Some properties of weighted hyperbolic polynomials (Q1907692)

The author studies relations between transfinite diametres, Chebyshev constants and equilibrium potentials of compact subsets of the unit disk \(U\) with respect to the ``distance'' \(\frac{|z- \zeta|}{|1- z\overline\zeta|} w(z) w(\zeta)\), where \(w\) is a positive function such that \(\log w\) is continuous on \(U\). He uses methods which imitate those of earlier papers [see e.g. \textit{H. N. Maskhar} and \textit{E. B. Saff}, Constructive Approximation 8, No. 1, 105-124 (1992; Zbl 0747.31001)], where such problems were studied for compact subsets of the complex plane with respect to the ``distance'' \(|z- \zeta|w(z) w(\zeta)\). \{Reviewer's remark: Proposition 3.1 of the paper is not true. Indeed, take \(w(z):= \max\{|z|, \varepsilon\}\), where \(0< \varepsilon< \frac 15\) and let \(E:= \{|z|= \frac 12\}\). Then \(\text{Chh}(w, E)\leq \varepsilon/4\leq \frac 1{20}\) and \(\text{Trh}(w, E)\geq 1/16\), which shows that the hyperbolic transfinite diameter of \(E\) is different from the hyperbolic Chebyshev constant of \(E\), contrary to the claim of the Proposition 3.1.\}.