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International Conference on Complex Analysis Dedicated to the memory of Andrei Gonchar and Anatoliy Vitushkin
October 8, 2018 11:00–11:50, Moscow, Steklov Mathematical Institute of RAS, Conference hall, 9th floor

Bernstein and Markov inequalities on Jordan arcs

V. Totikab

a Bolyai Institute, University of Szeged
b University of South Florida, Department of Mathematics
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V. Totik
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Abstract: Let $\Gamma$ be a smooth Jordan arc and $x_0\in \Gamma$ a point that is different from the endpoints of $\Gamma$. The talk will be about the smallest constant $B_{x_0}$ for which
$$ |P_n'(x_0)|\le B_{x_0}(1+o(1))n\|P_n\|_\Gamma $$
for all polynomials $P_n$ of degree $n=1,2,…$, where $o(1)$ tends to 0 (uniformly in $P_n$) as $n\to\infty$. Thus, this $B_{x_0}$ is the asymptotically sharp Bernstein factor at the point $x_0$. It turns out that $B_{x_0}=\max (g'_+(x_0),g'_-(x_0))$, where $g$ is the Green's function of $\overline C\setminus \Gamma$ with pole at infinity, and $g'_\pm(x_0)$ are the normal derivatives of $g$ at $x_0$ with respect to the two normals to $\Gamma$ at $x_0$. The proof uses in an essential way a result of Gonchar and Grigorian on the supremum norm of the sum of the principal parts of a meromorphic function on the boundary of the given domain in terms of the supremum norm of the function itself.
The asymptotically best Markov factor $M=M_\Gamma$, i.e. the smallest $M$ for which
$$ \|P_n'\|_\Gamma \le M(1+o(1))n^2\|P_n\|_\Gamma $$
is true, is also expressed in terms of the normal derivative of the associated Green's function. Similar results are established for rational functions provided the poles lie in a closed set disjoint from $\Gamma$.
This is a joint work with Sergei Kalmykov and Béla Nagy.

Language: english

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