Extremal problems in geometric function theory

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-01-01 DOI:10.4213/rm10076e

Farit Gabidinovich Avkhadiev, Ilgiz Rifatovich Kayumov, Semen Rafailovich Nasyrov

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Abstract

This survey is devoted to a number of achievements in the theory of extremal problems in geometric function theory. The approaches to the solution of problems under consideration and the methods used are based on conformal isomorphisms and on the theory of univalent functions developed since the beginning of the 20th century. Results on integral means of conformal mappings of a disc are presented and, in particular, Dolzenko's inequality for rational functions is extended to arbitrary domains with rectifiable boundaries. Investigations in the field of Bohr-type inequalities are described. An emphasis is made on integral inequalities of Hardy and Rellich type, in which the analytic properties of inequalities are intertwined with geometric characteristics of the boundaries of domains. Results related to the solution of the Vuorinen problem on the behaviour of conformal moduli under unlimited dilations of the plane are presented. Formulae for the variation of Robin capacity are obtained. One-parameter families of rational and elliptic functions whose critical values vary in accordance with a prescribed law are characterized. The last results on Smale's conjecture and Smale's dual conjecture are described. Bibliography: 149 titles.

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几何函数理论中的极值问题

本文综述了几何函数理论中关于极值问题的一些研究成果。所考虑的问题的解决方法和所使用的方法是基于共形同构和自20世纪初以来发展起来的一元函数理论。给出了圆盘共形映射的积分均值的结果，并将有理函数的Dolzenko不等式推广到具有可整流边界的任意区域。描述了玻尔型不等式领域的研究。重点讨论了Hardy和Rellich类型的积分不等式，其中不等式的解析性质与域边界的几何特征交织在一起。给出了平面无限膨胀下共形模量行为的Vuorinen问题的解的有关结果。得到了Robin容量的变化公式。对临界值按一定规律变化的有理函数和椭圆函数的单参数族进行了刻画。给出了关于斯梅尔猜想和斯梅尔对偶猜想的最后结果。参考书目:149篇。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.