On maximum modulus of polynomials (Q2769251)

Let \(p(z)\) denote a polynomial of degree \(n\geq 3\), \(q(z)= 2^n\overline{p(\overline z^{-1})}\) and let \(M(f,r)= \max[|f(z):|z|= r|]\). An objective of this note is to show that for each positive integer \(s\), \(R\geq 0\), \(\theta\in \langle 0,2\pi\rangle\) there holds the inequality NEWLINE\[NEWLINE\begin{multlined} |p(\text{Re}^{i\theta})|^s+|q(\text{Re}^{i\theta})|^s\leq (R^{ns}+ 1) M^s(p,1)-\\ \Biggl({R^{ns}- 1\over ns}- {R^{ns-2}- 1\over ns- 2}\Biggr) \|p'(0)|-|q'(0)\|sM^{s-1}(p, 1).\end{multlined}NEWLINE\]NEWLINE{}.