  Using functional calculi theory, we obtain several estimates for
$\|\psi(A)g(A)\|$, where $\psi$ is a Bernstein function, $g$ is a bounded
completely monotone function and $-A$ is the generator of a holomorphic
$C_0$-semigroup on a Banach space, bounded on $[0,\infty)$. Such estimates are
of value, in particular, in approximation theory of operator semigroups. As a
corollary, we obtain a new proof of the fact that $-\psi(A)$ generates a
holomorphic semigroup whenever $-A$ does, established recently in [8] by a
different approach.
