SCHWEITZER 1980 Problem 7

Let $n \geq 2$ be a natural number and $p(x)$ a real polynomial of degree at most $n$ for which $$\max _{ -1 \leq x \leq 1} |p(x)| \leq 1, p(-1)=p(1)=0 \ .$$ Prove that then $$|p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1-x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\; \left( -\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \ \right).$$

J. Szabados