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

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?

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

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