The growth of Laguerre matrix polynomials on bounded intervals

Let A be a matrix in C-rxr such that Re(z) > -1/2 for all the eigenvalues o f A and let {IIn(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. this paper, it is proved that IIn(A,1/2) (x) = O(n(alpha>(*) over bar * (A)/2)ln(r-1)(n)) and IIn+1(A,1/2) (x) = O(n((a lpha>(*) over bar * (A)-1)/2)In(r-1)(n)) uniformly on bounded intervals, wh ere alpha>(*) over bar * (A) = max{Re(z); z eigenvalue of A}. (C) 2000 Else vier Science Ltd. All rights reserved.