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 $g_n$, $n=1,2,\dots$, 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 $L_p[-1,1]$, $p\ge 1$.