Hölder functions are operator-Hölder

Let $A$ and $B$ be selfadjoint operators in a separable Hilbert space such that $A-B$ is bounded. If a function $f$ satisfies the Hölder condition of order $\alpha$, $0<\alpha<1$, i.e., $|f(x)-f(y)|\leq L|x-y|^\alpha$, then $\|f(A)-f(B)\|\leq CL\|A-B\|^\alpha$, where $C$ is a constant, specifically, $C=2^{1-\alpha}+2\pi\sqrt 8\frac1{(1-2^{\alpha-1})^2}$. This result is a consequence of a general inequality in which the norm of $f(A)-f(B)$ is controlled in terms of the continuity modulus of $f$. Similar results are true for the quasicommutators $f(A)K-Kf(B)$, where $K$ is a bounded operator.