We study a class of probability densities with very thin upper tails.
These densities generate exponential families which are asymptotically
normal. Furthermore the class is closed under convolution. In this pa
per we shall be concerned with Abelian and strong Tauberian theorems f
or moment generating functions and Laplace transforms with respect to
these densities. We obtain a duality relation between this class of de
nsities and the associated class of moment generating functions which
is closely related to the duality relation for convex functions.