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?