On the modified entropy equation

The object of this paper is to solve the so--called modified entropy equation $$f<x, y, z>=f<x, y+z, \mathbf{0}>+ \mu<y+z>f<\mathbf{0}, \frac{y}{y+z}, \frac{z}{y+z}>,$$ on the positive cone of $\R^{k}$, where $\mu$ is a given multiplicative function on this cone. After that the regular solutions of this equation are determined. Furthermore we investigate its connection between the entropy equation and other equations, as well.