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arXiv · 2411.19811

Certain Bernstein-type $L_p$ inequalities for polynomials

N. A. Rather·Aijaz Bhat·Suhail Guzlar
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Abstract

Let $P(z)$ be a polynomial of degree $n,$ then it is known that for $α\in\mathbb{C}$ with $|α|\leq \frac{n}{2},$ \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 $|P^\prime(z)|$ 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 $α$ can be relaxed. We also prove similar inequalities for polynomials with restricted zeros.

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N. A. Rather, Aijaz Bhat, Suhail Guzlar. 2024-11-29. Certain Bernstein-type $L_p$ inequalities for polynomials. https://arxiv.org/abs/2411.19811

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