A Markov-type inequality for arbitrary plane continua

Eremenko, Alexandre

doi:10.1090/S0002-9939-06-08640-0

A Markov-type inequality for arbitrary plane continua
HTML articles powered by AMS MathViewer

by Alexandre Eremenko PDF

Proc. Amer. Math. Soc. 135 (2007), 1505-1510 Request permission

Abstract:

Markov’s inequality is \[ \sup _{[-1,1]}|f’|\leq (\deg f)^2\sup _{[-1,1]}|f|,\] for all polynomials $f$. We prove a precise version of this inequality with an arbitrary continuum in the complex plane instead of the interval $[-1,1]$.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Adrien Douady, Prolongement de mouvements holomorphes (d’après Słodkowski et autres), Astérisque 227 (1995), Exp. No. 775, 3, 7–20 (French, with French summary). Séminaire Bourbaki, Vol. 1993/94. MR 1321641
A. È. Erëmenko and G. M. Levin, Estimation of the characteristic exponents of a polynomial, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 58 (1992), 30–40 (1993) (Russian); English transl., J. Math. Sci. (New York) 85 (1997), no. 5, 2164–2171. MR 1258059, DOI 10.1007/BF02355764
A. Erëmenko and L. Lempert, An extremal problem for polynomials, Proc. Amer. Math. Soc. 122 (1994), no. 1, 191–193. MR 1207536, DOI 10.1090/S0002-9939-1994-1207536-1
Alexandre Eremenko and Walter Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999), no. 2, 409–415. MR 1704189, DOI 10.1307/mmj/1030132418
A.A. Markov, On a question of Mendeleiev, Petersb. Abhandl. LXII (1890) 1–24.
Qazi Ibadur Rahman and Gerhard Schmeisser, Les inégalités de Markoff et de Bernstein, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 86, Presses de l’Université de Montréal, Montreal, QC, 1983 (French). MR 729316
G. Polya, How to Solve It. A New Aspect of Mathematical Method, Princeton University Press, Princeton, N. J., 1945. MR 0011666, DOI 10.1515/9781400828678
Ch. Pommerenke, On the derivative of a polynomial, Michigan Math. J. 6 (1959), 373–375. MR 109208, DOI 10.1307/mmj/1028998284