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We prove that for an arbitrary real rational function r of degree n , a measure of the set $$\{x\in \mathbb{R}: |r'(x)/r(x)|\ge n\}$$ { x ∈ R : | r… Click to show full abstract
We prove that for an arbitrary real rational function r of degree n , a measure of the set $$\{x\in \mathbb{R}: |r'(x)/r(x)|\ge n\}$$ { x ∈ R : | r ′ ( x ) / r ( x ) | ≥ n } is at most $$2\pi\Theta$$ 2 π Θ ( $$\Theta\approx 1.347$$ Θ ≈ 1.347 is the weak $$(1,1)$$ ( 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.
Keywords:
line;
logarithmic derivative;
derivative rational;
function;
distribution logarithmic;
rational function
Journal Title: Acta Mathematica Hungarica
Year Published: 2020
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Journal Title: Acta Mathematica Hungarica
Year Published: 2020
Link to full text (if available)
Share on Social Media: Sign Up to like & get
recommendations!