Distribution of the logarithmic derivative of a rational function on the line

Abstract

We prove that for an arbitrary real rational function r of degree n, a measure of the set $$\{x\in \mathbb{R}: |r'(x)/r(x)|\ge n\}$$ is at most $$2\pi\Theta$$ ( $$\Theta\approx 1.347$$ is the weak $$(1,1)$$ -norm of the Hilbert transform), and this bound is extremal. A problem of rational approximations on the whole real line is also considered.