On the zeros of certain composite polynomials and an operator preserving inequalities
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917853385457664 |
|---|---|
| author | Rather, N. A. Dar, Ishfaq Gulzar, Suhail |
| author_facet | Rather, N. A. Dar, Ishfaq Gulzar, Suhail |
| contents | If all the zeros of $n$th degree polynomials $f(z)$ and $g(z) = \sum_{k=0}^{n}λ_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-σ|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm} that all the zeros of the polynomial $h(z)= \sum_{k=0}^{n}λ_k f^{(k)}(z) \frac{(σz)^k}{k!}$ lie in the circle $|z| \leq r ~ \max(1,s)$. In this paper, we relax the condition that $f(z)$ and $g(z)$ are of the same degree and instead assume that $f(z)$ and $g(z)$ are polynomials of arbitrary degree $n$ and $m$ respectively, $m\leq n,$ and obtain a generalization of this result. As an application, we also introduce a linear operator which preserve Bernstein type polynomial inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the zeros of certain composite polynomials and an operator preserving inequalities Rather, N. A. Dar, Ishfaq Gulzar, Suhail Complex Variables If all the zeros of $n$th degree polynomials $f(z)$ and $g(z) = \sum_{k=0}^{n}λ_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-σ|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm} that all the zeros of the polynomial $h(z)= \sum_{k=0}^{n}λ_k f^{(k)}(z) \frac{(σz)^k}{k!}$ lie in the circle $|z| \leq r ~ \max(1,s)$. In this paper, we relax the condition that $f(z)$ and $g(z)$ are of the same degree and instead assume that $f(z)$ and $g(z)$ are polynomials of arbitrary degree $n$ and $m$ respectively, $m\leq n,$ and obtain a generalization of this result. As an application, we also introduce a linear operator which preserve Bernstein type polynomial inequalities. |
| title | On the zeros of certain composite polynomials and an operator preserving inequalities |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2412.01088 |