  In this article densities (and their derivatives) of subordinators and
inverse subordinators are considered. Under minor restrictions, generally
milder than the existing in the literature, using a useful modification of the
saddle point method, we obtain the large asymptotic behaviour of these
densities (and their derivatives) for a specific region of space and time and
quantify how the ratio between time and space affects the explicit speed of
convergence. The asymptotics is governed by an exponential term depending on
the Laplace exponent of the subordinator and the region represents the
behaviour of the subordinator when it is atypically small (the inverse one is
larger than usual). As a result a route to the derivation of novel general or
particular fine estimates for densities with explicit constants in the speed of
convergence in the region of the lower envelope/the law of iterated logarithm
is available. Furthermore, under mild conditions, we provide a power series
representation for densities (and their derivatives) of subordinators and
inverse subordinators. This representation is explicit and based on the
derivatives of the convolution of the tails of the corresponding L\'evy
measure, whose smoothness is also investigated. In this context the methods
adopted are based on Laplace inversion and strongly rely on the theory of
Bernstein functions extended to the cut complex plane. As a result, smoothness
properties of densities (and their derivatives) and their behaviour near zero
immediately follow.
