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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 4(406), Pages 3–88 (Mi umn9488)  

This article is cited in 11 scientific papers (total in 11 papers)

Methods of geometric function theory in classical and modern problems for polynomials

V. N. Dubinin

Far Eastern Federal University, Vladivostok

Abstract: This paper gives a survey of classical and modern theorems on polynomials, proved using methods of geometric function theory. Most of the paper is devoted to results of the author and his students, established by applying majorization principles for holomorphic functions, the theory of univalent functions, the theory of capacities, and symmetrization. Auxiliary results and the proofs of some of the theorems are presented.
Bibliography: 124 titles.

Keywords: majorization principles, Schwarz's lemma, capacities, univalent functions, symmetrization, inequalities, polynomials, critical points, critical values, rational functions.

DOI: https://doi.org/10.4213/rm9488

Full text: PDF file (1212 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2012, 67:4, 599–684

Bibliographic databases:

Document Type: Article
UDC: 517.5
MSC: Primary 30C10; Secondary 30C50, 30C85
Received: 12.09.2011

Citation: V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials”, Uspekhi Mat. Nauk, 67:4(406) (2012), 3–88; Russian Math. Surveys, 67:4 (2012), 599–684

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Mehmet Ali Akturk, A. Lukashov, “Weighted analogues of Bernstein-type inequalities on several intervals”, J. Inequal. Appl., 2013:1 (2013), 487, 8 pp.  crossref  mathscinet  zmath  isi  scopus
    2. V. N. Dubinin, “Four-point distortion theorem for complex polynomials”, Complex Var. Elliptic Equ., 59:1 (2014), 59–66  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. V. Olesov, “Inequalities for majorizing analytic functions and their applications to rational trigonometric functions and polynomials”, Sb. Math., 205:10 (2014), 1413–1441  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. V. N. Dubinin, “On the reduced modulus of the complex sphere”, Siberian Math. J., 55:5 (2014), 882–892  mathnet  crossref  mathscinet  isi
    5. V. Totik, “Bernstein- and Markov-type inequalities for trigonometric polynomials on general sets”, International Mathematics Research Notices, 2015:11 (2015), 2986–3020  crossref  mathscinet  zmath  isi  elib  scopus
    6. J. Math. Sci. (N. Y.), 222:5 (2017), 645–689  mathnet  crossref  mathscinet
    7. S. Kalmykov, B. Nagy, V. Totik, “Bernstein- and Markov-type inequalities for rational functions”, Acta Math., 219:1 (2017), 21–63  crossref  mathscinet  zmath  isi  scopus
    8. P. A. Pugach, V. A. Shlyk, “Vesovye moduli i emkosti na rimanovoi poverkhnosti”, Analiticheskaya teoriya chisel i teoriya funktsii. 33, Posvyaschaetsya pamyati Galiny Vasilevny KUZMINOI, Zap. nauchn. sem. POMI, 458, POMI, SPb., 2017, 164–217  mathnet
    9. Yu. V. Dymchenko, V. A. Shlyk, “On a Problem of Dubinin for the Capacity of a Condenser with a Finite Number of Plates”, Math. Notes, 103:6 (2018), 901–910  mathnet  crossref  crossref  isi  elib
    10. 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, eds. Agranovsky M., Golberg A., Jacobzon F., Shoikhet D., Zalcman L., Birkhauser Verlag Ag, 2018, 81–92  crossref  mathscinet  isi  scopus
    11. E. G. Ganenkova, V. V. Starkov, “The Möbius Transformation and Smirnov's Inequality for Polynomials”, Math. Notes, 105:2 (2019), 216–226  mathnet  crossref  crossref
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