Tur\'an type oscillation inequalities in $L^q$ norm on the boundary of convex domains

TL;DR

This paper extends Turán's classical inequalities to Lq norms on convex domain boundaries, establishing that the oscillation order is n for a broad class of convex domains, including smooth and certain polygonal shapes.

Contribution

It proves that in Lq norms, the oscillation order matches that of the disk and interval cases for a wide class of convex domains, filling a gap in Turán-type inequalities.

Findings

Oscillation order is n for smooth convex domains.

Oscillation order is n for convex polygons without acute angles.

Results generalize Turán's inequalities to Lq norms on convex boundaries.

Abstract

Some 76 years ago P. Tur\'an was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I:=[-1,1] and on the unit disk D:={z : |z| <= 1} under the normalization condition that the zeroes of the polynomial all lie in the interval or in the disk, respectively. He proved that with n:=deg p tending to infinity, the precise growth order of the minimal possible ratio of the derivative norm and the norm is square-root{n} for I and n for D. J. Er\"od continued the work of Tur\'an and extended his results to several other domains. The growth of the minimal possible ratio of the infinity norm of the derivative and the polynomial itself was proved to be of order n for all compact convex domains a decade ago. Although Tur\'an himself gave comments about the above oscillation question in Lq norms, till recently results were known only for D and I. Here we prove that in Lq norm the oscillation order is again n for a certain class of convex domains, including all smooth convex domains and also convex polygonal domains having no acute angles at their vertices.