Note on the lower estimate of optimal Lebesgue constants (Q1333038)

Let \(X: -1\leq x_{n- 1}< x_{n- 2}<\cdots< x_ 0\leq 1\) denote an array of \(n\) arbitrary points in \([- 1,1]\). Let \(\ell_ k(X, x)= \displaystyle{\prod^{n- 1}_{{\begin{smallmatrix} i= 0\\ i\neq k\end{smallmatrix}}}} {x- x_ i\over x_ k- x_ i}\), \[ \begin{aligned} L_{n- 1}(X, x) & = \sum^{n- 1}_{k= 0} | \ell_ k(X, x)|,\\ \Lambda_{n- 1}(X) & = \max_{-1\leq x\leq 1} L_{n- 1} (X, x),\\ \Lambda^*_{n- 1} & = \min_ X \Lambda_{n- 1}(x)\qquad\text{and}\\ \chi & = {2\over \pi} \biggl(\gamma+ \log{4\over \pi}\biggr),\end{aligned} \] where \(\gamma\) is the Euler's constant. In this note the author proves that \(\Lambda^*_{n- 1}- {2\over \pi}\log n- \chi> {\pi\over 18 n^ 2}- {49 \pi^ 3\over 10800 n^ 4}\). At the end he mentions that his result was also proved independently by \textit{Stolzmann} [Diplomarbeit Universität Osnabrück, 1992].