On sufficient conditions for the total positivity and for the multiple positivity of matrices

Abstract

The following theorem is proved. Theorem Suppose M = (a i,j ) be a k × k matrix with positive entries and a i , j a i + 1 , j + 1 > 4 cos 2 π k + 1 a i , j + 1 a i + 1 , j ( 1 ⩽ i ⩽ k - 1 , 1 ⩽ j ⩽ k - 1 ) . Then det M > 0. The constant 4 cos 2 π k + 1 in this theorem is sharp. A few other results concerning totally positive and multiply positive matrices are obtained.