Inequalities on generalized matrix functions

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 $d_χ, d_ξ$ such as the determinant and permanent, etc., we have \begin{eqnarray*}&&\left(d_χ(A_1+B_1+C_1)d_ξ(A_2+B_2+C_2)\right)^r \\ &&\hskip 1in + \left(d_χ(A_1)d_ξ(A_2)\right)^r + \left(d_χ(B_1)d_ξ(B_2)\right)^r + \left(d_χ(C_1)d_ξ(C_2)\right)^r \\ & \ge &\left(d_χ(A_1+B_1 )d_ξ(A_2+B_2 )\right)^r + \left(d_χ(A_1+ C_1)d_ξ(A_2+ C_2)\right)^r + \left(d_χ( B_1+C_1)d_ξ( 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.

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