Question 50
Let $n$ be an integer with $n \ge 2$, and let $a_0, a_1, a_2, \dots, a_n$ be complex numbers with $a_n \ne 0$. Prove that the following statements are equivalent:
(P) $|a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0| \le |a_n + a_0|$, for every complex number $z$ of modulus 1,
(Q) $a_1 = a_2 = \dots = a_{n-1} = 0$ and $\frac{a_0}{a_n} \in [0, \infty)$.
(P) $|a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0| \le |a_n + a_0|$, for every complex number $z$ of modulus 1,
(Q) $a_1 = a_2 = \dots = a_{n-1} = 0$ and $\frac{a_0}{a_n} \in [0, \infty)$.