A book I'm reading gives the following definition for quasi-conformal maps:



> 
> If $f$ is a homeomorphism of a metric space X to itself, $f$ is *K-quasi-conformal* if and only if for all $z \in X$:
> 
> 
> $$
> \limsup\_{r\to0}\frac{\sup\_{x,y \in S\_r(z)} d (f(x), f(y))}
> {\inf\_{x,y \in S\_r(z)} d (f(x), f(y))} \leq K
> $$
>  where $S\_r(z)$ is the sphere of radius $r$ around $z$, and $x$ and $y$ are diametrically
>  opposite.
> 
> 
> 


However, I can't see why this is equivalent to some other definitions that are given elsewhere.


In Ahlfors's *Lectures on Quasi-conformal Mappings*, the author gives two equivalent conditions for a homeomorphism $f$ of two open subsets of $\mathbb{C}$ to be K-quasi-conformal. The first one is that $f$ has locally integrable distributional derivatives, which satisfies $\,f\_\overline{z} \leq K \cdot f\_z$; the second one is that the modules of quadrilaterals are K-quasi-invariant under $f$. 


I can't see why the definition above, when restricted to open subsets of $\mathbb{C}$, is equivalent to the two definitions in Ahlfors's book. Could you please help me? Actually I only need the direction from the above definition to the properties in Ahlfors's book. Thanks a lot!



About a definition of quasi-conformal maps
For the equivalence of definitions of quasiconformal maps the reference is
J. Heinonen, Lectures on analysis on metric spaces, Springer 2001. Notice that the $K$
in the definiton you cite is not the same $K$ as in the Ahlfors definitions.
So your definition of quasiconformality is equivalent to the usual one, but with a different $K$. 
 Another reference is Lehto and Virtanen, Quasiconformal mappings in the plane, Springer 1973. The equivalence of all these definitions is a non-trivial fact.


