Spring 2018, problem 56

Let $\mathbb{Q}_+$ denote the set of all positive rational numbers and define $f:\mathbb{Q}_+\to \mathbb{R}$ to be a function such that the following two inequalities hold

$$f(xy)\le f(x)f(y), \quad \quad f(x)+f(y)\le f(x+y).$$

Show that if $f$ has a fixed point with value strictly greater than $1$, that $f(x)=x$ for all $x\in \mathbb{Q}_+$. Show by example that the requirement that on the value of the fixed point is necessary.

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