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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