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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!

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  • $\begingroup$ What does it mean "diametraly opposite" in an arbitrary metric space? $\endgroup$ – Alexandre Eremenko Dec 25 '13 at 20:59
  • $\begingroup$ @AlexandreEremenko: I'm not sure about that, but this definition is only applied to manifolds, so it's not very crucial in the context. $\endgroup$ – Boyu Zhang Dec 25 '13 at 23:55
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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.

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