Inequalities on generalized matrix functions

Abstract

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\ B_i,\ C_i\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_\chi, d_\xi$ such as the determinant and permanent, etc., we have \begin{eqnarray*}&&\left(d_\chi(A_1+B_1+C_1)d_\xi(A_2+B_2+C_2)\right)^r \\ &&\hskip 1in + \left(d_\chi(A_1)d_\xi(A_2)\right)^r + \left(d_\chi(B_1)d_\xi(B_2)\right)^r + \left(d_\chi(C_1)d_\xi(C_2)\right)^r \\ & \ge &\left(d_\chi(A_1+B_1 )d_\xi(A_2+B_2 )\right)^r + \left(d_\chi(A_1+ C_1)d_\xi(A_2+ C_2)\right)^r + \left(d_\chi( B_1+C_1)d_\xi( B_2+C_2)\right)^r\,.\end{eqnarray*} A general scheme is introduced to prove more general inequalities involving $m$ positive semi-definite matrices for $m \ge 3$ that extend the results of other authors.