Let $P(z)=\sum_{k=0}^na_kz^k$ be a polynomial of degree $n$ having all its zeros on the unit circle. Let $M=\max_{0\leq k\leq n}\lvert a_k\rvert$. The polynomial $P(z)=z^n+1$  has  $\max_{\lvert z\rvert=1}\lvert P(z)\rvert= 2$.  Intuitively, it appears that  $\max_{\lvert z\rvert=1}\lvert p(z)\rvert\geq 2M$.  Kindly share your opinion on the validity of this bound $2M$. Can it be further sharpened?