Thinnest covering of the Euclidean plane with incongruent circles (Q1689539)

For \(0 < q \leq 1\), let \(D(q)\) denote the smallest (lower) density of a covering of the Euclidean plane by circles with radii from the interval \([q,1]\). \textit{L. Fejes Tóth} and \textit{J. Molnár} conjectured that \[ D(q) \geq S(q):= \min\limits_{0 < x \leq 1} \frac{\pi-2\left(1-q^2\right)\arctan\left(\sqrt{1-x^2}/(x+q)\right)}{2\sqrt{1-x^2}(x+q)} \] [Math. Nachr. 18, 235--243 (1958; Zbl 0083.38501)]. In the paper under review that conjecture was made accessible to a computer-aided numerical confirmation. Moreover, bounds of \(S(q)\) are given.