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This article is cited in 4 scientific papers (total in 4 papers)
Steiner symmetrization and the initial coefficients of univalent functions
V. N. Dubinin Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences
Abstract:
We establish the inequality $|a_1|^2-\operatorname{Re}a_1a_{-1}\ge
|a_1^*|^2-\operatorname{Re}a_1^*a_{-1}^*$ for the initial
coefficients of any function $f(z)=a_1z+a_0+{a_{-1}}/z+\dotsb$
meromorphic and univalent in the domain $D=ż\colon |z|>1\}$,
where $a_1^*$ and $a_{-1}^*$ are the corresponding coefficients
in the expansion of the function $f^*(z)$ that maps the domain $D$
conformally and univalently onto the exterior of the result
of the Steiner symmetrization with respect to the real axis
of the complement of the set $f(D)$. The Pólya–Szegő
inequality $|a_1|\ge |a_1^*|$ is already known.
We describe some applications of our inequality
to functions of class $\Sigma$.
Keywords:
Steiner symmetrization, capacity of a set, univalent function, covering theorem.
DOI:
https://doi.org/10.4213/im4080
Full text:
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English version:
Izvestiya: Mathematics, 2010, 74:4, 735–742
Bibliographic databases:
UDC:
517.54
MSC: Primary 30C50; Secondary 30C85 Received: 27.01.2009
Citation:
V. N. Dubinin, “Steiner symmetrization and the initial coefficients of univalent functions”, Izv. RAN. Ser. Mat., 74:4 (2010), 75–82; Izv. Math., 74:4 (2010), 735–742
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http://mi.mathnet.ru/eng/izv4080https://doi.org/10.4213/im4080 http://mi.mathnet.ru/eng/izv/v74/i4/p75
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
This publication is cited in the following articles:
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V. N. Dubinin, “Lower bounds for the half-plane capacity of compact sets and symmetrization”, Sb. Math., 201:11 (2010), 1635–1646
-
V. N. Dubinin, “Asymptotic Behavior of the Capacity of a Condenser as Some of Its Plates Contract to Points”, Math. Notes, 96:2 (2014), 187–198
-
V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Russian Math. Surveys, 72:3 (2017), 479–511
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Dubinin V.N., “Some Unsolved Problems About Condenser Capacities on the Plane”, Complex Analysis and Dynamical Systems: New Trends and Open Problems, Trends in Mathematics, ed. Agranovsky M. Golberg A. Jacobzon F. Shoikhet D. Zalcman L., Birkhauser Verlag Ag, 2018, 81–92
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