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A Markov-type inequality for arbitrary plane continua


Author: Alexandre Eremenko
Journal: Proc. Amer. Math. Soc. 135 (2007), 1505-1510
MSC (2000): Primary 41A17, 26D05
DOI: https://doi.org/10.1090/S0002-9939-06-08640-0
Published electronically: November 29, 2006
MathSciNet review: 2276660
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Abstract | References | Similar Articles | Additional Information

Abstract: Markov's inequality is

$\displaystyle \sup_{[-1,1]}\vert f'\vert\leq(\deg f)^2\sup_{[-1,1]}\vert f\vert,$

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 [Enhancements On Off] (What's this?)

  • 1. L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Toronto 1966. MR 0200442 (34:336)
  • 2. L. Ahlfors, Conformal Invariants. Topics in geometric function theory, McGraw-Hill, NY 1973. MR 0357743 (50:10211)
  • 3. A. Douady, Prolongement de movements holomorphes, Sém. Bourbaki 1993/4. Astérisques 227 (1995) Exp. No. 775 7-20. MR 1321641 (95m:58104)
  • 4. A. Eremenko and G. Levin, Estimates of the characteristic exponents of a polynomial. (Russian) Teor. Funkt. Funktsional. Anal. i Prilozhen. No. 58, 30-40 (1993); English transl. J. Math. Sci. (New York) 85 (1997) no. 5 2164-2171.MR 1258059 (95e:30027)
  • 5. A. Eremenko and L. Lempert, An extremal problem for polynomials. Proc. AMS 122 (1994) 191-193.MR 1207536 (94k:30007)
  • 6. A. Eremenko and W. Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999) 409-415. MR 1704189 (2000k:30001)
  • 7. A.A. Markov, On a question of Mendeleiev, Petersb. Abhandl. LXII (1890) 1-24.
  • 8. Q. Rahman and G. Schmeisser, Les inégalités de Markoff et de Bernstein, Sém. Math. Supér., Presses de l'Université de Montréal, Montréal, QC, 1983.MR 0729316 (85f:41009)
  • 9. G. Pólya, How to solve it? Princeton UP, Princeton NJ 1945.MR 0011666 (6:198f)
  • 10. Ch. Pommerenke, On the derivative of a polynomial, Michigan Math. J. 6 (1959) 373-375.MR 0109208 (22:95)

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Additional Information

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: eremenko@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08640-0
Received by editor(s): September 1, 2005
Received by editor(s) in revised form: January 3, 2006
Published electronically: November 29, 2006
Additional Notes: The author was supported by NSF grants DMS-0100512 and DMS-0244421.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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