Certain Bernstein-type $$L_p$$ inequalities for polynomials

Abstract

Let P(z) be a polynomial of degree n, then it is known that for $$\alpha \in {\mathbb {C}}$$ with $$|\alpha |\le \frac{n}{2},$$ $$\begin{aligned} \underset{|z|=1}{\max }|\left| zP^{\prime }(z)-\alpha P(z)\right| \le \left| n-\alpha \right| \underset{|z|=1}{\max }|P(z)|. \end{aligned}$$ This inequality includes Bernstein’s inequality, concerning the estimate for $$|P^\prime (z)|$$ over $$|z|\le 1,$$ as a special case. In this paper, we extend this inequality to $$L_p$$ norm which among other things shows that the condition on $$\alpha $$ can be relaxed. We also prove similar inequalities for polynomials with restricted zeros.