Modulus of continuity of operator functions

Let $A$ and $B$ be bounded selfadjoint operators on a separable Hilbert space, and let $f$ be a continuous function defined on an interval $[a,b]$ containing the spectra of $A$ and $B$. If $\omega _f$ denotes the modulus of continuity of $f$, then
$$ \|f(A)-f(B)\|\leq 4\Big[\log\Big(\frac{b-a}{\|A-B\|}+1\Big)+1\Big]^2\cdot\omega _f(\|A-B\|). $$
A similar result is true for unbounded selfadjoint operators, under some natural assumptions on the growth of $f$.