{"title":"内半径积的上估计值在单值函数变形定理中的应用","authors":"I. Denega, Yaroslav V. Zabolotnyi","doi":"10.30970/ms.60.2.138-144","DOIUrl":null,"url":null,"abstract":"In 1934 Lavrentiev solved the problem of maximum ofproduct of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form$$T_{n}:={\\prod\\limits_{k=1}^nr(B_{k},a_{k})}{\\bigg(\\prod\\limits_{1\\leqslant k<p\\leqslant n}|a_{k}-a_{p}|\\bigg)^{-\\frac{2}{n-1}}},$$where $r(B,a)$ denotes the inner radius of the domain $B$ with respect to the point $a$ (for an infinitely distant point under the corresponding factor we understand the unit).In 1951 Goluzin for $n=3$ obtained an accurate evaluation for $T_{3}$.In 1980 Kuzmina showedthat the problem of the evaluation of $T_{4}$ isreduced to the smallest capacity problem in the certain continuumfamily and obtained the exact inequality for $T_{4}$.No other ultimate results in this problem for $n \\geqslant 5$ are known at present.In 2021 \\cite{Bakhtin2021,BahDen22} effective upper estimates are obtained for $T_{n}$, $n \\geqslant 2$.Among the possible applications of the obtained results in other tasks of the function theory are the so-called distortion theorems.In the paper we consider an application of upper estimates for products of inner radii to distortion theorems for univalent functionsin disk $U$, which map it onto a star-shaped domains relative to the origin.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":"102 3","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Application of upper estimates for products of inner radii to distortion theorems for univalent functions\",\"authors\":\"I. Denega, Yaroslav V. Zabolotnyi\",\"doi\":\"10.30970/ms.60.2.138-144\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1934 Lavrentiev solved the problem of maximum ofproduct of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form$$T_{n}:={\\\\prod\\\\limits_{k=1}^nr(B_{k},a_{k})}{\\\\bigg(\\\\prod\\\\limits_{1\\\\leqslant k<p\\\\leqslant n}|a_{k}-a_{p}|\\\\bigg)^{-\\\\frac{2}{n-1}}},$$where $r(B,a)$ denotes the inner radius of the domain $B$ with respect to the point $a$ (for an infinitely distant point under the corresponding factor we understand the unit).In 1951 Goluzin for $n=3$ obtained an accurate evaluation for $T_{3}$.In 1980 Kuzmina showedthat the problem of the evaluation of $T_{4}$ isreduced to the smallest capacity problem in the certain continuumfamily and obtained the exact inequality for $T_{4}$.No other ultimate results in this problem for $n \\\\geqslant 5$ are known at present.In 2021 \\\\cite{Bakhtin2021,BahDen22} effective upper estimates are obtained for $T_{n}$, $n \\\\geqslant 2$.Among the possible applications of the obtained results in other tasks of the function theory are the so-called distortion theorems.In the paper we consider an application of upper estimates for products of inner radii to distortion theorems for univalent functionsin disk $U$, which map it onto a star-shaped domains relative to the origin.\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":\"102 3\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-18\",\"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.60.2.138-144\",\"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.60.2.138-144","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
Application of upper estimates for products of inner radii to distortion theorems for univalent functions
In 1934 Lavrentiev solved the problem of maximum ofproduct of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form$$T_{n}:={\prod\limits_{k=1}^nr(B_{k},a_{k})}{\bigg(\prod\limits_{1\leqslant k
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