About an efficiency functional implementing the principle of least effort

A probabilistic functional of efficiency has been proposed recently in order to implement the principle of least effort and to derive Zipf-Pareto laws with a calculus of variation. This work is a further investigation of this efficiency measure from mathematical point of view. We address some key mathematical properties of this functional such as its unicity, its robustness against small variation of probability distribution and its relationship with inequality as well as probabilistic uncertainty. In passing, a method for calculating non-negative continuous (differential) entropy is proposed based upon a generalized definition of informational entropy called varentropy.