Consider a twice-differentiable PDF $f(x)$. If $f(x)$ is log-concave, then so is its CDF, $F(x)$. As a consequence, the Mills' ratio or inverse hazard function $[1-F(x)]/f(x)$ is monotonic decreasing.
All of these assertions are found in Bagnoli and Bergstrom 1989 "Log-Concave Probability and Its Applications".
Question: For what family of distributions is the Mill's ratio/inverse hazard function also log convex?
I know I can simply impose the second derivative to be positive. I am interested in knowing if there is a name for this family of distributions.
Additional question: The same for the ratio $F(x)/f(x)$. For the same family of distributions, will $F(x)/f(x)$ be log-convex?