions were obtained: 1) If ${\\bf b}>0$ and$\\sum\\limits_{\\|(n)\\|=k_0}^{+\\infty}(\\lambda_{(n)},{\\bf b})|f_{(n)}|\\le (h,{\\bf b})$then $\\partial_{{\\bf b}}F( {s})\\not=0$ in $\\Pi_0:=\\{s\\colon \\text{Re}\\,s<0\\}$, i.e. $F$ is conformal in $\\Pi_0$ in the direction ${\\bf b}$ (Proposition 1).2) We say that function $F$ is pseudostarlike of the order $\\alpha\\in [0,\\,(h,{\\bf b}))$ and the type$\\beta >0$ in the direction ${\\bf b}$ if$\\Big|\\frac{\\partial_{{\\bf b}}F( {s})}{F(s)}-(h, {\\bf b})\\Big|<\\beta\\Big|\\frac{\\partial_{{\\bf b}}F( {s})}{F(s)}-(2\\alpha-(h, {\\bf b}))\\Big|,\\quad s\\in \\Pi_0.$Let $0\\le \\alpha<(h,{\\bf b})$ and $\\beta>0$. In order that the function $F$ ispseudostarlike of the order $\\alpha$ and the type $\\beta$ in the direction ${\\bf b}> 0$, it is sufficient and in the case, when all $f_{(n)}\\le 0$, it is necessary that$\\sum\\limits_{\\|(n)\\|=k_0}^{+\\infty}\\{((1+\\beta)\\lambda_{(n)}-(1-\\beta)h,{\\bf b})-2\\beta\\alpha\\}|f_{(n)}|\\le 2\\beta ((h,{\\bf b})-\\alpha)$ (Theorem 1).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series\",\"authors\":\"M. Sheremeta, O. Skaskiv\",\"doi\":\"10.30970/ms.58.2.182-200\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The article introduces the concepts of pseudostarlikeness and pseudoconvexity in the direction of absolutely converges in $\\\\Pi_0=\\\\{s\\\\in\\\\mathbb{C}^p\\\\colon \\\\text{Re}\\\\,s<0\\\\}$, $p\\\\in\\\\mathbb{N},$ the multiple Dirichlet series of the form$$ F(s)=e^{(h,s)}+\\\\sum\\\\nolimits_{\\\\|(n)\\\\|=\\\\|(n^0)\\\\|}^{+\\\\infty}f_{(n)}\\\\exp\\\\{(\\\\lambda_{(n)},s)\\\\}, \\\\quad s=(s_1,...,s_p)\\\\in {\\\\mathbb C}^p,\\\\quad p\\\\geq 1,$$where $ \\\\lambda_{(n^0)}>h$, $\\\\text{Re}\\\\,s<0\\\\Longleftrightarrow (\\\\text{Re}\\\\,s_1<0,...,\\\\text{Re}\\\\,s_p<0)$,$h=(h_1,...,h_p)\\\\in {\\\\mathbb R}^p_+$, $(n)=(n_1,...,n_p)\\\\in {\\\\mathbb N}^p$, $(n^0)=(n^0_1,...,n^0_p)\\\\in {\\\\mathbb N}^p$, $\\\\|(n)\\\\|=n_1+...+n_p$ and the sequences$\\\\lambda_{(n)}=(\\\\lambda^{(1)}_{n_1},...,\\\\lambda^{(p)}_{n_p})$ are such that $0c$ if $a_j\\\\ge c_j$ for all $1\\\\le j\\\\le p$ and there exists at least one $j$ such that $a_j> c_j$. Let ${\\\\bf b}=(b_1,...,b_p)$ and $\\\\partial_{{\\\\bf b}}F( {s})=\\\\sum\\\\limits_{j=1}^p b_j\\\\dfrac{\\\\partial F( {s})}{\\\\partial {s}_j}$ be the derivative of $F$ in the direction ${\\\\bf b}$. In this paper, in particular, the following assertions were obtained: 1) If ${\\\\bf b}>0$ and$\\\\sum\\\\limits_{\\\\|(n)\\\\|=k_0}^{+\\\\infty}(\\\\lambda_{(n)},{\\\\bf b})|f_{(n)}|\\\\le (h,{\\\\bf b})$then $\\\\partial_{{\\\\bf b}}F( {s})\\\\not=0$ in $\\\\Pi_0:=\\\\{s\\\\colon \\\\text{Re}\\\\,s<0\\\\}$, i.e. $F$ is conformal in $\\\\Pi_0$ in the direction ${\\\\bf b}$ (Proposition 1).2) We say that function $F$ is pseudostarlike of the order $\\\\alpha\\\\in [0,\\\\,(h,{\\\\bf b}))$ and the type$\\\\beta >0$ in the direction ${\\\\bf b}$ if$\\\\Big|\\\\frac{\\\\partial_{{\\\\bf b}}F( {s})}{F(s)}-(h, {\\\\bf b})\\\\Big|<\\\\beta\\\\Big|\\\\frac{\\\\partial_{{\\\\bf b}}F( {s})}{F(s)}-(2\\\\alpha-(h, {\\\\bf b}))\\\\Big|,\\\\quad s\\\\in \\\\Pi_0.$Let $0\\\\le \\\\alpha<(h,{\\\\bf b})$ and $\\\\beta>0$. 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引用次数: 1

