For an n n interval matrix A = (A i j ), we say that A is majorized by the point matrix A = (a i j ) if a i j = A i j when the jth column of A has the property that there exists a power A m containing in the same jth column at least one interval not degenerated to a point interval, and a i j = A i j otherwise. Denoting the generalized spectral radius (in the sense of Daubechies and Lagarias) of A by ρ(A), and the usual spectral radius of A by ρ(A), it is proved that if A is majorized by A then ρ(A) ρ(A). This inequality sheds light on the asymptotic stability theory of discrete-time linear interval systems.