Abstract. Let $A, B$, and $T$ be bounded operators on a separable Hilbert space. Suppose that $A$ and $B$ are normal and that $f$ is a Hölder$(\alpha)$ function with Hölder constant $\lbrack f \rbrack$ defined on the square $D$ containing the spectra of both $A$ and $B$. Then $$\eqalign{\|f(A)T-Tf(B)\| &\leq8\lbrack f \rbrack2^{1-\alpha} \frac{(\sqrt{2}+1)^{\alpha}}{\alpha+1} \|T\|^{1-\alpha} \|AT-TB\|^{\alpha}\times\cr &\hskip30pt \times\big(\log\big(\frac{d\|T\|}{\|AT-TB\|}+1\big)+2\big)^{\alpha+2}, }$$ where $d$ is the side of the square $D$.

AMS Subject Classification (1991): 47B35

Received December 28, 2004. (Registered under 5898/2009.)