Geodesic
Asymptotic Markov inequality on Jordan arcs
Sbornik. Mathematics, Tome 208 (2017) no. 3, pp. 413-432

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Markov's inequality for the derivative of algebraic polynomials is considered on $C^2$-smooth Jordan arcs. The asymptotically best estimate is given for the $k$th derivative for all $k=1,2,\dots$ . The best constant is related to the behaviour around the endpoints of the arc of the normal derivative of the Green's function of the complementary domain. The result is deduced from the asymptotically sharp Bernstein inequality for the $k$th derivative at inner points of a Jordan arc, which is derived from a recent result of Kalmykov and Nagy on the Bernstein inequality on analytic arcs. In the course of the proof we shall also need to reduce the analyticity condition in this last result to $C^2$-smoothness. Bibliography: 21 titles.
Keywords: Markov inequality, normal derivative of Green's function.
Mots-clés : Jordan arc 
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     author = {V. Totik},
     title = {Asymptotic {Markov} inequality on {Jordan} arcs},
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V. Totik. Asymptotic Markov inequality on Jordan arcs. Sbornik. Mathematics, Tome 208 (2017) no. 3, pp. 413-432. http://geodesic.mathdoc.fr/item/SM_2017_208_3_a6/