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

Abstract:

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 $\left | {Q(z)} \right |{\text {on}}\left | z \right | = r$ 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.