1.3. MODULUS ESTIMATES 11A more detailed reasoning yields the following sharp estimate for the constant
c in (1.3.5), namely c = >..(K) where(1.3.6) >..(K) = (~ e.,,.K/^2 - e-.,,.K/^2 )
2
+ b(K), 0 < b(K) < e-.,,.K.See Anderson-Vamanamurthy-Vuorinen [ 11 ] and Lehto-Virtanen-Viiisiilii [118].
COROLLARY 1.3.7. If f: R^2 --t R^2 is K-quasiconformal and if
(1.3.8) lz2 -zo I :::; 2k lz1 - zo I
where k is an integer, k 2". 0, then
(1.3.9) lf(z2) - f(zo)I:::; c(c+ l)k lf(z1) - f(zo)I
where c = e^8 K.
PROOF. By Theorem 1.3.4, (1.3.8) implies (1.3.9) when k = 0. Suppose this
implication is true for some k 2". 0 and set z = ~ ( z 2 + z 0 ). Then
lz2 -z l = lz -zol:::; 2k lz1 -zol
and
lf(z2) - f(z)I:::; c lf(z) -f(zo)I
again by Theorem 1.3.4. Since
lf(z) -f(zo)I :::; c(c + l)k lf(z1) -f(zo)I
by hypothesis, we obtain
If ( z2) - f ( zo) I :::; If ( z2) - f ( z) I + If ( z) - f ( zo) I
:::; (c + 1) lf(z) -f(zo)I
:::; c(c + l)k+^1 lf(zi) -f(zo)I.
Thus (1.3.8) implies (1.3.9) for k + 1 and hence for all k by induction. D
Theorem 1.3.4 and its corollary are also consequences of the following general
result (Gehring-Hag [ 57 ]), the proof of which is less elementary and depends on
theorems due to Teichmiiller [157] and Agard [l].
THEOREM 1.3.10. If f: R^2 --t R^2 is K- quasiconformal, then
lf(z2) -f(zo)I + 1 < 16 x-1 (lz2 - zol + 1 )K
lf(z1) -f(zo)I - lz1 - zolfor zo, z 1 , z2 E R^2. The coefficient 15K -l cannot be replaced by any smaller con-
stant.
The property in Theorem 1.3.10 is called quasisymmetry (Heinonen [8 0 ], Astala-
Iwaniec-Martin [ 16 ]).
We conclude by listing two properties of qu asiconformal mappings that we will
need in what follows. See, for example, Lehto-Virtanen [117].
THEOREM 1.3.11. If f: D --t D' is quasiconformal and if D and D' are Jordan
domains, then f has a homeomorphic extension which maps D onto D'.
THEOREM 1.3.12. Suppose that EC D is closed and contained in a countable
union of rectifiable curves. If f : D --t D' is a homeomorphism which is K -
quasiconformal in each component of D \ E , then f is K -quasiconformal in D.