The purpose of this paper is to revisit the proof of the Gearhart- Priiss-Huang-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 parallel to S(t)parallel to 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 some of these results has been subsequently published 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.