TITLE: Generalizations of Oppenheim's inequality
QUESTION [5 upvotes]: The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$. 
There has been a lot of beautiful work done extending it to cases when $A$ or $B$ or both of them are $M$-matrices or their inverses, or totally nonnegative. 
My question is: do you know of other extensions, in which $A$ is non-symmetric in an "interesting" way?

REPLY [4 votes]: This is not a generalization to other matrix classes, but replacing determinant by permanent. Actually, it is a conjecture made by Bapat and Sunder: Under the same conditions $per(A \circ B) \leq (\prod{a_{ii}})per(B)$.
...but the following result due to Jiao [On a conjecture of H. Minc, Linear and Multilinear Algebra 32 (1992) 103–105.] couldn't surprise me more
$$per(A \circ B)+per (A) per (B) \geq (\prod{a_{ii}})per(B)+(\prod{b_{ii}})per(A).$$