Putnam

It is obvious that for any $c>0$, the function $f(x) = x^c$ has the desired property; we will prove that conversely, any function with the desired property has this form for some $c$. Define the function $g: (0, \infty) \to (0, \infty)$ given by $g(x) = \log f(e^x)$; this function has the property that if $x,y \in (0, \infty)$ and $2x \leq y \leq 3x$, then $2g(x) \leq g(y) \leq 3g(x)$. It will suffice to show that there exists $c>0$ such that $g(x) = cx$ for all $x >0$. Similarly, define the function $\text{h:}$ given by $h(x) = \log g(e^x)$; this function has the property that if $\text{x,y}$ and $x + \log 2 \leq y \leq x + \log 3$, then $h(x) + \log 2 \leq h(y) \leq h(x) + \log 3$. It will suffice to show that there exists $c>0$ such that $h(x) = x + c$ for all $\text{x}$ (as then $h(x) = e^c x$ for all $x>0$). By interchanging the roles of $x$ and $y$, we may restate the condition on $h$ as follows: if $x - \log 3 \leq y \leq x - \log 2$, then $h(x) - \log 3 \leq h(y) \leq h(x) - \log 2$. This gives us the cases $a+b=0,1$ of the following statement, which we will establish in full by induction on $a+b$: for any nonnegative integers $a,b$, for all $\text{x,y}$ such that $$x + a \log 2 - b \log 3 \leq y \leq x + a \log 3 - b \log 2,$$ we have $$h(x) + a \log 2 - b \log 3 \leq h(y) \leq h(x) + a \log 3 - b \log 2.$$ To this end, suppose that $a+b>0$ and that the claim is known for all smaller values of $a+b$. In particular, either $a>0$ or $b>0$; the two cases are similar, so we treat only the first one. Define the function $$j(t) = \frac{(a+b-1)t - b(\log 2 + \log 3)}{a+b},$$ so that $$j(a \log 2 - b \log 3) = (a-1) \log 2 - b \log 3,$$ $$j(a \log 3 - b \log 2) = (a-1) \log 3 - b \log 2.$$ For $t \in [a \log 2 - b \log 3, a \log 3 - b \log 2]$ and $y = x+t$, we have $\log 2 \leq t-j(t) \leq \log 3$ and hence $$(a-1) \log 2 - b \log 3 \leq h(x+j(t)) - h(x) \leq (a-1) \log 3 - b \log 2$$ $$\log 2 \leq h(y)-h(x+j(t)) \leq \log 3;$$ this completes the induction. Now fix two values $\text{x,y}$ with $x \leq y$. Since $\log 2$ and $\log 3$ are linearly independent over $\text{}$, the fractional parts of the nonnegative integer multiples of $\log 3/\log 2$ are dense in $[0,1)$. (This result is due to Kronecker; a stronger result of Weyl shows that the fractional parts are uniformly distributed in $[0,1)$. In particular, for any $\epsilon > 0$ and any $N > 0$, we can find integers $a,b > N$ such that $$y-x < a \log 3 - b \log 2 < y-x + \epsilon.$$ By writing $$a \log 2 - b \log 3 = \frac{\log 2}{\log 3}(a \log 3 - b \log 2) - b \frac{(\log 3)^2 - (\log 2)^2}{\log 3},$$ we see that this quantity tends to $-\infty$ as $N \to \infty$; in particular, for $N$ sufficiently large we have that $a \log 2 - b \log 3 < y-x$. We thus have $h(y) \leq h(x) + a \log 2 - b \log 3 < y-x + \epsilon$; since $\epsilon>0$ was chosen arbitrarily, we deduce that $h(y)-h(x) \leq y-x$. A similar argument shows that $h(y)-h(x) \geq y-x$; we deduce that $h(y) - h(x) = y-x$, or equivalently $h(y)-y = h(x) - x$. In other words, the function $x \mapsto h(x) - x$ is constant, as desired.