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 Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 75–82 (Mi izv4080)

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

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English version:
Izvestiya: Mathematics, 2010, 74:4, 735–742

Bibliographic databases:

UDC: 517.54
MSC: Primary 30C50; Secondary 30C85

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv4080
• https://doi.org/10.4213/im4080
• http://mi.mathnet.ru/eng/izv/v74/i4/p75

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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:
1. V. N. Dubinin, “Lower bounds for the half-plane capacity of compact sets and symmetrization”, Sb. Math., 201:11 (2010), 1635–1646
2. 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
3. V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Russian Math. Surveys, 72:3 (2017), 479–511
4. 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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