Multiplicative type functional equations arising from characterization problems

Abstract

We give the general and the so-called density function solutions of equation

$$\begin{array}{lll}f_{U}(x)f_{V}(y)=f_{X}\left(\frac{1-y}{1-xy} \right) f_{Y} (1-xy) \frac{y}{1-xy} \qquad \left( (x, y) \in (0,1)^2 \right)\end{array}$$

and the density function solutions of equation

$$\begin{array}{lll}f \left(x \right)g\left(y \right)=p\left(x+y\right)q\left( \frac{x}{y} \right) \qquad \left((x, y)\in \mathbb{R}^{2}_{+} \right).\end{array}$$