Let
f
f
be a Teichmüller self-mapping of the unit disk
Δ
\Delta
corresponding to a holomorphic quadratic differential
φ
\varphi
. If
φ
\varphi
satisfies the growth condition
A
(
r
,
φ
)
=
∬
|
z
|
>
r
|
φ
|
d
x
d
y
=
O
(
(
1
−
r
)
−
s
)
A(r,\varphi )=\iint _{|z|>r}|\varphi |dxdy=O((1-r)^{-s})
(as
r
→
1
r\to 1
), for any given
s
>
0
s>0
, then
f
f
is extremal, and for any given
a
∈
(
0
,
1
)
a\in (0,1)
, there exists a subsequence
{
n
k
}
\{n_k\}
of
N
\mathbb {N}
such that
{
φ
(
a
1
/
2
n
k
z
)
∬
Δ
|
φ
(
a
1
/
2
n
k
z
)
|
d
x
d
y
}
\begin{equation*} \Big \{\frac {\varphi (a^{1/2^{n_k}}z)} {\iint _\Delta |\varphi (a^{1/2^{n_k}}z)|dxdy}\Big \} \end{equation*}
is a Hamilton sequence. In addition, it is shown that there exists
φ
\varphi
with bounded Bers norm such that the corresponding Teichmüller mapping is not extremal, which gives a negative answer to a conjecture by Huang in 1995.