Let F:R_+ →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 the is a constant C such that |f_0~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_0-semigroups are also sharp.
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