The values of the generalized matrix functions of $3 × 3$ matrices

Abstract

When A is a $3 x 3$ positive semi-definite Hermitian matrix, Schur’s inequality and the permanental dominance conjecture are known to hold. In Sharp inequalities for the permanental dominance conjecture, we determined the possible positions of the normalized generalized matrix functions relative to the determinant and the permanent except in the case that the order of the subgroup is 2. The purpose of this paper is to determine the possible positions in the last open case.