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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.00328 |
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| _version_ | 1866916753767923712 |
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| author | Kumar, Deepak Tripathi, D. Hans, Sunil |
| author_facet | Kumar, Deepak Tripathi, D. Hans, Sunil |
| contents | Let $P(z)$ be a polynomial of degree $n$. In $2004$, Aziz and Rather \cite{aziz2004some} investigated the dependence of \[\bigg|P(Rz)-αP(z)+β\biggl\{\biggl(\frac{R+1}{2}\biggr)^n-|α|\biggr\}P(z)\bigg|, \ \text{for} \ z \in B(\mathbb{D}),\] on $\max_{z\in B(\mathbb{D})}|P(z)|$, for every real and complex number $α, β$ satisfying $|α| \leq 1$, $|β| \leq 1$, and $R \geq 1$. This paper presents a compact generalization of several well-known polynomial inequalities using modified Smirnov operator, demonstrating that the operator preserves inequalities between polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00328 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Compact Generalization of Bernstein-Type Inequalities Preserved by Modified Smirnov Operator Kumar, Deepak Tripathi, D. Hans, Sunil Complex Variables 30C10, 30A10, 30C15, 30C80 Let $P(z)$ be a polynomial of degree $n$. In $2004$, Aziz and Rather \cite{aziz2004some} investigated the dependence of \[\bigg|P(Rz)-αP(z)+β\biggl\{\biggl(\frac{R+1}{2}\biggr)^n-|α|\biggr\}P(z)\bigg|, \ \text{for} \ z \in B(\mathbb{D}),\] on $\max_{z\in B(\mathbb{D})}|P(z)|$, for every real and complex number $α, β$ satisfying $|α| \leq 1$, $|β| \leq 1$, and $R \geq 1$. This paper presents a compact generalization of several well-known polynomial inequalities using modified Smirnov operator, demonstrating that the operator preserves inequalities between polynomials. |
| title | Some Compact Generalization of Bernstein-Type Inequalities Preserved by Modified Smirnov Operator |
| topic | Complex Variables 30C10, 30A10, 30C15, 30C80 |
| url | https://arxiv.org/abs/2503.00328 |