Generalization of Some Well-Known Polynomial Inequalities for the Modified Smirnov Operator
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| Format: | Preprint |
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2025
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| _version_ | 1866917979268055040 |
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| author | Kumar, Deepak Tripathi, Dinesh Hans, Sunil |
| author_facet | Kumar, Deepak Tripathi, Dinesh Hans, Sunil |
| contents | Let $P(z)$ be a polynomial of degree $n$. In this paper, we consider the modified Smirnov operator, which carries a polynomial $P(z)$ into $\tilde{\mathbb{S}}_a[P](z)=(1+az)P'(z)-naP(z),$ where $a$ is an arbitrary number in $\overline{\mathbb{D}}$. We estimate minimum and maximum moduli of modified Smirnov operator of $P(z)$ on the unit circle with restricted zeros and thereby obtain a generalization of some results of Dewan and Hans \cite{dewan2013some}. This study includes compact generalization of some well-known polynomial inequalities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalization of Some Well-Known Polynomial Inequalities for the Modified Smirnov Operator Kumar, Deepak Tripathi, Dinesh Hans, Sunil Complex Variables 30C10, 30A10, 30C15, 30C80 Let $P(z)$ be a polynomial of degree $n$. In this paper, we consider the modified Smirnov operator, which carries a polynomial $P(z)$ into $\tilde{\mathbb{S}}_a[P](z)=(1+az)P'(z)-naP(z),$ where $a$ is an arbitrary number in $\overline{\mathbb{D}}$. We estimate minimum and maximum moduli of modified Smirnov operator of $P(z)$ on the unit circle with restricted zeros and thereby obtain a generalization of some results of Dewan and Hans \cite{dewan2013some}. This study includes compact generalization of some well-known polynomial inequalities. |
| title | Generalization of Some Well-Known Polynomial Inequalities for the Modified Smirnov Operator |
| topic | Complex Variables 30C10, 30A10, 30C15, 30C80 |
| url | https://arxiv.org/abs/2504.05195 |