SCHWEITZER 1953 Problem 3

Denoting by $E$ the class of trigonometric polynomials of the form $f(x)=c_{0}+c_{1}cos(x)+\dots +c_{n} cos(nx)$ , where $c_{0} \geq c_{1} \geq \dots \geq c_{n}>0$ , prove that $(1-\frac{2}{\pi})\frac{1}{n+1}\leq min_{{f\epsilon E}}( \frac{max_{\frac{\pi}{2}\leq x\leq \pi} \left | f(x) \right |}{max_{0\leq x\leq 2\pi} \left | f(x) \right |})\leq (\frac{1}{2}+\frac{1}{\sqrt{2}})\frac{1}{n+1}$ . (S. 24)