On the bounds of eigenvalues of matrix polynomials

Let M(z) = A(m)z(m) + A(m-1)z(m-1) + ... + A(1)z + A(0) be a matrix polynomial, whose coefficients A(k) is an element of C-nxn, for all k = 0, 1,...,m, satisfying the following dominant property parallel to A(m)parallel to > parallel to A(k)parallel to, for all k = 0,1,...,m - 1, then it is known that all eigenvalues lambda of M(z) locate in the open disk vertical bar lambda vertical bar < 1 + parallel to A(m)parallel to parallel to A(m)(-1)parallel to. 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.