A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial
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| Format: | Preprint |
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2025
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| _version_ | 1866915730428002304 |
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| author | Rather, N. A. Bhat, Danish Rashid Bhat, Tanveer |
| author_facet | Rather, N. A. Bhat, Danish Rashid Bhat, Tanveer |
| contents | Let \( P(z) \) be a polynomial of degree \( n \) and $α\in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_αP(z) \) and is defined by $D_αP(z): = nP(z) + (α-z)P'(z)$. The polar derivative \( D_αP(z) \) is a polynomial of degree at most \( n - 1 \) and it generalizes the ordinary derivative \( P'(z) \). In this paper, we establish some \( L_p \) inequalities for the polar derivative of a polynomial with all its zeros located within a prescribed disk. Our results refine and generalize previously known findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_06539 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial Rather, N. A. Bhat, Danish Rashid Bhat, Tanveer Complex Variables 26D10, 41A17, 30A10, 26D05 Let \( P(z) \) be a polynomial of degree \( n \) and $α\in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_αP(z) \) and is defined by $D_αP(z): = nP(z) + (α-z)P'(z)$. The polar derivative \( D_αP(z) \) is a polynomial of degree at most \( n - 1 \) and it generalizes the ordinary derivative \( P'(z) \). In this paper, we establish some \( L_p \) inequalities for the polar derivative of a polynomial with all its zeros located within a prescribed disk. Our results refine and generalize previously known findings. |
| title | A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial |
| topic | Complex Variables 26D10, 41A17, 30A10, 26D05 |
| url | https://arxiv.org/abs/2505.06539 |