{"title":"关于在半平面内绝对收敛的某类实系数狄利克列数列","authors":"M. Sheremeta","doi":"10.30970/ms.61.1.35-50","DOIUrl":null,"url":null,"abstract":" For $h>0$, $\\alpha\\in [0,h)$ and $\\mu\\in {\\mathbb R}$ denote by $SD_h(\\mu, \\alpha)$ a class \nof absolutely convergent in the half-plane $\\Pi_0=\\{s:\\, \\text{Re}\\,s<0\\}$ Dirichlet series \n$F(s)=e^{sh}+\\sum_{k=1}^{\\infty}f_k\\exp\\{s\\lambda_k\\}$ such that \n \n\\smallskip\\centerline{$\\text{Re}\\left\\{\\frac{(\\mu-1)F'(s)-\\mu F''(s)/h}{(\\mu-1)F(s)-\\mu F'(s)/h}\\right\\}>\\alpha$ for all $s\\in \\Pi_0$,} \n \n\\smallskip\\noi and \nlet $\\Sigma D_h(\\mu, \\alpha)$ be a class of absolutely convergent in half-plane $\\Pi_0$ Dirichlet series \n$F(s)=e^{-sh}+\\sum_{k=1}^{\\infty}f_k\\exp\\{s\\lambda_k\\}$ such that \n \n\\smallskip\\centerline{$\\text{Re}\\left\\{\\frac{(\\mu-1)F'(s)+\\mu F''(s)/h}{(\\mu-1)F(s)+\\mu F'(s)/h}\\right\\}<-\\alpha$ for all $s\\in \\Pi_0$.} \n \n\\smallskip\\noi \nThen $SD_h(0, \\alpha)$ consists of pseudostarlike functions of order $\\alpha$ and $SD_h(1, \\alpha)$ consists of pseudoconvex functions of order $\\alpha$. \n \nFor functions from the classes $SD_h(\\mu, \\alpha)$ and $\\Sigma D_h(\\mu, \\alpha)$, estimates for the coefficients and growth estimates are obtained. {In particular, it is proved the following statements: 1) In order that function $F(s)=e^{sh}+\\sum_{k=1}^{\\infty}f_k\\exp\\{s\\lambda_k\\}$ belongs to \n$SD_h(\\mu, \\alpha)$, it is \nsufficient, and in the case when $f_k(\\mu\\lambda_k/h-\\mu+1)\\le 0$ for all $k\\ge 1$, it is necessary that} \n \n\\smallskip\\centerline{$ \n\\sum\\limits_{k=1}^{\\infty}\\big|f_k\\big(\\frac{\\mu\\lambda_k}{h}-\\mu+1\\big)\\big|(\\lambda_k-\\alpha)\\le h-\\alpha,$} \n \n\\noi {where $h>0, \\alpha\\in [0, h)$ (Theorem 1).} \n \n\\noi 2) {In order that function $F(s)=e^{-sh}+\\sum_{k=1}^{\\infty}f_k\\exp\\{s\\lambda_k\\}$ belongs to $\\Sigma D_h(\\mu, \\alpha)$, it is \nsufficient, and in the case when $f_k(\\mu\\lambda_k/h+\\mu-1)\\le 0$ for all $k\\ge 1$, it is necessary that \n \n\\smallskip\\centerline{$\\sum\\limits_{k=1}^{\\infty}\\big|f_k\\big(\\frac{\\mu\\lambda_k}{h}+\\mu-1\\big)\\big|(\\lambda_k+\\alpha)\\le h-\\alpha,$} \n \n\\noi where $h>0, \\alpha\\in [0, h)$ (Theorem~2).} Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus $m$ were also studied.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On certain classes of Dirichlet series with real coefficients absolute convergent in a half-plane\",\"authors\":\"M. Sheremeta\",\"doi\":\"10.30970/ms.61.1.35-50\",\"DOIUrl\":null,\"url\":null,\"abstract\":\" For $h>0$, $\\\\alpha\\\\in [0,h)$ and $\\\\mu\\\\in {\\\\mathbb R}$ denote by $SD_h(\\\\mu, \\\\alpha)$ a class \\nof absolutely convergent in the half-plane $\\\\Pi_0=\\\\{s:\\\\, \\\\text{Re}\\\\,s<0\\\\}$ Dirichlet series \\n$F(s)=e^{sh}+\\\\sum_{k=1}^{\\\\infty}f_k\\\\exp\\\\{s\\\\lambda_k\\\\}$ such that \\n \\n\\\\smallskip\\\\centerline{$\\\\text{Re}\\\\left\\\\{\\\\frac{(\\\\mu-1)F'(s)-\\\\mu F''(s)/h}{(\\\\mu-1)F(s)-\\\\mu F'(s)/h}\\\\right\\\\}>\\\\alpha$ for all $s\\\\in \\\\Pi_0$,} \\n \\n\\\\smallskip\\\\noi and \\nlet $\\\\Sigma D_h(\\\\mu, \\\\alpha)$ be a class of absolutely convergent in half-plane $\\\\Pi_0$ Dirichlet series \\n$F(s)=e^{-sh}+\\\\sum_{k=1}^{\\\\infty}f_k\\\\exp\\\\{s\\\\lambda_k\\\\}$ such that \\n \\n\\\\smallskip\\\\centerline{$\\\\text{Re}\\\\left\\\\{\\\\frac{(\\\\mu-1)F'(s)+\\\\mu F''(s)/h}{(\\\\mu-1)F(s)+\\\\mu F'(s)/h}\\\\right\\\\}<-\\\\alpha$ for all $s\\\\in \\\\Pi_0$.