Log-concavity, total positivity and other properties of the product of a finite Toeplitz matrix and its transpose

If a0,a1,…,an is a log-concave sequence of real numbers, so is (mij)j=1n+1, mij:=∑k=0min(i,j)−1akak+|j−i| for every i∈{1,2,…,n+1}. In other words, the rows (and columns) of the matrix (mij)i,j=1n+1 constitute log-concave sequences. If additionally a0,a1,…,an are the coefficients of a polynomial all of whose roots are real and negative then the matrix is strictly totally positive and the author conjectures that the sequence of minors (M(k+1k+2⋯k+rjj+1⋯j+r−1))j=1n+2−r is log-concave for all r∈{1,2,…,n−r+1},k∈{0,1,…,n+2−r}. Using the Variation Diminishing Property of strictly totally positive matrices, we prove the case for r=2,k=0,1,…,n+1−r and with the additional condition that the roots are distinct, for k=n+2−r also. Further, we prove, again using the Variation Diminishing Property, that the row-sums (or column-sums) of the matrix constitute a log-concave sequence. If a0,a1,…,an is merely log-concave, we show that the sums of the rows are unimodal.