On an operator preserving inequalities between polynomials View Full Text


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Article Info

DATE

2021-09-20

AUTHORS

Imtiaz Hussain, A. Liman

ABSTRACT

Let P(z) be a polynomial of degree at most n. We consider an operator N, which carries a polynomial P(z) into N[P](z):=∑j=0mλj(nz2)jP(j)(z)j!,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N[P](z):=\sum \limits _{j=0}^{m}\lambda _j\bigg (\frac{nz}{2}\bigg )^j\frac{P^{(j)}(z)}{j!}, \end{aligned}$$\end{document}where λ0,λ1,…,λm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _0,\lambda _1,\ldots ,\lambda _m$$\end{document} are such that all the zeros of u(z)=∑j=0mnjλjzj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u(z)=\sum \limits _{j=0}^{m}\left( {\begin{array}{c}n\\ j\end{array}}\right) \lambda _jz^j \end{aligned}$$\end{document}lie in the half plane |z|≤|z-n2|.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |z|\le \bigg |z-\frac{n}{2}\bigg |. \end{aligned}$$\end{document}In this paper, we estimate the minimum and maximum modulii of N[P(z)] on |z|=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|=1$$\end{document} with restrictions on the zeros of P(z) and thereby obtain compact generalizations of some well known polynomial inequalities. More... »

PAGES

285-292

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11565-021-00375-5

DOI

http://dx.doi.org/10.1007/s11565-021-00375-5

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1141239888


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