Certain Bernstein-type $L_p$ inequalities for polynomials

Let $P(z)$ be a polynomial of degree $n,$ then it is known that for $\alpha \in \mathbb{C}$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le \frac{n}{2},$ \begin{align*} \underset{|z|=1}{\max}|\left|zP^{\prime}(z)-αP(z)\right|\leq \left|n-α\right|\underset{|z|=1}{\max}|P(z)|. \end{align*} This inequality includes Bernstein’s inequality, concerning the estimate for $\mathrm{\mid }{P}^{\mathrm{\prime }}(z)\mathrm{\mid }$ over $\mathrm{\mid }z\mathrm{\mid }\le 1,$ 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.