Bounded Laplace transforms, primitives and semigroup orbits

Let f : ℝ<inf>+</inf> → ℂ be an exponentially bounded, measurable function whose Laplace transform has a bounded holomorphic extension to the open right half-plane. It is known that there is a constant C such that |∫<inf>0</inf>^t f (s) ds| ≦ C (1 + t) for all t ≧ 0. We show that this estimate is sharp. Furthermore, the corresponding estimates for orbits of C<inf>0</inf>-semigroups are also sharp.