Hook immanantal inequalities for Hadamard's function

For an n x n positive semi-definite (psd) matrix A, Peter Heyfron showed in [9] that the normalized hook immanants, (d) over bar(k), k = 1, ..., n, sa tisfy the dominance ordering per(A) = (d) over bar(n)(A) greater than or equal to (d) over bar(n-1)(A) g reater than or equal to ... greater than or equal to (d) over bar(2)(A) gre ater than or equal to (d) over bar(1) (A) = det(A). (a) The classical Hadamard-Marcus inequalities assert that for an n x n psd mat rix A = [a(ij)], [GRAPHICS] In view of the Hadamard-Marcus inequalities, it is natural to ask where the term Pi(i=1)(n) a(ii) sits in the family of descending normalized hook imm anants in (a). More specifically, for each n x n pad A one wishes to determ ine the smallest kappa(A) such that [GRAPHICS] Heyfron [10] (see also [11, 17]) established for all n x n psd A that kappa (A) greater than or equal to min{n - 2, 1 + root n-1}.