This seems like a standard problem, but unable to find a solution online.

Suppose we have two singular PSD matrices A and B with the following assumptions:

$ 0 < x \leq ||A|| \leq y$

$ 0 < ||B|| = z$

$||A - B|| \leq \delta$

Can an upper bound be provided for:

$ ||A^\dagger - B^\dagger || $

The final bound would be useful with any specific norm. If an upper bound is only possible with additional assumptions, that would be useful to know as well.

I'm interested in using this for an application where $B$ is the expected value of a random graph Laplacian and the bounds on $A$ are a high probability interval.