Far Eastern Mathematical Journal

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On the distortion theorems for algebraic polynomials


V. N. Dubinin

2011, issue 1, Ñ. 28–36


Abstract
The applications of a boundary Schwarz lemma and the properties of the condenser capacity to some inequalities for polynomials and their derivatives are considered. We prove a new Bernstein-type inequality for the polynomials on a circle, two-sided estimates for the polynomials with constraints on their critical values, and two-sided estimates of the average distortion computed at zeros of the polynomials.

Keywords:
polynomials, critical points, critical values, Chebyshev polynomial, Bernstein-type inequality, distortion theorem, condenser capacity

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References

[1] G. V. Milovanovic?, D. S. Mitrinovic?, Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., Singapore, 1994
[2] P. Borwein, T. Erdelyi, Polynomials and polynomial inequalities, Grad. Texts in Math, 161, Springer-Verlag, New York, 1995
[3] Q. I. Rahman, G. Schmeisser, Analytic theory of polynomials, London Math. Soc. Monographs, New Series, 26, Clarendon Press, Oxford, 2002
[4] V. N. Dubinin, V. Ju. Kim, “Privedennye moduli i neravenstva dlja polinomov”, Zap. nauchn. semin. POMI, 263, 2000, 70–83
[5] V. N. Dubinin, “Teoremy iskazhenija dlja polinomov na okruzhnosti”, Matem. sb., 191:12 (2000), 51–60
[6] V. N. Dubinin, “Konformnye otobrazhenija i neravenstva dlja algebraicheskih polinomov”, Algebra i analiz, 13:5 (2001), 16–43
[7] V. N. Dubinin, “Konformnye otobrazhenija i neravenstva dlja algebraicheskih polinomov. II”, Zap. nauchn. semin. POMI, 302, 2003, 18–37
[8]. V. N. Dubinin, “O polinomah s kriticheskimi znachenijami na otrezke”, Mat. zametki, 78:6 (2005), 827–832
[9] T. Sheil-Small, “An inequality for the modulus of a polynomial evaluated at the roots of unity”, Bull. London Math. Soc., 40 (2008), 956–964
[10] S. Smale, “The fundamental theorem of algebra and complexity theory”, Bull. Amer. Math. Soc., 4:1 (1981), 1–36
[11] V. N. Dubinin, Emkosti kondensatorov i simmetrizacija v geometricheskoj teorii funkcij kompleksnogo peremennogo, Dal'nauka, Vladivostok, 2009
[12] P. Duren, Univalent functions, Springer-Verlag, New York, 1983
[13] I. Schur, “Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten”, Math. Zeit., 1 (1918), 377–402

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