  Given matrices $A$ and $B$ such that $B=f(A)$, where $f(z)$ is a holomorphic
function, we analyze the relation between the singular values of the
off-diagonal submatrices of $A$ and $B$. We provide family of bounds which
depend on the interplay between the spectrum of the argument $A$ and the
singularities of the function. In particular, these bounds guarantee the
numerical preservation of quasiseparable structures under mild hypotheses. We
extend the Dunford-Cauchy integral formula to the case in which some poles are
contained inside the contour of integration. We use this tool together with the
technology of hierarchical matrices ($\mathcal H$-matrices) for the effective
computation of matrix functions with quasiseparable arguments.
