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Does there exist a concave function $f:(0,\infty)\to(0,\infty)$ with the following properties?

  1. $f$ is $r$-homogeneous for some $r>0$, i.e., $f(\lambda x)=\lambda^r f(x)$ for all $x>0$

  2. $\lim_{x\to 0}f(x)=0$

  3. $\lim_{x\to\infty}f(x)/x=0$

  4. $f$ is not a monomial, i.e., $f$ is not of the form $f(x)=x^m$ for some $m$

We also put $f(0)=0$.

user3816
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1 Answers1

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Condition 1 alone gives $$\tag{1} f(x)=f(x\cdot 1) = Cx^r, \qquad \forall x>0, $$ for $C=f(1)$. This is not exactly a monomial because $r$ needs not be integer, though.

Condition 2 implies that $r>0$ and condition 3 implies that $r<1$. So the only functions with the properties you seek are of the form (1) for some $C>0$ and some $r\in (0, 1)$.

Giuseppe Negro
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    I am not sure this is the answer you expected. You might want to expand on your question, as Cameron Williams also suggests. – Giuseppe Negro Feb 04 '21 at 14:43