Tur\'an type converse Markov inequalities in $L^q$ on a generalized Er\H{o}d class of convex domains

TL;DR

This paper extends Turán and Erf3d's classical polynomial derivative inequalities from simple domains to a broader class of convex domains in the context of $L^q$ norms, providing new bounds and insights.

Contribution

It introduces new Turán-type inequalities for polynomial derivatives on generalized convex domains in $L^q$ spaces, expanding the scope beyond classical domains.

Findings

Derived lower bounds for polynomial derivatives on convex domains

Extended Turán and Erf3d inequalities to new domain classes

Provided asymptotic growth orders for derivative norms

Abstract

P. Tur\'an was the first to derive lower estimations on the uniform norm of the derivatives of polynomials $p$ of uniform norm $1$ on the interval I:=[-1,1] and the disk D:=$\{z \in C~:~|z| \le 1\}$, under the normalization condition that the zeroes of the polynomial p in question all lie in I or D, 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 $\sqrt{n}$ for I and n for D. Already the same year J. Er\H{o}d considered the problem on other domains. In his most general formulation, he extended Tur\'an's order n result on D 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. Tur\'an himself gave comments about the above oscillation question in $L^q$ norm on D. Nevertheless, till recently results were known only for I, D and so-called R-circular domains. Continuing our recent work, also here we investigate the Tur\'an-Er\H{o}d problem on general classes of domains.