On a paper of Erod and Turan-Markov inequalities for non-flat convex domains

TL;DR

This paper reviews Erdős and Turán-Markov inequalities for convex domains, highlighting historical results and recent advances, especially regarding boundary curvature's influence on polynomial derivative bounds.

Contribution

It revisits Erdős's classical work and discusses recent developments on curvature effects in Markov inequalities for convex sets.

Findings

Erdős's work on polynomial inequalities is foundational and historically significant.

Recent research emphasizes the role of boundary curvature in lower bounds of polynomial derivatives.

The paper consolidates old and new results, advancing understanding of polynomial behavior in convex domains.

Abstract

For a convex domain K in the complex plane C, the well-known general Markov inequality asserting that a polynomial p of degree n ||p'|| < c(K) n^2 ||p|| holds. On the other hand for polynomials in general, ||p'|| can be arbitrarily small as compared to ||p||. The situation changes when we assume that the polynomials have all their zeroes in the convex body K. This problem of lower bound for Markov factors was first investigated by Tur\'an in 1939. Tur\'an showed ||p'|| \ge n/2 ||p|| for the unit disk D and ||p'|| > c \sqrt{n} ||p|| for the unit interval I:=[-1,1]. Soon after that, J. Er\H od published a long article, discussing various extensions of the results and methods of Tur\'an. For decades, Er\H od's paper was quoted only for the explicit calculation of the exact constant of the interval case. However, in recent years Levenberg and Poletsky, Erd\'elyi and also the author investigated Tur\'an's problem for various sets - basically, convex domains. In this context the much richer content of Er\H od's work is to be realized again. Thus, the aim of the paper is twofold. On the one hand we give an account of the half-forgotten, old Hungarian article of Er\H od, also commemorating its author. On the other hand we report on recent developments with particular emphasis on development of one of the key observations of Er\H od, namely, the role of the curvature of the boundary curve in the estimation of the lower bound of Markov factors.