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

2022/09/14 by М. А. Комаров, Komarov, Mikhail A.
Mathematics · #Mathematical functions and polynomials #Analytic and geometric function theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2209.06689

Abstract

Let g_n, n=1,2,…, 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-z₁)^-1+…+(z-z_n)^-1, |z₁|=…=|z_n|=1. For any p>0, we establish the bound ∫_-1¹ |g_n(x)|^p dxgt;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≥ 1.

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