  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.
