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What are the strictly monotone functions $f\colon (0,\infty)\to (0,\infty)$ which satisfy $x= f(\tfrac{x^2}{f(x)})$ for $x>0$. I cannot find any other than $f(x)=x$.

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When $f(x) = a x$, $a>0$, then

$$f \left ( \frac{x^2}{f(x)} \right ) = a \frac{x^2}{f(x)} = a \frac{x^2}{a x} = x $$

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Are there any other solutions? – user60253 Jan 31 '13 at 15:30
It doesn't look likely; of course this must be proven. You know that no other power will work (try $x^b$). Otherwise, prove by contradiction; assume some other function that is not $a x$ works. What then? – Ron Gordon Jan 31 '13 at 15:32

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