Remarks on the Entropy Equation

Z. Daróczy

In the paper [4] A. Kamicute{n}ski and J. Mikusicute{n}ski have proved the following Theorem: If a function $H(x,y,z)$ is continuous, symmetric and positively homogeneous (of order 1) in the domain $$D={(x,y,z)|x,y,z\geq 0, xy+yz+zx>0}$$ and satisfied in the order of $D$ the functional equation $$H(x,y,z)=H(x+y,0,z)+H(x,y,0)$$ then $$H(x,y,z)=c[(x+y+z)n(x+y+z)-xn x-yn y= zn z],$$ where $c$ is a real constant and $0\ln 0=0$