On Visser's inequality concerning coefficient estimates for a polynomial
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917852190081024 |
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| author | Gulzar, Suhail Rather, N. A. Wani, M. S |
| author_facet | Gulzar, Suhail Rather, N. A. Wani, M. S |
| contents | If $P(z)=\sum_{j=0}^{n}a_jz^j$ is a polynomial of degree $n$ having no zero in $|z|<1,$ then it was recently proved that for every $p\in[0,+\infty]$ and $s=0,1,\ldots,n-1,$ \begin{align*} \left\|a_nz+\frac{a_s}{\binom{n}{s}}\right\|_{p}\leq \frac{\left\|z+δ_{0s}\right\|_p}{\left\|1+z\right\|_p}\left\|P\right\|_{p}, \end{align*} where $δ_{0s}$ is the Kronecker delta. In this paper, we consider the class of polynomials having no zero in $|z|<ρ,$ $ρ\geq 1$ and obtain some generalizations of above inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19831 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Visser's inequality concerning coefficient estimates for a polynomial Gulzar, Suhail Rather, N. A. Wani, M. S Complex Variables If $P(z)=\sum_{j=0}^{n}a_jz^j$ is a polynomial of degree $n$ having no zero in $|z|<1,$ then it was recently proved that for every $p\in[0,+\infty]$ and $s=0,1,\ldots,n-1,$ \begin{align*} \left\|a_nz+\frac{a_s}{\binom{n}{s}}\right\|_{p}\leq \frac{\left\|z+δ_{0s}\right\|_p}{\left\|1+z\right\|_p}\left\|P\right\|_{p}, \end{align*} where $δ_{0s}$ is the Kronecker delta. In this paper, we consider the class of polynomials having no zero in $|z|<ρ,$ $ρ\geq 1$ and obtain some generalizations of above inequality. |
| title | On Visser's inequality concerning coefficient estimates for a polynomial |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2411.19831 |