On a Functional Equation Characterizing Some Probability Distributions

We find all nonnegative solutions f of the equation \(f\left(x\right)=\prod_{j=1}^{n}f{\left({s}_{j}x\right)}^{{p}_{j}},\) $$f\left(x\right)=\prod _{j=1}^{n}f{\left(s_j,x\right)}^{p_j},$$

defined in a one-sided vicinity of 0 and having a prescribed asymptotic at 0. The main theorem extends a result obtained by J. A. Baker [Proc. Amer. Math. Soc., 121, 767 (1994)].