1549055259-Ubiquitous_Quasidisk__The__Gehring_

4.1. MEROMORPHIC FUNCTIONS 61

The conclusion in Theorem 4.1.7 can be strengthened when Dis a disk or half-

plane. In this case f has a continuous extension to D and f (D) is either a Jordan

domain or the image of the strip domain

{z=x+iy: JxJ <oo,JyJ < 1}

under a Mobius transformation (Gehring-Pommerenke [69]). It would be interest-

ing to know what the situation is for other domains D.

The following result characterizes the domains D for which a-(D) > 0.

THEOREM 4.1.9 (Ahlfors [5], Gehring [49]). A simply connected domain D is

a K-quasidisk if and only if a-(D) > 0, where K and a-(D) depend only on each

other.

Theorem 4.1.9 allows us to obtain a still stronger conclusion in Theorem 4.1.7

when (4.1.8) holds with strict inequality.

THEOREM 4.1.10. If D is a quasidisk and if f is meromorphic in D with

sup JS1(z)J PD(z)-^2 ::::; c < a-(D),

zED

then f has a K-quasiconformal extension to R

2

where K depends only on a-(D)

and c.

PROOF. Let D' = f(D) and suppose that g is meromorphic in D' with

JS 9 (w)JPD'(w)-^2 ::::; a-(D)-c.

Then

SgoJ(z) = S 9 (f(z) ) f'(z )^2 + S1(z), PD(z ) = PD'(f(z)) Jf'(z)J,

whence

JSgoJ(z)J PD(z)-^2 ::::; JS 9 (f(z))J PD'(f(z))-^2 + JS1(z)J PD(z)-^2 ::::; a-(D).

Thus go f is injective in D, g is injective in D', and

a-(D') 2 a-(D) - c > 0.

Hence D and D' are K'-quasidisks, where K' depends only on a-(D) and a-(D) - c,

and f has a homeomorphic extension f* which maps D onto D'.

Next there exist K'-quasiconformal self-mappings ¢ and 'ljJ of R

2

which map

the upper half-plane H onto D and D', respectively. If r denotes reflection in the

real axis, then h 1 = ¢or o ¢-^1 and h 2 = 'ljJ or o 7/J-^1 are quasiconformal reflections

in 8D and 8D' and h 2 of o h1^1 defines a K-quasiconformal extension off to R

2

where K = (K')^4. D

We consider next counterparts of the above results for the pre-Schwarzian de-

rivative

f"

T1=f'

of a function f analytic and locally injective in D C R^2. In this case TJ = 0 in D
if and only if f is a similarity mapping, in which case f is injective.
We then have the following analogue for the pre-Schwarzian derivative of The-
orem 4.1.3.