Assume that $f:[0,2\pi]\to [0,2\pi]$ is a diffeomorphism, and let $\underline f = \min f'$ and $\overline f = \max f'$ and define $$g(s) = \int_0^s e^{if(t)} dt.$$ Assume that $g(0)=g(2\pi)$. It seems that $$\underline f \le \frac{|g(x)-g(y)|}{|e^{ix}-e^{iy}|}\le \overline f,$$ but I do not have the proof jet.

Assume that $f:[0,2\pi]\to [0,2\pi]$ is a an increasing diffeomorphism, and let $\underline f = \min f'$ and $\overline f = \max f'$ and define $$g(s) = \int_0^s e^{if(t)} dt.$$ Assume that $g(0)=g(2\pi)$. It seems that $$\underline f \le \frac{|g(x)-g(y)|}{|e^{ix}-e^{iy}|}\le \overline f,$$ but I do not have the proof jet.