Let
f
∈
Σ
f \in \Sigma
, the class of all analytic univalent functions defined in
γ
=
{
z
:
|
z
|
>
1
}
\gamma = \{ z:|z| > 1\}
. For
f
,
g
∈
Σ
f,g \in \Sigma
define
h
h
in
γ
\gamma
by
h
(
z
)
=
f
(
z
)
1
−
α
g
(
z
)
α
,
0
>
α
>
1
h(z) = f{(z)^{1 - \alpha }}g{(z)^\alpha },0 > \alpha > 1
. If
h
(
z
)
=
z
+
Σ
n
=
0
∞
c
n
z
−
n
h(z) = z + \Sigma _{n = 0}^\infty {c_n}{z^{ - n}}
, it is shown that
Σ
n
=
1
∞
n
|
c
n
|
2
>
∞
\Sigma _{n = 1}^\infty n|{c_n}{|^2} > \infty
. This result is used to show that if
B
α
{B_\alpha }
denotes the class of all meromorphic Bazilevič functions of type
α
\alpha
and
f
∈
B
α
f \in {B_\alpha }
with
f
(
z
)
=
z
+
Σ
n
=
0
∞
a
n
z
−
n
f(z) = z + \Sigma _{n = 0}^\infty {a_n}{z^{ - n}}
, then
n
a
n
=
O
(
1
)
n{a_n} = O(1)
as
n
→
∞
n \to \infty
, the result being best possible.