On the connection of Posse's \(L_ 1\)- and Zolotarev's maximum-norm problem (Q1178501)

The Posse's problem is to describe the numbers \(a,b\in\mathbb{R}\), \(1<a<b\), for which the functional \(\int^ 1_{-1}| P_ n|+(-1)^ n\int^ b_ a P_ n\) attains its minimum on the set of polynomials of degree \(n\) with leading coefficient one. The Zolotarev's problem is to find among all polynomials of the form \(x^ n-n\sigma x^{n-1}+a_ 2x^{n-1}+\cdots+a_ n\), where \(\sigma\in\mathbb{R}\) is given and \((a_ 2,a_ 3,\dots,a_ n)\in\mathbb{R}^{n- 1}\), the one which deviates least from zero on [-1,1] in the maximum norm. The aim is to demonstrate that the solution of Zolotarev's problem contains the solution of Posse's problem.