An inequality for derivatives of polynomials whose zeros are in a half-plane

Let $Q$ be a real polynomial of degree $N$ all of whose zeros lie in the half-plane $\operatorname{Re} z \leqslant 0$. Let $M(r,Q)$ be the maximum of $\left\vert {Q(z)} \right\vert{\text{on}}\left\vert z \right\vert = r$ and $n(r,0)$ the counting function of the zeros of $Q$. It is shown that the inequality $M(r,Q') \leqslant {(2r)^{ - 1}}\left\{ {N + n(r,0)} \right\}M(r,Q)$ holds for $r > 0$. It is also shown that Bernstein's inequality characterizes polynomials.