We consider nonnegative infinitely divisible random variables whose Levy measures are either absolutely continuous or supported by the integers. Necessary conditions are found ensuring that such distributions are $\log$-concave or $\log$-convex.
Bjorn G. Hansen. "On Log-Concave and Log-Convex Infinitely Divisible Sequences and Densities." Ann. Probab. 16 (4) 1832 - 1839, October, 1988. https://doi.org/10.1214/aop/1176991600