{"title":"一元函数类中泛函系统的值集","authors":"D. V. Prokhorov","doi":"10.1070/SM1992V071N02ABEH001407","DOIUrl":null,"url":null,"abstract":"The problem of describing the set of values of a system of functional {f(Z), ..., f(n)(Z)} in the class of univalent functions holomorphic in the disk is formalized as a problem of constructing the set of attainability for a control system generated by the Loewner equation. In this problem the maximum principle turns out to be a necessary and sufficient condition for optimality. An algorithm for finding this set for a generalized Loewner equation with constant coefficients and continuous control is constructed. The results are extended to classes of bounded univalent functions.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"3 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"20","resultStr":"{\"title\":\"Sets of Values of Systems of Functionals in Classes of Univalent Functions\",\"authors\":\"D. V. Prokhorov\",\"doi\":\"10.1070/SM1992V071N02ABEH001407\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The problem of describing the set of values of a system of functional {f(Z), ..., f(n)(Z)} in the class of univalent functions holomorphic in the disk is formalized as a problem of constructing the set of attainability for a control system generated by the Loewner equation. In this problem the maximum principle turns out to be a necessary and sufficient condition for optimality. An algorithm for finding this set for a generalized Loewner equation with constant coefficients and continuous control is constructed. The results are extended to classes of bounded univalent functions.\",\"PeriodicalId\":208776,\"journal\":{\"name\":\"Mathematics of The Ussr-sbornik\",\"volume\":\"3 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1992-02-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"20\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematics of The Ussr-sbornik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1070/SM1992V071N02ABEH001407\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of The Ussr-sbornik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1070/SM1992V071N02ABEH001407","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Sets of Values of Systems of Functionals in Classes of Univalent Functions
The problem of describing the set of values of a system of functional {f(Z), ..., f(n)(Z)} in the class of univalent functions holomorphic in the disk is formalized as a problem of constructing the set of attainability for a control system generated by the Loewner equation. In this problem the maximum principle turns out to be a necessary and sufficient condition for optimality. An algorithm for finding this set for a generalized Loewner equation with constant coefficients and continuous control is constructed. The results are extended to classes of bounded univalent functions.
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