Let $A$ and $B$ be two positive, self-adjoint and traceclass operators mapping from some Hilbert space $H$ to it self. Note that we have for any $a, b > 0$, $p>1$
$$(a+b)^p\leq 2^{p-1}(a^p+b^p)$$
I would like to use
$$ \left\| A^p B^{-p} \right\| = \left\| (A-B+B)^pB^{-p} \right\| \leq 2^{p-1} \left( 1 + \left\|(A-B)^pB^{-p} \right\| \right) $$
Is something like this possible? Normally, I would try to argue with an SVD argument but I don't see how to start here.