Analysis of a abstract function

Question

The known function $f:\mathbb{Z}^+ \to \mathbb{Z}^+$ is a monotone non decreasing function. For any coprime positive integers m and n satisfy $f(mn)=f(m)f(n)$. Compare the size of $f(2)f(8)$ and $f(3)f(5)$.

Answer 1

Paul Erdős showed that every increasing multiplicative arithmetic function $f:\mathbb{Z}_{>0}\to\mathbb{R}$ is of the form $$f(n)=n^\alpha$$ for all $n\in\mathbb{Z}_{>0}$ and $\alpha > 0$ is fixed. A proof can be seen here. Here is another link.