  The purpose of this paper is to revisit the proof of the
Gearhardt-Pr\"uss-Hwang-Greiner theorem for a semigroup $S(t)$, following the
general idea of the proofs that we have seen in the literature and to get an
explicit estimate on $\Vert S(t) \Vert$ in terms of bounds on the resolvent of
the generator. A first version of this paper was presented by the two authors
in ArXiv (2010) together with applications in semi-classical analysis and a
part of these results has been published later in two books written by the
authors. Our aim is to present new improvements, partially motivated by a paper
of D. Wei. On the way we discuss optimization problems confirming the
optimality of our results.
