We prove that for an arbitrary real rational functionr 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.

中文翻译：

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在线上有理函数的对数导数的分布

我们证明，对于 n 次任意实有理函数，集合 $$\{x\in \mathbb{R} 的度量： |r'(x)/r(x)|\ge n\}$$至多是 $$2\pi\Theta$$（$$\Theta\approx 1.347$$ 是弱 $$(1,1)$$ - Hilbert 变换的范数），并且这个界限是极值。还考虑了整条实线上的有理逼近问题。