This is a continuation of our paper \cite{AP2}. We prove that for functions
$f$ in the H\"older class $\L_\a(\R)$ and $1<p<\be$, the operator $f(A)-f(B)$
belongs to $\bS_{p/\a}$, whenever $A$ and $B$ are self-adjoint operators with
$A-B\in\bS_p$. We also obtain sharp estimates for the Schatten--von Neumann
norms $\big\|f(A)-f(B)\big\|_{\bS_{p/\a}}$ in terms of $\|A-B\|_{\bS_p}$ and
establish similar results for other operator ideals. We also estimate
Schatten--von Neumann norms of higher order differences
$\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$.