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# Matrix inequality (A-B)^2 <= c (A+B)^2 ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.

My question is: Is there a constant c (independent of A and B and the dimension) such that

$$(A-B)^2 \leq c (A+B)^2?$$

Thanks.