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Extremal Problems and Inequalities of Markov-Bernstein Type for Polynomials

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Book cover Analytic and Geometric Inequalities and Applications

Part of the book series: Mathematics and Its Applications ((MAIA,volume 478))

Abstract

The classical Markov (1889) and Bernstein (1912) inequalities and corresponding extremal problems were generalized for various domains, various norms and for various subclasses for polynomials, both algebraic and trigonometric. Beside some classical results in uniform norm, we give a short account of L r inequalities of Markov type for algebraic polynomials, with a special attention to the case r = 2. We also study extremal problems of Markov’s type

$$Cn,m = \begin{array}{*{20}{c}} {\sup } \\ {P \in {P_n}} \end{array}\frac{{\left\| {\mathcal{D}mP} \right\|}}{{\left\| {{A^{m/2}}P}\right\|}}$$

, where P n is the class of all algebraic polynomials of degree at most n, dλ(t) = w(t)dt is a nonnegative measure corresponding to the classical orthogonal polynomials, A ∈ P 2, and D m is a differential operator defined by

$$\mathcal{D}mP = \frac{{{d^m}}}{{d{t^m}}}\left[ {{A^m}P} \right]\left( {P \in {P_{n,}}m \geqslant 1} \right) $$

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Electronic Engineering, P. O. Box 73, 18000, Niš, Yugoslavia

    Gradimir V. Milovanović

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  1. Gradimir V. Milovanović
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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, Greece

    Themistocles M. Rassias

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    Hari M. Srivastava

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© 1999 Springer Science+Business Media Dordrecht

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Milovanović, G.V. (1999). Extremal Problems and Inequalities of Markov-Bernstein Type for Polynomials. In: Rassias, T.M., Srivastava, H.M. (eds) Analytic and Geometric Inequalities and Applications. Mathematics and Its Applications, vol 478. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4577-0_15

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  • DOI: https://doi.org/10.1007/978-94-011-4577-0_15

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5938-1

  • Online ISBN: 978-94-011-4577-0

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