Asymptotic Markov inequality on Jordan arcs

Abstract

Markov's inequality for the derivative of algebraic polynomials is considered on -smooth Jordan arcs. The asymptotically best estimate is given for the th derivative for all . 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 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 -smoothness. Bibliography: 21 titles.