I am working on a problem where I need to know the Lipschitz constant for the matrix square root function. The function is given by $f(A)=\sqrt{A}$
$ \Vert f(X) -f(Y) \Vert \leq L\Vert X-Y \Vert$
Where X and Y are positive definite matrices. How can I determine or estimate the Lipschitz constant L for the matrix square root function in this setting? Are there known bounds or perturbation results in matrix analysis that can provide this constant? (when the eigenvalue is bounded away from zero and two positive definites there may exist a local Lipschitz constant?)