  Under appropriate conditions, we obtain smoothness and convexity properties
of $q$-scale functions for spectrally negative L\'evy processes. Our method
appeals directly to very recent developments in the theory of potential
analysis of subordinators. As an application of the latter results to scale
functions, we are able to continue the very recent work of \cite{APP2007} and
\cite{Loe}. We strengthen their collective conclusions by showing, amongst
other results, that whenever the L\'evy measure has a density which is log
convex then for $q>0$ the scale function $W^{(q)}$ is convex on some half line
$(a^*,\infty)$ where $a^*$ is the largest value at which $W^{(q)\prime}$
attains its global minimum. As a consequence we deduce that de Finetti's
classical actuarial control problem is solved by a barrier strategy where the
barrier is positioned at height $a^*$.
