Let
f
(
z
)
f(z)
be a function analytic in the disk
E
{
z
:
|
z
|
>
1
}
E\{ z:|z| > 1\}
and for some real number
n
>
0
n > 0
let
|
f
(
z
)
|
≦
(
1
−
|
z
|
2
)
−
n
,
z
∈
E
|f(z)| \leqq {(1 - |z{|^2})^{ - n}},z \in E
. In this paper it is shown that
\[
|
f
′
(
z
)
|
≦
(
n
+
1
)
n
+
1
n
n
[
1
−
(
n
n
+
1
)
2
n
(
1
−
|
z
|
2
)
2
n
|
f
(
z
)
|
2
]
÷
(
1
−
|
z
|
2
)
n
+
1
,
|f’(z)| \leqq \frac {{{{(n + 1)}^{n + 1}}}}{{{n^n}}}\left [ {1 - {{\left ( {\frac {n}{{n + 1}}} \right )}^{2n}}{{(1 - |z{|^2})}^{2n}}|f(z){|^2}} \right ] \div {(1 - |z{|^2})^{n + 1}},
\]
z
∈
E
z \in E
. In the special case
n
=
1
n = 1
there is a constant
K
,
3
≦
K
≦
4
K,3 \leqq K \leqq 4
, so that
\[
f
′
(
z
)
|
+
|
f
(
z
)
|
2
≦
K
(
1
−
|
z
|
2
)
−
2
.
f’(z)| + |f(z){|^2} \leqq K{(1 - |z{|^2})^{ - 2}}.
\]
This result has application in univalent function theory.