1. Introduction. We will be interested in Tauberian theorems concerning the limiting behaviour of a monotone function U and its Laplace transform
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00008683/resource/name/S0008414X00008683_eqn1.gif?pub-status=live)
A famous theorem of Karamata concerns the case in which the function U is regularly varying (i.e., U(tx)/U(t) → xα(t → ∞) for x > 0). Here we will consider functions U that grow faster, in fact our conditions will be in terms of log U rather than on U itself. So it is convenient to write the Laplace transform in terms of q = log U. For a function q:R+ → R such that exp q is locally integrable and
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00008683/resource/name/S0008414X00008683_eqn2.gif?pub-status=live)
we define the function
by the relation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00008683/resource/name/S0008414X00008683_eqn3.gif?pub-status=live)