The subclasses of class $L$ probability distributions recently studied by K. Urbanik are characterized by requiring that certain functions be convex and have derivatives of some fixed order. The extreme points of certain compact convex sets of probability measures are determined, and this information is then used to obtain a representation of the characteristic functions of the probability distributions in those classes, in the same manner as Urbanik has proceeded for the class $L$.
"Characterization of Subclasses of Class $L$ Probability Distributions." Ann. Probab. 6 (2) 279 - 293, April, 1978. https://doi.org/10.1214/aop/1176995573