*(English)*Zbl 1152.39019

The author presents solutions (both general and continuous) of the functional equation

for $x,y,z$ in the positive cone of a $k$-dimensional Euclidean space, where $\mu $ is a given multiplicative function on this cone.

The statements of the main results, Theorems 2.1 and 3.1, can be enhanced so that their converses are also true. For example, in Theorem 2.1 in the case $\mu $ is a projection one can find in the proof (using $f(x,y,0)=F(x,y)$) that $f(x,y,0)=\mu \left(x\right)l\left(x\right)+\mu \left(y\right)l\left(y\right)+{\psi}_{1}(x+y)$. Substitution into the functional equation (1.1) yields also ${\psi}_{1}\left(1\right)=0$. If one adds these two statements to this case of Theorem 2.1, then the converse is also true. Similar comments apply to the other two cases, which can be combined into a single case. There one can deduce the additional conditions $f(x,y,0)=b\mu \left(x\right)+b\mu \left(y\right)+{\psi}_{3}(x+y)$ and ${\psi}_{3}\left(1\right)=-b$, and with these the converse is again true. Note that the terms $-b\mu (y+z)$, $-b\mu \left(y\right)$ are missing from the right hand sides of equations (2.8), respectively (2.9).