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.
Citation:
V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials”, Russian Math. Surveys, 67:4 (2012), 599–684
\Bibitem{Dub12}
\by V.~N.~Dubinin
\paper Methods of geometric function theory in classical and modern problems for polynomials
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 4
\pages 599--684
\mathnet{http://mi.mathnet.ru/eng/rm9488}
\crossref{https://doi.org/10.1070/RM2012v067n04ABEH004803}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3013845}
\zmath{https://zbmath.org/?q=an:1267.30012}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000310789000001}
\elib{https://elibrary.ru/item.asp?id=20423456}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84868642061}
Linking options:
https://www.mathnet.ru/eng/rm9488
https://doi.org/10.1070/RM2012v067n04ABEH004803
https://www.mathnet.ru/eng/rm/v67/i4/p3
This publication is cited in the following 26 articles:
Natalia N. Rybakova, “Chebyshev polynomials with zeros outside the open arc segment”, Zhurn. SFU. Ser. Matem. i fiz., 17:1 (2024), 18–26
D. Dmitrishin, D. Gray, A. Stokolos, I. Tarasenko, “Extremal problems for typically real odd polynomials”, Acta Math. Hungar., 173:1 (2024), 1
A. Mir, T. Fayaz, “Inequalities for a Rational Function with Prescribed Poles”, Sib Math J, 65:4 (2024), 899
F. G. Avkhadiev, I. R. Kayumov, S. R. Nasyrov, “Extremal problems in geometric function theory”, Russian Math. Surveys, 78:2 (2023), 211–271
E. G. Kompaneets, V. V. Starkov, “On the Smirnov-Type Inequality for Polynomials”, Math. Notes, 111:3 (2022), 388–397
A. Mir, S. Hans, “Inequalities Concerning Rational Functions in the Complex Domain”, Sib Math J, 63:5 (2022), 1012
V. N. Dubinin, “Sharp Inequalities for Rational Functions on a Circle”, Math. Notes, 110:1 (2021), 41–47
V. N. Dubinin, “Some remarks on rotation theorems for complex polynomials”, Sib. elektron. matem. izv., 18:1 (2021), 369–376
Milovanovic V G., Mir A., Ahmad A., “Estimates For the Maximal Modulus of Rational Functions With Prescribed Poles”, Filomat, 35:5 (2021), 1511–1517
E. G. Ganenkova, V. V. Starkov, “The Möbius Transformation and Smirnov's Inequality for Polynomials”, Math. Notes, 105:2 (2019), 216–226
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
V. N. Dubinin, “On the Dual Mean-Value Conjecture for Complex Polynomials”, Math. Notes, 106:1 (2019), 133–135
V. N. Dubinin, “On holomorphic self-mappings of the unit disk”, Sib. elektron. matem. izv., 16 (2019), 1633–1639
Dmitrishin D., Smorodin A., Stokolos A., “on a Family of Extremal Polynomials”, C. R. Math., 357:7 (2019), 591–596
Kompaneets E., Starkov V., “Generalization of the Smirnov Operator and Differential Inequalities For Polynomials”, Lobachevskii J. Math., 40:12 (2019), 2043–2051
Hinkkanen A., Kayumov I.R., Khammatova D.M., “Dual Smale'S Mean Value Conjecture”, Proc. Amer. Math. Soc., 147:12 (2019), 5227–5237
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
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
S. Kalmykov, B. Nagy, V. Totik, “Bernstein- and Markov-type inequalities for rational functions”, Acta Math., 219:1 (2017), 21–63
P. A. Pugach, V. A. Shlyk, “Weighted modules and capacities on a Riemann surface”, J. Math. Sci. (N. Y.), 234:5 (2018), 701–736
Citation in format AMSBIB
Contact us: