Tur\'an-Er\H{o}d type converse Markov inequalities on general convex domains of the plane in $\boldsymbol{L^q}$

TL;DR

This paper extends Turán's classical inequalities to general convex domains in the plane within the $L^q$ norm setting, establishing a lower bound of order $n/ ext{log} n$ for the derivative norms of polynomials.

Contribution

It proves a new lower bound of order $n/ ext{log} n$ for the derivative norms of polynomials on convex domains in the $L^q$ norm, generalizing previous results.

Findings

Lower bound of order $n/ ext{log} n$ for convex domains in $L^q$ norm

Extension of Turán-Erőd inequalities to general convex domains

Confirmation of conjectured growth order for polynomial derivatives

Abstract

In 1939 P. Tur\'an started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and $D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (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 $n:=\mathop{\mathrm{deg}} p$ tending to infinity, the precise growth order of the minimal possible derivative norm is $\sqrt{n}$ for $I$ and $n$ for $D$. J. Er\H{o}d continued the work of Tur\'an considering other domains. Finally, a decade ago the growth of the minimal possible $\infty$-norm of the derivative was proved to be of order $n$ for all compact convex domains. Although Tur\'an himself gave comments about the above oscillation question in $L^q$ 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 $L^q$ norm the oscillation order is at least $n/\log n$ for all compact convex domains.