  The purpose of this paper is to revisit the proof of the
Gearhart-Pr\"uss-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 the operator norm of $S(t)$ in terms of bounds on the
resolvent of the generator. In \cite{HelSj} by the first two authors, this was
done and some applications in semiclassical analysis were given. Some of these
results have been subsequently published in two books written by the two first
authors \cite{He1,Sj2}. A second work \cite{HeSj21} by the first two authors
presents new improvements partially motivated by a paper of D. Wei \cite{W}.
  In this third paper, we continue the discussion on whether the aforementioned
results are optimal, and whether one can improve these results through
iteration. Numerical computations will illustrate some of the abstract results.
