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Extensions of Schur’s Inequality for the Leading Coefficient of Bounded Polynomials with Two Prescribed Zeros

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 41)

Abstract

We extend Schur’s Chebyshev-type inequality([18], p. 285) for the leading coefficient of polynomials that are uniformly bounded on the interval [ − 1, 1] and vanish at its endpoints. Our extension is threefold: We obtain sharp V.A. Markov-type estimates for all single coefficients as well as sharp Szegö-type estimates for consecutive pairs of coefficients of such polynomials, and both these estimates imply Schur’s inequality for the leading coefficient. Thirdly, we consider a larger class of admissible polynomials by replacing uniform with pointwise boundedness on [ − 1, 1].

References

  1. 1.
    P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics Vol. 161, Springer, New York, 1995Google Scholar
  2. 2.
    P.L. Chebyshev, Théorie des mécanismes connus sous le nom de parallélogrammes, Mem. Acad. Sci. St. Petersburg 7, 539–568 (1854); available at http://www.math.technion.ac.il/hat/fpapers/cheb11.pdf
  3. 3.
    D.P. Dryanov, M.A. Qazi and Q.I. Rahman, Certain extremal problems for polynomials, Proc. Am. Math. Soc. 131, 2741–2751 (2003)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    R.J. Duffin and A.C. Schaeffer, A refinement of an inequality of the brothers Markoff, Trans. Am. Math. Soc. 50, 517–528 (1941)MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Erdös and G. Szegö, On a problem of I. Schur, Ann. Math. (2) 43, 451–470 (1942)Google Scholar
  6. 6.
    P. Erdös, Some remarks on polynomials, Bull. Am. Math. Soc. 53, 1169–1176 (1947)MATHCrossRefGoogle Scholar
  7. 7.
    W. Markoff (V.A. Markov), Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen, Math. Ann. 77, 213–258 (1916); Russian original of 1892 available at http://www.math.technion.ac.il/hat/fpapers/vmar.pdf
  8. 8.
    L. Milev and G. Nikolov, On the inequality of I. Schur, J. Math. Anal. Appl. 216, 421–437 (1997)Google Scholar
  9. 9.
    G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in Polynomials - Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994MATHCrossRefGoogle Scholar
  10. 10.
    M.A. Qazi and Q.I. Rahman, Some coefficient estimates for polynomials on the unit interval, Serdica Math. J. 33, 449–474 (2007)MathSciNetMATHGoogle Scholar
  11. 11.
    H.-J. Rack, On V.A. Markov’s and G. Szegö’s inequality for the coefficients of polynomials in one and several variables, East J. Approx. 14, 319–352 (2008)Google Scholar
  12. 12.
    H.-J. Rack, On the length and height of Chebyshev polynomials in one and two variables, East J. Approx. 16, 35–91 (2010)MathSciNetGoogle Scholar
  13. 13.
    Q.I. Rahman and G. Schmeisser, Inequalities for polynomials on the unit interval, Trans. Am. Math. Soc. 231, 93–100 (1977)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs New Series Vol. 26, Oxford, 2002Google Scholar
  15. 15.
    Th.J. Rivlin, Chebyshev Polynomials - From Approximation Theory to Algebra and Number Theory, Second Edition, J. Wiley and Sons, New York, 1990MATHGoogle Scholar
  16. 16.
    W.W. Rogosinski, Some elementary inequalities for polynomials, Math. Gaz. 39, 7–12 (1955)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    A. Schönhage, Approximationstheorie, Walter de Gruyter, Berlin, 1971MATHCrossRefGoogle Scholar
  18. 18.
    I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall, Math. Zeitschr. 4, 271–287 (1919)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    I. Schur, Gesammelte Abhandlungen, Band I, II, III (A. Brauer and H. Rohrbach, eds.), Springer, New York, 1973Google Scholar
  20. 20.
    J.A. Shohat, On some properties of polynomials, Math. Zeitschr. 29, 684–695 (1929)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Studies in Memory of I. Schur, Progress in Mathematics Vol. 210 (A. Joseph, A. Melnikov and R. Rentschler, eds.), Birkhäuser, Boston, 2003Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.HagenGermany

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