Take $f(x)=k\log(x+1)$ where $k=2\pi/\log(2\pi+1)$, hence $f$ is a diffeomorphism. The maximum of the derivate is atained when $x=0$ and it is $k\approx 3.1644$. On the other hand, taking $x=\pi/3$, $y=\pi/3-1$ we obtain

$$ \frac{|g(x)-g(y)|}{|e^{\mathrm{i}x}-e^{\mathrm{i}y}|}> \frac{2}{0.2}=10.$$