{"title":"用mittagleffler函数系统对级数的相对增长作了初步说明","authors":"O. Mulyava","doi":"10.31861/bmj2022.01.03","DOIUrl":null,"url":null,"abstract":"For a regularly converging in ${\\Bbb C}$ series $F_{\\varrho}(z)=\\sum\\limits_{n=1}^{\\infty} a_n E_{\\varrho}(\\lambda_nz)$, where\n$0<\\varrho<+\\infty$ and $E_{\\varrho}(z)=\\sum\\limits_{k=0}^{\\infty}\\frac{z^k}{\\Gamma(1+k/\\varrho)}$\nis the Mittag-Leffler function, it is investigated the asymptotic behavior of the function $E_{\\varrho}^{-1} (M_{F_{\\varrho}}(r))$, where $M_f(r)=\\max\\{|f(z)|:\\,|z|=r\\}$. For example, it is proved that if $\\varlimsup\\limits_{n\\to \\infty}\\frac{\\ln\\,\\ln\\,n}{\\ln\\,\\lambda_n}\\le \\varrho$ and $a_n\\ge 0$ for all $n\\ge 1$, then $\\varlimsup\\limits_{r\\to+\\infty}\\frac{\\ln\\,E^{-1}_{\\varrho}(M_{F_{\\varrho}}(r))}{\\ln\\,r}=\\frac{1}{1-\\overline{\\gamma}\\varrho}$, where\n$\\overline{\\gamma}=\\varlimsup\\limits_{n\\to\\infty}\\frac{\\ln\\,\\lambda_n}{\\ln\\,\\ln\\,(1/a_n)}$.\n\nA similar result is obtained for the Laplace-Stiltjes type integral $I_{\\varrho}(r) = \\int\\limits_{0}^{\\infty}a(x)E_{\\varrho}(r x) d F(x)$.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"ELEMENTARY REMARKS TO THE RELATIVE GROWTH OF SERIES BY THE SYSTEM OF MITTAG-LEFFLER FUNCTIONS\",\"authors\":\"O. Mulyava\",\"doi\":\"10.31861/bmj2022.01.03\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a regularly converging in ${\\\\Bbb C}$ series $F_{\\\\varrho}(z)=\\\\sum\\\\limits_{n=1}^{\\\\infty} a_n E_{\\\\varrho}(\\\\lambda_nz)$, where\\n$0<\\\\varrho<+\\\\infty$ and $E_{\\\\varrho}(z)=\\\\sum\\\\limits_{k=0}^{\\\\infty}\\\\frac{z^k}{\\\\Gamma(1+k/\\\\varrho)}$\\nis the Mittag-Leffler function, it is investigated the asymptotic behavior of the function $E_{\\\\varrho}^{-1} (M_{F_{\\\\varrho}}(r))$, where $M_f(r)=\\\\max\\\\{|f(z)|:\\\\,|z|=r\\\\}$. For example, it is proved that if $\\\\varlimsup\\\\limits_{n\\\\to \\\\infty}\\\\frac{\\\\ln\\\\,\\\\ln\\\\,n}{\\\\ln\\\\,\\\\lambda_n}\\\\le \\\\varrho$ and $a_n\\\\ge 0$ for all $n\\\\ge 1$, then $\\\\varlimsup\\\\limits_{r\\\\to+\\\\infty}\\\\frac{\\\\ln\\\\,E^{-1}_{\\\\varrho}(M_{F_{\\\\varrho}}(r))}{\\\\ln\\\\,r}=\\\\frac{1}{1-\\\\overline{\\\\gamma}\\\\varrho}$, where\\n$\\\\overline{\\\\gamma}=\\\\varlimsup\\\\limits_{n\\\\to\\\\infty}\\\\frac{\\\\ln\\\\,\\\\lambda_n}{\\\\ln\\\\,\\\\ln\\\\,(1/a_n)}$.\\n\\nA similar result is obtained for the Laplace-Stiltjes type integral $I_{\\\\varrho}(r) = \\\\int\\\\limits_{0}^{\\\\infty}a(x)E_{\\\\varrho}(r x) d F(x)$.\",\"PeriodicalId\":196726,\"journal\":{\"name\":\"Bukovinian Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1900-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bukovinian Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.31861/bmj2022.01.03\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bukovinian Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31861/bmj2022.01.03","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
ELEMENTARY REMARKS TO THE RELATIVE GROWTH OF SERIES BY THE SYSTEM OF MITTAG-LEFFLER FUNCTIONS
For a regularly converging in ${\Bbb C}$ series $F_{\varrho}(z)=\sum\limits_{n=1}^{\infty} a_n E_{\varrho}(\lambda_nz)$, where
$0<\varrho<+\infty$ and $E_{\varrho}(z)=\sum\limits_{k=0}^{\infty}\frac{z^k}{\Gamma(1+k/\varrho)}$
is the Mittag-Leffler function, it is investigated the asymptotic behavior of the function $E_{\varrho}^{-1} (M_{F_{\varrho}}(r))$, where $M_f(r)=\max\{|f(z)|:\,|z|=r\}$. For example, it is proved that if $\varlimsup\limits_{n\to \infty}\frac{\ln\,\ln\,n}{\ln\,\lambda_n}\le \varrho$ and $a_n\ge 0$ for all $n\ge 1$, then $\varlimsup\limits_{r\to+\infty}\frac{\ln\,E^{-1}_{\varrho}(M_{F_{\varrho}}(r))}{\ln\,r}=\frac{1}{1-\overline{\gamma}\varrho}$, where
$\overline{\gamma}=\varlimsup\limits_{n\to\infty}\frac{\ln\,\lambda_n}{\ln\,\ln\,(1/a_n)}$.
A similar result is obtained for the Laplace-Stiltjes type integral $I_{\varrho}(r) = \int\limits_{0}^{\infty}a(x)E_{\varrho}(r x) d F(x)$.