Bound-Preserving Operators and Bernstein Type Inequalities

Abstract

Let D be the unit disk in the complex plane ℂ and $$\Vert p\Vert:=\max_{z \in \partial {\rm D}}\mid p(z)\mid$$, where \(p(z)=\sum_{k=0}^{n}a_k(p)z^k\) is a polynomial of degree at most n and a k(p) ∈ ℂ;. The following sharpening of Bernstein’s inequality $$\Vert p^{\prime}\Vert+{2n\over n+2}\mid a_0(p)\mid \le n \Vert p\Vert$$ has been proved by Ruscheweyh. Our main contribution concerns the case of equality which has remained unsolved since 1982. We prove another inequality of Bernstein type that leads to an improvement of the upper bound for \(\Vert p^{\prime}\Vert\) under some additional condition.