On Lipschitz functions of normal operators

It is shown that if $N$ and $M$ are normal operators on a separable, complex Hilbert space $H$, and $f$ is a Lipschitz function on $\Omega = \sigma (N) \cup \sigma (M)$ (i.e., $\left\vert {f(z) - f(w)} \right\vert \leqslant k\left\vert {z - w} \right\vert$ for some positive constant $k$ and all $z,w \in \Omega )$, then ${\left\Vert {f(N)X - Xf(M)} \right\Vert _2} \leqslant k{\left\Vert {NX - XM} \right\Vert _2}$ for any operator $X$ on $H$. In particular, ${\left\Vert {f(N) - f(M)} \right\Vert _2} \leqslant k{\left\Vert {N - M} \right\Vert _2}$.