} \\n \\n\\\\smallskip\\\\noi \\nThen $SD_h(0, \\\\alpha)$ consists of pseudostarlike functions of order $\\\\alpha$ and $SD_h(1, \\\\alpha)$ consists of pseudoconvex functions of order $\\\\alpha$. \\n \\nFor functions from the classes $SD_h(\\\\mu, \\\\alpha)$ and $\\\\Sigma D_h(\\\\mu, \\\\alpha)$, estimates for the coefficients and growth estimates are obtained. {In particular, it is proved the following statements: 1) In order that function $F(s)=e^{sh}+\\\\sum_{k=1}^{\\\\infty}f_k\\\\exp\\\\{s\\\\lambda_k\\\\}$ belongs to \\n$SD_h(\\\\mu, \\\\alpha)$, it is \\nsufficient, and in the case when $f_k(\\\\mu\\\\lambda_k/h-\\\\mu+1)\\\\le 0$ for all $k\\\\ge 1$, it is necessary that} \\n \\n\\\\smallskip\\\\centerline{$ \\n\\\\sum\\\\limits_{k=1}^{\\\\infty}\\\\big|f_k\\\\big(\\\\frac{\\\\mu\\\\lambda_k}{h}-\\\\mu+1\\\\big)\\\\big|(\\\\lambda_k-\\\\alpha)\\\\le h-\\\\alpha,$} \\n \\n\\\\noi {where $h>0, \\\\alpha\\\\in [0, h)$ (Theorem 1).} \\n \\n\\\\noi 2) {In order that function $F(s)=e^{-sh}+\\\\sum_{k=1}^{\\\\infty}f_k\\\\exp\\\\{s\\\\lambda_k\\\\}$ belongs to $\\\\Sigma D_h(\\\\mu, \\\\alpha)$, it is \\nsufficient, and in the case when $f_k(\\\\mu\\\\lambda_k/h+\\\\mu-1)\\\\le 0$ for all $k\\\\ge 1$, it is necessary that \\n \\n\\\\smallskip\\\\centerline{$\\\\sum\\\\limits_{k=1}^{\\\\infty}\\\\big|f_k\\\\big(\\\\frac{\\\\mu\\\\lambda_k}{h}+\\\\mu-1\\\\big)\\\\big|(\\\\lambda_k+\\\\alpha)\\\\le h-\\\\alpha,$} \\n \\n\\\\noi where $h>0, \\\\alpha\\\\in [0, h)$ (Theorem~2).} Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus $m$ were also studied.\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-19\",\"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.61.1.35-50\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.61.1.35-50","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
On certain classes of Dirichlet series with real coefficients absolute convergent in a half-plane
For $h>0$, $\alpha\in [0,h)$ and $\mu\in {\mathbb R}$ denote by $SD_h(\mu, \alpha)$ a class
of absolutely convergent in the half-plane $\Pi_0=\{s:\, \text{Re}\,s<0\}$ Dirichlet series
$F(s)=e^{sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ such that
\smallskip\centerline{$\text{Re}\left\{\frac{(\mu-1)F'(s)-\mu F''(s)/h}{(\mu-1)F(s)-\mu F'(s)/h}\right\}>\alpha$ for all $s\in \Pi_0$,}
\smallskip\noi and
let $\Sigma D_h(\mu, \alpha)$ be a class of absolutely convergent in half-plane $\Pi_0$ Dirichlet series
$F(s)=e^{-sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ such that
\smallskip\centerline{$\text{Re}\left\{\frac{(\mu-1)F'(s)+\mu F''(s)/h}{(\mu-1)F(s)+\mu F'(s)/h}\right\}<-\alpha$ for all $s\in \Pi_0$.}
\smallskip\noi
Then $SD_h(0, \alpha)$ consists of pseudostarlike functions of order $\alpha$ and $SD_h(1, \alpha)$ consists of pseudoconvex functions of order $\alpha$.
For functions from the classes $SD_h(\mu, \alpha)$ and $\Sigma D_h(\mu, \alpha)$, estimates for the coefficients and growth estimates are obtained. {In particular, it is proved the following statements: 1) In order that function $F(s)=e^{sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ belongs to
$SD_h(\mu, \alpha)$, it is
sufficient, and in the case when $f_k(\mu\lambda_k/h-\mu+1)\le 0$ for all $k\ge 1$, it is necessary that}
\smallskip\centerline{$
\sum\limits_{k=1}^{\infty}\big|f_k\big(\frac{\mu\lambda_k}{h}-\mu+1\big)\big|(\lambda_k-\alpha)\le h-\alpha,$}
\noi {where $h>0, \alpha\in [0, h)$ (Theorem 1).}
\noi 2) {In order that function $F(s)=e^{-sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ belongs to $\Sigma D_h(\mu, \alpha)$, it is
sufficient, and in the case when $f_k(\mu\lambda_k/h+\mu-1)\le 0$ for all $k\ge 1$, it is necessary that
\smallskip\centerline{$\sum\limits_{k=1}^{\infty}\big|f_k\big(\frac{\mu\lambda_k}{h}+\mu-1\big)\big|(\lambda_k+\alpha)\le h-\alpha,$}
\noi where $h>0, \alpha\in [0, h)$ (Theorem~2).} Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus $m$ were also studied.