If
f
f
is a holomorphic self-map of the open unit disc and
1
≤
p
>
∞
1 \leq p > \infty
, then the following are equivalent.
(
1
)
h
∘
f
∈
H
2
p
(1)\,\,\,\, h\circ f \in H^{2p}
for all Bloch functions
h
h
.
(2)
s
u
p
r
∫
0
2
π
(
l
o
g
1
1
−
|
f
(
r
e
i
θ
)
|
2
)
p
d
θ
>
∞
.
\begin{equation*}\underset {{r} }{sup} \int _{0}^{2\pi } \left ( log \frac {1}{1 - \vert f(re^{i\theta })\vert ^{2}}\right )^{p} \,d\theta \,\, > \infty . \tag {2}\end{equation*}
(3)
∫
0
2
π
(
∫
0
1
(
f
#
)
2
(
r
e
i
θ
)
(
1
−
r
)
d
r
)
p
d
θ
>
∞
,
\begin{equation*}\int _{0}^{2\pi } \left ( \int _{0}^{1} (f^{\#})^{2}(re^{i\theta })\, (1-r) dr \right )^{p} d\theta > \infty , \tag {3}\end{equation*}
where
f
#
f^{\#}
is the hyperbolic derivative of
f
f
:
f
#
=
|
f
′
|
/
(
1
−
|
f
|
2
)
f^{\#} = \vert f’\vert / (1-\vert f\vert ^{2})
.