I have nonnegative matrices $A,B,C$ such that $A = B \circ C$. It seems that $A^N \leq B^N \circ C^N$ (i.e. $(B\circ C)^N \leq B^N \circ C^N$) is true element-wise. I would like to show that this it true or provide a counterexample otherwise. I've tried writing out the matrix multiplication as a sum without much success. Any help is appreciated.
Edit: I should also note that my matrices are all substochastic, in particular, the elements of matrices $A,B,C$ are bounded between $0$ and $1$.