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

This article is cited in 19 scientific papers (total in 19 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.


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

Bibliographic databases:

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

Citation in format AMSBIB
\by V.~N.~Dubinin
\paper Methods of geometric function theory in classical and modern problems for polynomials
\jour Uspekhi Mat. Nauk
\yr 2012
\vol 67
\issue 4(406)
\pages 3--88
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 4
\pages 599--684

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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, “Weighted modules and capacities on a Riemann surface”, J. Math. Sci. (N. Y.), 234:5 (2018), 701–736  mathnet  crossref
    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  mathscinet  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  mathscinet  isi  elib
    12. Ganenkova E.G., Starkov V.V., “Variations on a Theme of the Marden and Smirnov Operators, Differential Inequalities For Polynomials”, J. Math. Anal. Appl., 476:2 (2019), 696–714  crossref  mathscinet  zmath  isi  scopus
    13. V. N. Dubinin, “On the Dual Mean-Value Conjecture for Complex Polynomials”, Math. Notes, 106:1 (2019), 133–135  mathnet  crossref  crossref  mathscinet  isi  elib
    14. V. N. Dubinin, “On holomorphic self-mappings of the unit disk”, Sib. elektron. matem. izv., 16 (2019), 1633–1639  mathnet  crossref
    15. Dmitrishin D., Smorodin A., Stokolos A., “on a Family of Extremal Polynomials”, C. R. Math., 357:7 (2019), 591–596  crossref  isi
    16. Kompaneets E., Starkov V., “Generalization of the Smirnov Operator and Differential Inequalities For Polynomials”, Lobachevskii J. Math., 40:12 (2019), 2043–2051  crossref  isi
    17. Hinkkanen A., Kayumov I.R., Khammatova D.M., “Dual Smale'S Mean Value Conjecture”, Proc. Amer. Math. Soc., 147:12 (2019), 5227–5237  crossref  isi
    18. V. N. Dubinin, “Sharp Inequalities for Rational Functions on a Circle”, Math. Notes, 110:1 (2021), 41–47  mathnet  crossref  crossref  isi  elib
    19. V. N. Dubinin, “Some remarks on rotation theorems for complex polynomials”, Sib. elektron. matem. izv., 18:1 (2021), 369–376  mathnet  crossref
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