REAL

Turán—Erőd Type Converse Markov Inequalities on General Convex Domains of the Plane in the Boundary Lq Norm

Glazyrina, Polina Yu. and Révész, Szilárd (2018) Turán—Erőd Type Converse Markov Inequalities on General Convex Domains of the Plane in the Boundary Lq Norm. Proceedings of the Steklov Institute of Mathematics, 303 (1). pp. 78-104. ISSN 0081-5438

[img]
Preview
Text
1805.04822v1.pdf

Download (655kB) | Preview

Abstract

In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on I:=[−1,1] (interval) and D:={z∈C:|z|≤1} (disk) under the normalization condition that the zeroes of the polynomial in question all lie in I or D, respectively. For the maximum norm he found that with n:= deg p tending to infinity, the precise growth order of the minimal possible derivative norm is √n for I and n for D. J. Erőd continued the work of Turán considering other domains. Finally, about a decade ago the growth of the minimal possible ∞-norm of the derivative was proved to be of order n for all compact convex domains. Although Turán himself gave comments about the above oscillation question in Lq norms, till recently results were known only for D and I. Recently, we have found order n lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be n. Now we prove that in Lq norm the oscillation order is at least n/log n for all compact convex domains.

Item Type: Article
Subjects: Q Science / természettudomány > QA Mathematics / matematika
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 24 Apr 2019 08:49
Last Modified: 24 Apr 2019 08:49
URI: http://real.mtak.hu/id/eprint/92902

Actions (login required)

Edit Item Edit Item