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 \(L_p[-1,1]\), \(p\ge 1\).