Let A = (A1, A2, A3, . . .) be a random sequence of non-negative numbers that are ultimately zero with E[ ∑ Ai] = 1 and E [ ∑ Ai log Ai] ≤ 0. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation Φ(ψ) = E [ ∏ i Φ(ψAi)] , where Φ is the Laplace transform of a non-negative random variable. T...