Let $n$ be a non-zero natural number. Let $R \in \mathbb{C}_{n-1}[X]$ and $t \in [-1,1]$. Show that $$|R(t)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|R(x)\sqrt{1-x^2}\right|.$$ One may consider the polynomial $S_t(X) = R(tX)$.
Let $n$ be a non-zero natural number. Let $R \in \mathbb{C}_{n-1}[X]$ and $t \in [-1,1]$. Show that
$$|R(t)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|R(x)\sqrt{1-x^2}\right|.$$
One may consider the polynomial $S_t(X) = R(tX)$.