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.

A sufficient condition for a polynomial to be stable

“…In [8] it is also shown that in the statement of Theorem A the constant c n is the smallest possible not only in the class of matrices with positive entries but in the classes of Toeplitz matrices and of Hankel matrices. We recall that a matrix M is Toeplitz matrix if it is of the form M = (a j−i ) and a matrix M is Hankel matrix if it is of the form M = (a j+i ).…”

“…In [8] the authors of this note have proved that Theorem A remains valid if one replace the constantc by the constant c n := 4 cos 2 π n+1 . In [8] it is also shown that in the statement of Theorem A the constant c n is the smallest possible not only in the class of matrices with positive entries but in the classes of Toeplitz matrices and of Hankel matrices.…”

Turán-type inequalities and the distribution of zeros of entire functions

“…We begin with a result of J. I. Hutchinson [64] that has been the subject of in vestigations by several authors: T. Craven and G. Csordas [39], D. K. Dimitrov [50], D. C. Kurtz [76], and O. M. Katkova and A. M. Vishnykova [70]. We begin with a result of J. I. Hutchinson [64] that has been the subject of in vestigations by several authors: T. Craven and G. Csordas [39], D. K. Dimitrov [50], D. C. Kurtz [76], and O. M. Katkova and A. M. Vishnykova [70].…”

A remark about positive polynomials

“…In [4] it is also shown that the constant c m := 4 cos 2 π m+1 in the statement of Theorem A is the smallest possible not only in the class of m × m matrices with positive entries but in the classes of m × m Toeplitz matrices and of m × m Hankel matrices.…”

“…By Sylvester's Criterion for positive definiteness we need to show that all leading principal minors of the matrix M Qp are positive. To do this we will use the following theorem from [4].…”