Let $F,G$ be arbitrary distribution functions on the real line and
let $\widehat{F},\widehat{G}$ denote their respective bilateral Laplace
transforms. Let $\kappa > 0$ and let $h : \mathbb{R}^+ \to \mathbb{R}^+$ be
continuous, non-decreasing, and such that $h(u) \ge Au^4$ for some $A > 0$
and all $u \ge 0$. Under the assumptions that
display 1
we establish the bound
display 2
where $C$ is a constant depending at most on $\kappa$
and $A$, $Q_G$ is the concentration function of $G$, and $l := (\log L) /L +
(\log W) /W$ ,with $W$ any solution to $h(W) = 1/\epsilon$. Improving and
generalizing an estimate of Alladi, this result provides a Laplace transform
analogue to the Berry-Esseen inequality, related to Fourier transforms. The
dependence in $\epsilon$ is optimal up to the logarithmic factor log $W$. A
number-theoretic application, developed in detail elsewhere, is described. It
concerns so-called lexicographic integers, whose characterizing property is
that their divisors are ranked according to size and valuation of the largest
prime factor. The above inequality furnishes, among other informations, an
effective Erdös-Kac theorem for lexicographical integers.