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 $|\alpha|\leq \frac{n}{2},$ \begin{align*} \underset{|z|=1}{\max}|\left|zP^{\prime}(z)-\alpha P(z)\right|\leq \left|n-\alpha\right|\underset{|z|=1}{\max}|P(z)|. \end{align*} This inequality includes Bernstein's inequality, concerning the estimate for $|P^\prime(z)|$ over $|z|\leq 1,$ 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.