On a theorem of Bernstein (Q1073943)

The case of equality in the following theorem of Bernstein has been proved. Theorem B. Let P(z) and Q(z) be polynomials satisfying the conditions that Q(z) has all its zeros in \(| z| \leq 1\) and the degree of P(z) does not exceed that of Q(z). If \[ (1)\quad | P(z)| \leq | Q(z)| \quad on\quad | z| =1 \] then \[ (2)\quad | P'(z)| \leq | Q'(z)| \quad on\quad | z| =1. \] In this note it is proved that if there is equality in (2) at any point \(\mu\) on \(| z| =1\), where \(Q(\mu)\neq 0\) then \(P(z)=\alpha Q(z)\), \(| \alpha | =1\).