Let \[ p(z)=a_0+a_1z+\cdots+a_nz^n, \qquad s=\sqrt{1-|z|^2}, \qquad |z|\le1, \] and let \(e_1,\ldots,e_{n+1}\) be the standard basis of \(\mathbb C^{n+1}\).
Let \(U\) be the \((n+1)\times(n+1)\) matrix whose columns, in order, are \[ ze_1+se_2,\ e_3,\ldots,e_{n+1},\ se_1-\overline z e_2. \] Show that \(U\) is unitary, hence its eigenvalues \(\lambda_1,\ldots,\lambda_{n+1}\) have modulus \(1\).
Show that \(z^k=e_1^TU^ke_1\), and hence \(p(z)=e_1^Tp(U)e_1\).
Deduce \[ \max_{|z|\le1}|p(z)|\le\|p(U)\|. \]
If \(D=\operatorname{diag}(\lambda_1,\ldots,\lambda_{n+1})\), show \[ \|p(U)\|=\|p(D)\|=\max_i|p(\lambda_i)|. \]
Conclude that \[ \max_{|z|\le1}|p(z)|=\max_{|z|=1}|p(z)|. \]