Question 27
Let $P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_m z^m$ be polynomial with complex coefficients such that $a_m \neq 0$, $a_n \neq 0$ and $n \gt m$. Prove that
$$
\max_{|z|=1}\{|P(z)|\} \ge \sqrt{\sum_{k=m}^{n} |a_k|^2 + 2|a_m a_n|}.
$$
$$
\max_{|z|=1}\{|P(z)|\} \ge \sqrt{\sum_{k=m}^{n} |a_k|^2 + 2|a_m a_n|}.
$$