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

Abstract

Let $g_n$, $n=1,2,\ldots$, be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form $g_n(z)=(z-z_{1})^{-1}+\cdots +(z-z_{n})^{-1}$, $|z_1|=\cdots =|z_n|=1$. For any $p>0$, we establish the bound

$$\begin{aligned} \int _{-1}^1 |g_n(x)|^p\, dx>C_p\, n^{p-1}, \end{aligned}$$

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 Newman (Am Math Mon 79(9):1015–1016, 1972). The result also shows that the set $\{g_n\}$ is not dense in the spaces $L_p[-1,1]$, $p\ge 1$.