Let $0<\beta<1$ and $ f \colon [0,1] \to [0,1]$ be $\beta$ Hölder continuous with constant $C$. Let $H$ be a Hilbert space and $A,B$ be self adjoint operators on $H$, such that $\sigma(A+B),\sigma(A) \subset [0,1]$. Then we can define $f(A+B)$ and $f(B)$ by the continuous functional calculus. Do we then have the estimate
$$ \left \lvert \operatorname{tr} (f(A+B)-f(A)) \right \rvert \le C \lVert B \rVert_\beta^\beta$$
EDIT: The semi-norm $\lVert B \rVert_\beta$ is the Schatten von Neumann semi-norm.

This does hold for commutating operators $A,B$ and it seems to hold for 2x2 matrices, if i calculated correctly. There is also the stronger hypothesis, that for any unitary equivalent norm $\lVert \cdot \rVert$, we have the estimate
$$ \left \lVert f(A+B) - f(A) \right \rVert \le C \lVert \lvert B \rvert^\beta\rVert$$
I am aware of the question <https://mathoverflow.net/questions/308048/h%c3%b6lder-continuity-for-operators> and its answer, but this is different, as the trivial counter example does not hold. The special case $f(t)=t^\beta$ is stated as true in an answer to that question.