SCHWEITZER 1970 Problem 8

Let $\pi_n(x)$ be a polynomial of degree not exceeding $n$ with real coefficients such that $$|\pi_n(x)| \leq \sqrt{1-x^2} \;\textrm{for}\ \;-1\leq x \leq 1 \ .$$ Then $$|\pi'_n(x)| \leq 2(n-1).$$

P. Turan