4 1. PRELIMINARIES
z f
FIGURE 1.DEFINITION 1.1.3. A homeomorphism f : D---+ D' is K-quasiconformal where
1 :SK< oo if H1(z) < oo for every z ED\ {oo,f-^1 (00)} and
H1(z) :SK
almost everywhere in D.
The inequality in the above definition can be weakened significantly to yield
the same class of mappings. Letting.. L1(z, r)
hi(z)=hmmfl( r--tO f z, r )'
where l 1 and Lt are as in (1.1.1), Heinonen and Koskela [81] and Kallunki and
Koskela [96] obtained the following surprising result.
THEOREM 1.1.4. A homeomorphism f : D ---+ D' is K-quasiconformal, where
1 :SK< oo, if H1(z) < oo for every z ED\ {oo, 1-^1 (00)} and
h1(z) :SK
almost everywhere in D.
The next results, which can be found in Lehto-Virtanen [117], identify map-
pings which are 1-quasiconformal or which are the composition and inverses of
quasiconformal mappings.THEOREM 1.1.5. A homeomorphism f : D ---+ D' is 1-quasiconformal if and
only if f or its complex conjugate f is a conj ormal mapping, i.e., analytic in D \
{oo,f-^1 (00)}.
THEOREM 1.1.6. If f : D ---+ D' is K 1 -quasiconformal and g : D' ---+ D" is
K2-quasiconformal, then go f: D---+ D" is K 1 K2-quasiconformal. The inverse of
a K -quasiconformal mapping is K -quasiconformal.
Menchoff's theorem asserts that a sense-preserving homeomorphism f of D
is a conformal mapping if, except at a countable set of points z E D, f maps
infinitesimal circles about z onto infinitesimal circles about f(z). Theorems 1.1.
and 1.1.5 extend this result by first replacing the countable exceptional set where