Laboratoire de Mathématiques de Besançon - UMR 6623 CNRS

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6 février 2018: 1 événement

  • Séminaire d’Analyse Fonctionnelle

    Mardi 6 février 13:45-15:00 - Safoura Zadeh - IMPAN (Varsovie)

    Markov’s problem for k-absolutely monotone polynomials

    Résumé : Studying the specific gravity of a solution as a function of the percentage of the dissolved substance D. Mendeleev came across a problem that, in mathematical term, asks how large $|p’(x)|$ on $[-1,1]$ can be if $p(x)$ is a quadratic polynomial with $|p(x)|\leq 1$ on $[-1,1]$. Mendeleev showed that $|p’(x)|\leq 4$ and talked to A. A. Markov about the problem. Markov found the question fascinating and studied the problem for a real polynomial of degree at most n. He proved that $|p’(x)| \leq n^2$ on $[-1,1]$ when $|p(x)|\leq 1$ on $[-1,1]$. Later Markov’s brother V. A. Markov studied the problem for the $m$-th derivative, when $m<n$.
    In this talk we study Markov’s problem for $m$-th derivative of a family of polynomials called k-absolutely monotone polynomials. As a consequence, we obtain a sharp version of Bernstein inequality for monotone polynomials as well as a new simple proof of Markov’s inequality for monotone polynomials. This is based on a joint work with Oleksiy Klurman.

    En savoir plus : Séminaire d’Analyse Fonctionnelle