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?