  We prove inequalities on symmetric tensor sums of positive definite
operators. In particular, we prove multivariable operator inequalities inspired
by generalizations to the well-known Hlawka and Popoviciu inequalities. As
corollaries, we obtain generalized Hlawka and Popoviciu inequalities for
determinants, permanents, and generalized matrix functions. The new operator
inequalities and their corollaries contain a few recently published
inequalities on positive definite matrices as special cases.
