The class of infinitely divisible characteristic functions which have unimodal Levy spectral functions is determined. It is shown that membership in this class is related to solutions of the equations $\phi(u) = \phi^r(ru)\phi_r(u)$, where $r \in (0, 1)$ and $\phi$ and $\phi_r$ are characteristic functions. We point out how elements of this class can serve as limit laws as well as some connections between this class and the class of self-decomposable characteristic functions.
"Infinitely Divisible Distributions with Unimodal Levy Spectral Functions." Ann. Probab. 7 (3) 494 - 499, June, 1979. https://doi.org/10.1214/aop/1176995049