A Newman type bound for $L_p[-1,1]$-means of the logarithmic derivative of polynomials having all zeros on the unit circle

TL;DR

This paper establishes a sharp lower bound for the Lp-norms of the logarithmic derivatives of polynomials with zeros on the unit circle, revealing their non-density in Lp spaces and extending classical estimates.

Contribution

It provides a Newman-type lower bound for Lp-means of logarithmic derivatives of polynomials with zeros on the unit circle, for all p>0, with sharp order in n.

Findings

The bound rac{1}{n^{1-p}} for the Lp-norms of the derivatives.

The result is sharp in the order of n, confirming the growth rate.

The set of such derivatives is not dense in Lp[-1,1] for p ge 1.

Abstract

Let $g_n$, $n=1,2,\dots$, be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form $g_n(z)=(z-z_{1})^{-1}+\dots+(z-z_{n})^{-1}$, $|z_1|=\dots=|z_n|=1$. For any $p>0$, we establish the bound \[\int_{-1}^1 |g_n(x)|^p\, dx>C_p\, n^{p-1},\] sharp in the order of the quantity $n$, where $C_p>0$ is a constant, depending only on $p$. The particular case $p=1$ of this inequality can be considered as a stronger variant of the well-known estimate $\iint_{|z|<1} |g_n(z)|\,dxdy>c>0$ for the area integral of $g_n$, obtained by D.J. Newman (1972). The result also shows that the set $\{g_n\}$ is not dense in the spaces $L_p[-1,1]$, $p\ge 1$.