Certain Bernstein-type $L_p$ inequalities for polynomials
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909409028866048 |
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| author | Rather, N. A. Bhat, Aijaz Guzlar, Suhail |
| author_facet | Rather, N. A. Bhat, Aijaz Guzlar, Suhail |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Certain Bernstein-type $L_p$ inequalities for polynomials Rather, N. A. Bhat, Aijaz Guzlar, Suhail Complex Variables 30A10, 30C10, 41A17 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. |
| title | Certain Bernstein-type $L_p$ inequalities for polynomials |
| topic | Complex Variables 30A10, 30C10, 41A17 |
| url | https://arxiv.org/abs/2411.19811 |