On the bounds of eigenvalues of matrix polynomials

Let M(z)=Amzm+Am-1zm-1+⋯+A1z+A0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(z)=A_mz^m+A_{m-1}z^{m-1}+\cdots +A_1z+A_0$$\end{document} be a matrix polynomial, whose coefficients Ak∈Cn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_k\in {{\mathbb {C}}}^{n\times n}$$\end{document}, ∀k=0,1,…,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \, k=0,1,\ldots , m$$\end{document}, satisfying the following dominant property ‖Am‖>‖Ak‖,∀k=0,1,…,m-1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert A_m\Vert >\Vert A_k\Vert ,\,\forall \, k=0,1,\ldots ,m-1, \end{aligned}$$\end{document}then it is known that all eigenvalues λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} of M(z) locate in the open disk λ<1+‖Am‖‖Am-1‖.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \lambda \right| <1+\Vert A_m\Vert \Vert {A_m}^{-1}\Vert . \end{aligned}$$\end{document}In this paper, among other things, we prove some refinements of this result, which in particular provide refinements of some results concerning the distribution of zeros of polynomials in the complex plane.