Open Access

Schur-Type Inequalities for Complex Polynomials with no Zeros in the Unit Disk

Journal of Inequalities and Applications20072007:090526

https://doi.org/10.1155/2007/90526

Received: 20 March 2007

Accepted: 28 June 2007

Published: 13 December 2007

Abstract

Starting out from a question posed by T. Erdélyi and J. Szabados, we consider Schur-type inequalities for the classes of complex algebraic polynomials having no zeros within the unit disk . The class of polynomials with no zeros in —also known as Bernstein or Lorentz class—was studied in detail earlier. For real polynomials utilizing the Bernstein-Lorentz representation as convex combinations of fundamental polynomials , G. Lorentz, T. Erdélyi, and J. Szabados proved a number of improved versions of Schur- (and also Bernstein- and Markov-) type inequalities. Here we investigate the similar questions for complex polynomials. For complex polynomials, the above convex representation is not available. Even worse, the set of complex polynomials, having no zeros in the unit disk, does not form a convex set. Therefore, a possible proof must go along different lines. In fact, such a direct argument was asked for by Erdélyi and Szabados already for the real case. The sharp forms of the Bernstein- and Markov-type inequalities are known, and the right factors are worse for complex coefficients than for real ones. However, here it turns out that Schur-type inequalities hold unchanged even for complex polynomials and for all monotonic, continuous weight functions. As a consequence, it becomes possible to deduce the corresponding Markov inequality from the known Bernstein inequality and the new Schur-type inequality with logarithmic weight.

[123456789101112131415]

Authors’ Affiliations

(1)
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences

References

  1. Borwein P, Erdélyi T: Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics. Volume 161. Springer, New York, NY, USA; 1995:x+480.View ArticleGoogle Scholar
  2. Milovanović GV, Mitrinović DS, Rassias ThM: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, River Edge, NJ, USA; 1994:xiv+821.View ArticleMATHGoogle Scholar
  3. Pólya Gy, Szegö G: Aufgaben und Lehrsätze aus der Analysis, Vol. II, Die Grundlehren der matehmatischen Wissenschaften in Einzeldarstellungen. Volume 20. Springer, Berlin, Germany; 1925.View ArticleGoogle Scholar
  4. Lorentz GG: The degree of approximation by polynomials with positive coefficients. Mathematische Annalen 1963,151(3):239–251. 10.1007/BF01398235MathSciNetView ArticleMATHGoogle Scholar
  5. Erdélyi T: Estimates for the Lorentz degree of polynomials. Journal of Approximation Theory 1991,67(2):187–198. 10.1016/0021-9045(91)90017-5MathSciNetView ArticleMATHGoogle Scholar
  6. Kopotun KA: Uniform estimates of monotone and convex approximation of smooth functions. Journal of Approximation Theory 1995,80(1):76–107. 10.1006/jath.1995.1005MathSciNetView ArticleMATHGoogle Scholar
  7. Kopotun KA, Leviatan D, Shevchuk IA: Convex polynomial approximation in the uniform norm: conclusion. Canadian Journal of Mathematics 2005,57(6):1224–1248. 10.4153/CJM-2005-049-6MathSciNetView ArticleMATHGoogle Scholar
  8. Kopotun K, Leviatan D, Shevchuk IA: Coconvex approximation in the uniform norm: the final frontier. Acta Mathematica Hungarica 2006,110(1–2):117–151. 10.1007/s10474-006-0010-3MathSciNetView ArticleMATHGoogle Scholar
  9. Polyrakis IA: Finite-dimensional lattice-subspaces of and curves of . Transactions of the American Mathematical Society 1996,348(7):2793–2810. 10.1090/S0002-9947-96-01639-XMathSciNetView ArticleMATHGoogle Scholar
  10. Polyrakis IA: Lattice-subspaces of and positive bases. Journal of Mathematical Analysis and Applications 1994,184(1):1–18. 10.1006/jmaa.1994.1178MathSciNetView ArticleMATHGoogle Scholar
  11. Polyrakis IA: Minimal lattice-subspaces. Transactions of the American Mathematical Society 1999,351(10):4183–4203. 10.1090/S0002-9947-99-02384-3MathSciNetView ArticleMATHGoogle Scholar
  12. Farkas B, Révész SGy: Positive bases in spaces of Polynomials. preprint, 2007 http://arxiv.org/abs/0708.2883 preprint, 2007
  13. Erdélyi T, Szabados J: On polynomials with positive coefficients. Journal of Approximation Theory 1988,54(1):107–122. 10.1016/0021-9045(88)90119-0MathSciNetView ArticleMATHGoogle Scholar
  14. Scheick JT: Inequalities for derivatives of polynomials of special type. Journal of Approximation Theory 1972,6(4):354–358. 10.1016/0021-9045(72)90041-XMathSciNetView ArticleMATHGoogle Scholar
  15. Erdélyi T: Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients. Journal d'Analyse Mathématique 1998, 74: 165–181.View ArticleMATHGoogle Scholar

Copyright

© Szilárd Gy. Révész 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.