摘要

本文在$\Pi_0=\{s\in\mathbb{C}^p\colon \text{Re}\,sh$, $\text{Re}\,s c_j$中介绍了绝对收敛方向上的伪星形和伪凸性的概念。设${\bf b}=(b_1,...,b_p)$和$\partial_{{\bf b}}F( {s})=\sum\limits_{j=1}^p b_j\dfrac{\partial F( {s})}{\partial {s}_j}$是$F$在${\bf b}$方向上的导数。在本文中,特别得到了以下断言:1)如果${\bf b}>0$和$\sum\limits_{\|(n)\|=k_0}^{+\infty}(\lambda_{(n)},{\bf b})|f_{(n)}|\le (h,{\bf b})$则$\partial_{{\bf b}}F( {s})\not=0$在$\Pi_0:=\{s\colon \text{Re}\,s0$方向上${\bf b}$如果$\Big|\frac{\partial_{{\bf b}}F( {s})}{F(s)}-(h, {\bf b})\Big|0$。为了使函数$F$在${\bf b}> 0$方向上的阶为$\alpha$和类型为$\beta$的伪星型,在这种情况下,当所有$f_{(n)}\le 0$时,有必要$\sum\limits_{\|(n)\|=k_0}^{+\infty}\{((1+\beta)\lambda_{(n)}-(1-\beta)h,{\bf b})-2\beta\alpha\}|f_{(n)}|\le 2\beta ((h,{\bf b})-\alpha)$(定理1)。
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Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series
The article introduces the concepts of pseudostarlikeness and pseudoconvexity in the direction of absolutely converges in $\Pi_0=\{s\in\mathbb{C}^p\colon \text{Re}\,s<0\}$, $p\in\mathbb{N},$ the multiple Dirichlet series of the form$$ F(s)=e^{(h,s)}+\sum\nolimits_{\|(n)\|=\|(n^0)\|}^{+\infty}f_{(n)}\exp\{(\lambda_{(n)},s)\}, \quad s=(s_1,...,s_p)\in {\mathbb C}^p,\quad p\geq 1,$$where $ \lambda_{(n^0)}>h$, $\text{Re}\,s<0\Longleftrightarrow (\text{Re}\,s_1<0,...,\text{Re}\,s_p<0)$,$h=(h_1,...,h_p)\in {\mathbb R}^p_+$, $(n)=(n_1,...,n_p)\in {\mathbb N}^p$, $(n^0)=(n^0_1,...,n^0_p)\in {\mathbb N}^p$, $\|(n)\|=n_1+...+n_p$ and the sequences$\lambda_{(n)}=(\lambda^{(1)}_{n_1},...,\lambda^{(p)}_{n_p})$ are such that $0c$ if $a_j\ge c_j$ for all $1\le j\le p$ and there exists at least one $j$ such that $a_j> c_j$. Let ${\bf b}=(b_1,...,b_p)$ and $\partial_{{\bf b}}F( {s})=\sum\limits_{j=1}^p b_j\dfrac{\partial F( {s})}{\partial {s}_j}$ be the derivative of $F$ in the direction ${\bf b}$. In this paper, in particular, the following assertions were obtained: 1) If ${\bf b}>0$ and$\sum\limits_{\|(n)\|=k_0}^{+\infty}(\lambda_{(n)},{\bf b})|f_{(n)}|\le (h,{\bf b})$then $\partial_{{\bf b}}F( {s})\not=0$ in $\Pi_0:=\{s\colon \text{Re}\,s<0\}$, i.e. $F$ is conformal in $\Pi_0$ in the direction ${\bf b}$ (Proposition 1).2) We say that function $F$ is pseudostarlike of the order $\alpha\in [0,\,(h,{\bf b}))$ and the type$\beta >0$ in the direction ${\bf b}$ if$\Big|\frac{\partial_{{\bf b}}F( {s})}{F(s)}-(h, {\bf b})\Big|<\beta\Big|\frac{\partial_{{\bf b}}F( {s})}{F(s)}-(2\alpha-(h, {\bf b}))\Big|,\quad s\in \Pi_0.$Let $0\le \alpha<(h,{\bf b})$ and $\beta>0$. In order that the function $F$ ispseudostarlike of the order $\alpha$ and the type $\beta$ in the direction ${\bf b}> 0$, it is sufficient and in the case, when all $f_{(n)}\le 0$, it is necessary that$\sum\limits_{\|(n)\|=k_0}^{+\infty}\{((1+\beta)\lambda_{(n)}-(1-\beta)h,{\bf b})-2\beta\alpha\}|f_{(n)}|\le 2\beta ((h,{\bf b})-\alpha)$ (Theorem 1).
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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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