I'm trying to relate the Frobenius Norm of a Hadamard Product to a trace that does not include another Hadamard Product, if possible. In other words, if A and B are (sxr) matrices, with not all positive values, 

$\left||A\circ B \right||^2\_F = $?

I'm trying to relate the sum of the squares of all the entries of the Hadamard product. I know that the sum of all the entries of the Hadamard product are

$\sum\_i \sum\_j (A\circ B)\_{ij} = tr(A B^T) $

and also 

$\left||A\circ B \right||^2\_F = tr((A\circ B)^T(A\circ B)) $

But I am trying to get $\left||A\circ B \right||^2\_F $ in the form of the trace of some combination of A and B, without a Hadamard product. Even an inequality would help. I've tried various identities and inequalities related to Hadamard Product and Frobenius Norm, but I am not having any luck. 

Any suggestions would be appreciated.