The Journal of Analysis

Zhao, Xingjun

2021 The Journal of Analysis

doi: 10.1007/s41478-021-00356-z

Let P(z) be a polynomial of degree n which have no zeros in |z|<k,k≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|< k, k\ge 1$$\end{document}. Then it was proved by Govil (J Approx Theory 66:29–35, 1991), for 1≤s<n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1\le s < n$$\end{document}, then max|z|=1|P(s)(z)|≤n(n-1)…(n-s+1)1+ksmax|z|=1|P(z)|-min|z|=k|P(z)|.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \mathop {\text {max}}\limits _{|z|= 1}|P^{(s)}(z)| \le \frac{n (n-1)\ldots (n-s+1)}{1+k^{s}} \left( \mathop {\text {max}}\limits _{|z|= 1}|P(z)|-\mathop {\text {min}}\limits _{|z|= k}|P(z)|\right) . \end{aligned}$$\end{document}In this paper, we shall present an generalization of this result and an Lγ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_{\gamma }$$\end{document} analogue of our result, which improve some previous results.