I know there exists a Cauchy-Schwarz type upper bound for the trace of a product of matrices $A$ and $B$, but what about a lower bound?
Regarding the properties of the matrices $A$ and $B$ I am interested in: $A$ is a symmetric matrix, and $B$ has no special properties apart from the fact that it is a dense matrix with full rank and all of its elements are non-zero.
I would like to have a lower bound that separates $A$ and $B$, e.g., $$\text{Tr}(AB) \ge f(A)g(B)$$ for some functions $f$ and $g$.