Let $f$ be a function of bounded distortion with distortion constant $D$, i.e. \begin{align} \max_{x, y} \log \frac{Df^k(x)}{Df^k(y)} \leq D. \end{align} I've been told that this implies \begin{align} \frac{|A|}{|B|} \leq D \frac{|f(A)|}{|f(B)|} \end{align} but I don't know how to go about showing this true. Does anyone know how to show this?