Following Willie Wong's answer, I think I've found a characterization of the positive real functions w such that $w\leq w^\*$ or $w^\*\leq w$.

**Proposition**: Let $w:R\rightarrow R$ a positive real function, and let $f(x)=\log(w(x))$. Then

- $w^*\leq w \iff f \text{ subadditive}$
- $w\leq w^\* \iff w(0) \leq 1$

In particular
$w=w^\*\iff f \text{ subadditive and } w(0) \leq 1$

**Proof**:
1) ($\Rightarrow$) We have that

$f(x)=\log w(x) \geq \log w^\*(x)=\sup _y\left\{\log w(x+y)-\log w(y)\right\}=\sup _y\left\{f(x+y)-f(y)\right\}$ which implies that f is subadditive, like Willie Wong suggested.

($\Leftarrow$) f is subadditive, so

$w^\*(x) = \sup_y \frac{w(x+y)}{w(y)}=\sup_y \frac{e^{f(x+y)}}{e^{f(y)}}\leq \sup_y\frac{e^{f(x)+f(y)}}{e^{f(y)}}=e^{f(x)}=w(x)$.

2) ($\Rightarrow$) $w(0)\leq w^\*(0)=1$

($\Leftarrow$) $w^\*(x)=\sup_y\frac{w(x+y)}{w(y)}\geq\frac{w(x)}{w(0)}\geq w(x)$

Noting that, if C is a costant, that $(Cw)^\*=w^\*$ the following corollary is trivial:

**Corollary:**

- $w^*\leq C w \iff f + \log C \text{ subadditive}$
- $w\leq C w^\* \iff w(0) \leq C$