Let X be a complex Banach space, T : ℝ+ → B(X) and f : ℝ+ → X be bounded functions, and suppose that the singular points of the Laplace transforms of T and f do not coincide. Under various supplementary assumptions, we show that the convolution T * f is bounded. When T(t) = I, this is a classical result of Ingham. Our results are applied to mild solutions of inhomogeneous Cauchy problems on ℝ+: u′(t) = Au(t)+f(t) (t≥0), where A is the generator of a bounded C0-semigroup on X. For holomorphic semigroups, a result of this type has been obtained by Basit.