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

TL;DR

This paper proves that for convex polygonal domains, the maximum norms of polynomial derivatives on the boundary follow Turán type inequalities of order n, extending classical results to a broader class of domains and norms.

Contribution

The paper establishes Turán type oscillation inequalities of order n for all convex polygonal domains and all 0<q<∞, filling a gap in the understanding of polynomial behavior on these domains.

Findings

Proved order n inequalities for convex polygonal domains.

Extended Turán inequalities to all 0<q<∞ norms.

Confirmed conjecture for polygonal domains.

Abstract

In 1939 P\'al Tur\'an and J\'anos Er\H{o}d initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $K\Subset {\mathbb C}$. While Tur\'an obtained the first results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~ |z|\le 1\}$, Er\H{o}d extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex \emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality. For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$ results for various classes of domains. Further, in the generality of all convex, compact domains we could show a $c n/\log n$ lower bound together with an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order $n$ Tur\'an type inequality. Here we prove this for all \emph{polygonal} convex domains and any $0< q <\infty$.