IMC 2009 · Problem 4

Let $p(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^n$ be a complex polynomial. Suppose that $1 = c_0 \ge c_1 \ge \dots \ge c_n \ge 0$ is a sequence of real numbers which is convex (i.e. $2c_k \le c_{k-1} + c_{k+1}$ for every $k = 1, 2, \dots, n-1$), and consider the polynomial

$$q(z) = c_0 a_0 + c_1 a_1 z + c_2 a_2 z^2 + \dots + c_n a_n z^n.$$

Prove that

$$\max_{|z| \le 1} |q(z)| \le \max_{|z| \le 1} |p(z)|.$$