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 $z \in X$:

$$ \overline{lim}_{r→0}\frac{sup_{x,y \in S_r(z)} d (f(x), f(y))} {inf_{x,y \in S_r(z)} d (f(x), f(y))} ≤ K $$ 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!

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!