Turan type converse Markov inequalities in L-q on a generalized Erod class of convex domains

Glazyrina, P. Y. and Révész, Szilárd (2017) Turan type converse Markov inequalities in L-q on a generalized Erod class of convex domains. JOURNAL OF APPROXIMATION THEORY, 221. pp. 62-76. ISSN 0021-9045

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Abstract

P.Turan was the first to derive lower estimations on the uniform norm of the derivatives of polynomials p of uniform norm 1 on the disk D := {z is an element of C : vertical bar z vertical bar <= 1} and the interval I := [-1, 1], under the normalization condition that the zeros of the polynomial p in question all lie in D or I, resp. Namely, in 1939 he proved that with n := deg p tending to infinity, the precise growth order of the minimal possible derivative norm is n for D and root n for II. Already the same year J.Erod considered the problem on other domains. In his most general formulation, he extended Turan's order n result on 101 to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains. Turk himself gave comments about the above oscillation question in L-q norm on D. Nevertheless, till recently results were known only for 11),)1 and so-called R-circular domains. Continuing our recent work, also here we investigate the Turan-Erod problem on general classes of domains. (C) 2017 Elsevier Inc. All rights reserved.

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