CMO 1993 Problem 6

Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$ , we have $f(xy)\le f(x)f(y)$ . Show that for arbitrary positive real number $x$ and natural number $n$ , inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.