On polynomials normalized on an interval |
Kalmykov S.I. |
2018, issue 2, P. 261-266 |
Abstract |
In this short communication new covering theorems, two-point distortion theorems and coefficient estimates for polynomials with a curved majorant on an interval are presented. Extremal polynomials in these therems are Chebyshev polynomials of the the second, third and forth kinds. Proofs are based on a new version of the Schwarz lemma and a univalent condition for holomorphic functions suggested by Dubinin. |
Keywords: Chebyshev polynomials, Bernstein inequality, conformal mappings |
Download the article (PDF-file) |
References |
[1] P. Borwein, T. Erdelyi, Polynomials and polynomial inequalities, Grad. Texts in Math, v. 161, Springer-Verlag, New York, 1995. [2] Q.I. Rahman, G. Schmeisser, “Analytic theory of polynomials”, London Math. Soc. Monogr. (N.S.), v. 26, The Clarendon Press, Oxford; Oxford Univ. Press, 2002, xiv 742. [3] T. Sheil-Small, Complex polynomials, Cambridge Univ. Press, Cambridge, Cambridge Stud. Adv. Math., 75, 2002, xx 428 pp. [4] V.N. Dubinin, S.I. Kalmykov, “Ekstremal’nyye svoystva polinomov CHebysheva”, Dal’nevost. matem. zhurn., 5:2, (2004), 169–177. [5] V. N. Dubinin, A.V. Olesov, “O primenenii konformnykh otobrazheniy k neravenstvam dlya polinomov”, Analiticheskaya teoriya chisel i teoriya funktsiy. 18, Zap. nauchn. sem. POMI, t. 286, POMI, SPb., 2002, 85–102. [6] V.N. Dubinin, “Lemma SHvartsa i otsenki koeffitsiyentov dlya regulyarnykh funktsiy so svobodnoy oblast’yu opredeleniya”, Matem. sb., 196:11, (2005), 53–74. [7] M.A. Lachance, “Bernstein and Markov inequalities for constrained polynomials”, Lect. Notes Math, v. 1045, 1984, 125–135. [8] Q.I. Rahman, “On a problem of Turan about polynomials with curved majorants”, Trans. Amer. Math. Soc, 163, (1972), 447–455. [9] S.I. Kalmykov, B. Nagy, V. Totik, “Asymptotically Sharp Markov and Schur Inequalities on General Sets”, Complex Analysis and Operator Theory, 9:6, (2014), 1287–1302. [10] Ya.L. Geronimus, Teoriya ortogonal’nykh mnogochlenov, M.: GITTL, 1950. |