Best constants in one-sided weak-type inequalities (Q1270877)

This elegant paper deals with the inequalities of the form \[ m \biggl(\bigl\{x \in\mathbb{R}; r'(x)/r(x) \geq\lambda \bigr\} \biggr)\leq Cn/\lambda,\;\lambda>0 \tag{R} \] and \[ m\biggl(\bigl\{x\in\mathbb{R}; f(x)\geq\lambda \bigr\}\biggr) \leq C\| f\|_{L^1}/ \lambda,\tag{H} \] where in (R) \(r\) is an arbitrary rational function of degree \(n\) and in (H) \(\widetilde f\) is the Hilbert transform of the real function \(f\in L^1 (\mathbb{R})\). In both cases \(m\) stands for the Lebesgue measure on \(\mathbb{R}\). These are one-sided inequalities, as opposed to two-sided ones where \(r'/r\) and \(f\) appear under the sign of absolute value. Among several other related results it is shown that the best constant in (R) is \(C=2\pi\) and in (H) the best constant is \(C=1\) (and is attained with a non-zero \(f)\). The constant \(C=2\pi\) was conjectured earlier by \textit{P. Borwein}, \textit{E. A. Rakhmanov} and \textit{E. B. Saff} [Constructive Approximation 12, No. 2, 223-240 (1996; Zbl 0870.41010)].