{"title":"On certain subclass of Dirichlet series absolutely convergent in half-plane","authors":"M. Sheremeta","doi":"10.30970/ms.57.1.32-44","DOIUrl":null,"url":null,"abstract":"Denote by $\\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\\Pi_0=\\{s\\colon \\text{Re}\\,s<0\\}$ Dirichlet series$F(s)=e^{sh}-\\sum_{k=1}^{\\infty}f_k\\exp\\{s(\\lambda_k+h)\\},\\, s=\\sigma+it$, where $h> 0$, $h<\\lambda_k\\uparrow+\\infty$ and $f_k>0$.For $0\\le\\alpha<h$ and $l\\ge 0$ we say that $F$ belongs to the class $\\mathfrak{DF}_h(l,\\alpha)$ if and only if$\\text{Re}\\{e^{-hs}((1-l)F(s)+\\frac{l}{h}F'(s))\\}>\\frac{\\alpha}{h}$,and belongs to the class $\\mathfrak{DG}_h(l,\\alpha)$ if and only if$\\text{Re}\\{e^{-hs}((1-l)F'(s)+\\frac{l}{h}F''(s))\\}>\\alpha$ for all $s\\in \\Pi_0$. It is provedthat $F\\in \\mathfrak{DF}_h(l,\\alpha)$ if and only if $ \\sum_{k=1}^{\\infty}(h+l\\lambda_k)f_k\\le h-\\alpha$, and$F\\in \\mathfrak{DG}_h(l,\\alpha)$ if and only if $\\sum_{k=1}^{\\infty}(h+l\\lambda_k)(\\lambda_k+h)f_k\\le h(h-\\alpha)$. \nIf $F_j\\in \\mathfrak{DF}_h(l_j,\\alpha_j)$, $j=1, 2$, where $l_j\\ge0$ and $0\\le \\alpha_j<h$, then Hadamard composition$(F_1*F_2)\\in \\mathfrak{D}F_h(l,\\alpha)$, where $l=\\min\\{l_1,l_2\\}$ and$\\alpha=h-\\frac{(h-\\alpha_1)(h-\\alpha_2)}{h+l\\lambda_1}$. Similar statement is correct for the class $F_j\\in \\mathfrak{DG}_h(l,\\alpha)$. \nFor $j>0$ and $\\delta>0$ the neighborhood of the function $F\\in \\mathfrak{D}_0$ is defined as follows$O_{j,\\delta}(F)=\\{G(s)=e^{s}-\\sum_{k=1}^{\\infty}g_k\\exp\\{s\\lambda_k\\}\\in \\mathfrak{D}_0\\colon \\sum_{k=1}^{\\infty}\\lambda^j_k|g_k-f_k|\\le\\delta\\}$. It is described the neighborhoods of functions from classes $\\mathfrak{DF}_h(l,\\alpha)$ and $\\mathfrak{DG}_h(l,\\alpha)$. \nConditions on real parameters $\\gamma_0,\\,\\gamma_1,\\,\\gamma_2,\\,a_1$ and $a_2$ of the differential equation $w''+(\\gamma_0e^{2hs}+\\gamma_1e^{hs}+\\gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solutioneither in $\\mathfrak{DF}_h(l,\\alpha)$ or in $\\mathfrak{DG}_h(l,\\alpha)$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.57.1.32-44","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: 1
Abstract
Denote by $\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\Pi_0=\{s\colon \text{Re}\,s<0\}$ Dirichlet series$F(s)=e^{sh}-\sum_{k=1}^{\infty}f_k\exp\{s(\lambda_k+h)\},\, s=\sigma+it$, where $h> 0$, $h<\lambda_k\uparrow+\infty$ and $f_k>0$.For $0\le\alpha\frac{\alpha}{h}$,and belongs to the class $\mathfrak{DG}_h(l,\alpha)$ if and only if$\text{Re}\{e^{-hs}((1-l)F'(s)+\frac{l}{h}F''(s))\}>\alpha$ for all $s\in \Pi_0$. It is provedthat $F\in \mathfrak{DF}_h(l,\alpha)$ if and only if $ \sum_{k=1}^{\infty}(h+l\lambda_k)f_k\le h-\alpha$, and$F\in \mathfrak{DG}_h(l,\alpha)$ if and only if $\sum_{k=1}^{\infty}(h+l\lambda_k)(\lambda_k+h)f_k\le h(h-\alpha)$.
If $F_j\in \mathfrak{DF}_h(l_j,\alpha_j)$, $j=1, 2$, where $l_j\ge0$ and $0\le \alpha_j0$ and $\delta>0$ the neighborhood of the function $F\in \mathfrak{D}_0$ is defined as follows$O_{j,\delta}(F)=\{G(s)=e^{s}-\sum_{k=1}^{\infty}g_k\exp\{s\lambda_k\}\in \mathfrak{D}_0\colon \sum_{k=1}^{\infty}\lambda^j_k|g_k-f_k|\le\delta\}$. It is described the neighborhoods of functions from classes $\mathfrak{DF}_h(l,\alpha)$ and $\mathfrak{DG}_h(l,\alpha)$.
Conditions on real parameters $\gamma_0,\,\gamma_1,\,\gamma_2,\,a_1$ and $a_2$ of the differential equation $w''+(\gamma_0e^{2hs}+\gamma_1e^{hs}+\gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solutioneither in $\mathfrak{DF}_h(l,\alpha)$ or in $\mathfrak{DG}_h(l,\alpha)$.