Is this matrix function bounded from above by a norm
Solution 1:
From the answer of this question, and in view of the fact that $$d(A, B) \leq \|A^{1/2} - B^{1/2}\|_F^2$$ by the Araki-Lieb-Thirring inequality, the answer to my question is yes. Quoting from the linked page:
In fact, we can say much more: every $α$-Holder continuous function $F$ is operator Holder continuous ($0<α<1$) on the space of self-adjoint matrices.