On the distortion theorems for algebraic polynomials
Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 28-36.

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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.
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V. N. Dubinin. On the distortion theorems for algebraic polynomials. Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 28-36. http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a2/

[1] G. V. Milovanović, D. S. Mitrinović, Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., Singapore, 1994 | MR | Zbl

[2] P. Borwein, T. Erdelyi, Polynomials and polynomial inequalities, Grad. Texts in Math, 161, Springer-Verlag, New York, 1995 | DOI | MR

[3] Q. I. Rahman, G. Schmeisser, Analytic theory of polynomials, London Math. Soc. Monographs, New Series, 26, Clarendon Press, Oxford, 2002 | MR | Zbl

[4] V. N. Dubinin, V. Yu. Kim, “Privedennye moduli i neravenstva dlya polinomov”, Zap. nauchn. semin. POMI, 263, 2000, 70–83 | MR | Zbl

[5] V. N. Dubinin, “Teoremy iskazheniya dlya polinomov na okruzhnosti”, Matem. sb., 191:12 (2000), 51–60 | DOI | MR | Zbl

[6] V. N. Dubinin, “Konformnye otobrazheniya i neravenstva dlya algebraicheskikh polinomov”, Algebra i analiz, 13:5 (2001), 16–43 | MR | Zbl

[7] V. N. Dubinin, “Konformnye otobrazheniya i neravenstva dlya algebraicheskikh polinomov. II”, Zap. nauchn. semin. POMI, 302, 2003, 18–37

[8] V. N. Dubinin, “O polinomakh s kriticheskimi znacheniyami na otrezke”, Mat. zametki, 78:6 (2005), 827–832 | DOI | MR

[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 | DOI | MR | Zbl

[10] S. Smale, “The fundamental theorem of algebra and complexity theory”, Bull. Amer. Math. Soc., 4:1 (1981), 1–36 | DOI | MR | Zbl

[11] V. N. Dubinin, Emkosti kondensatorov i simmetrizatsiya v geometricheskoi teorii funktsii kompleksnogo peremennogo, Dalnauka, Vladivostok, 2009

[12] P. Duren, Univalent functions, Springer-Verlag, New York, 1983 | MR | Zbl

[13] I. Schur, “Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten”, Math. Zeit., 1 (1918), 377–402 | DOI | MR | Zbl