Let $f, g\colon S^1 \rightarrow S^1$ be two orientation-preserving circle homeomorphisms. We consider the usual rotation number $\rho \colon Homeo_{+}(S^1) \rightarrow \mathbb{R}/\mathbb{Z}$ defined by $\rho(f) = lim_{n\rightarrow \infty} \frac{1}{n}(F^{n}(x)-x) \mod 1$, where $F$ is a lift of $f$ to $\mathbb{R}$. This number is independent of $x$ and $\rho$ is continous in the $C^0$ topology.
The question is: what are some sufficient conditions for $\rho(f) > \rho(g)$?
Let $F, G$ be the lifts of $f$ and $g$, respectively. Clearly if $F(x) > G(x)$ for all $x$, we have $\rho(f) > \rho(g)$ immediately. However, that's the only result I've seen regarding this question. It seems reasonable to think that if $G$ is larger than $F$ but only on a small portion of the domain, and only by a small amount, then we still have $\rho(f) > \rho(g)$. However, using the usual definition of rotation number, I can't come up with any ways to find estimates on how much $G$ can exceed $F$ by.
I only had a couple of very vague ideas.
1) We can rewrite $\rho(f)$ as follows: let $j \colon S^1 \rightarrow \mathbb{R}$ be given by $j(x) = F(\pi^{-1}(x))-\pi^{-1}(x)$, where $\pi \colon \mathbb{R} \rightarrow S^1$ is the usual projection. Note that $j(x)$ is independent of the choice of $\pi^{-1}(x)$, since $F$ is such that $F(x + k) = F(x) + k$ for $k \in \mathbb{Z}$. In particular, we can choose $\pi^{-1}(x) = y \in [0,1)$. Then we have that $$F^{n}(x)-x = \sum_{i=0}^{n-1}(F^{i+1}(x) - F^{i}(x)) = \sum_{i=0}^{n-1}j(f^{i}(y)) $$ Then by the Birkoff Ergodic THeorem we get $$\rho(f) = \lim_{n \rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}j(f^{i}(y)) = \int_{S^1} j d\mu$$ where $\mu$ is a measure preserved by $j$.
This seems a bit better because with an integral we can maybe start comparing things, somehow, but the measure $\mu$ is not very useful for explicit computations from what I remember.
2) Suppose $f, g$ are actually part of a one-parameter family of circle maps, $f_t(x)$, where $t$ is the parameter and $\rho(f_0) = 0, \rho(f_1) = 1$. If this family varies continuously, then one might want to show $\rho(f_t)$ is non-decreasing in $t$. In this case it is enough to check that $\rho(f_t) \neq \rho(t_{t+\delta})$ for any $\delta > 0$ and $\rho(f_t)$ is irrational (if it is rational, then we can get intervals of constant value). Then if the maps in the family are $C^{1+BV}$ and $\rho(f_t) = \rho(f_{t+\delta})$, the maps are conjugate and so we can try looking for some invariant that may show a contradiction--I haven't been able to find one yet.