Applying a theorem of Bagemihl and Seidel (1953), we prove that if
S
2
{S_2}
is a set of second category in
(
α
,
β
)
(\alpha ,\beta )
, where
0
⩽
α
>
β
⩽
2
π
0 \leqslant \alpha > \beta \leqslant 2\pi
, and if
f
(
z
)
f(z)
is a function meromorphic in the sector
Δ
(
α
,
β
)
=
{
z
:
0
>
|
z
|
>
∞
,
α
>
arg
z
>
β
}
\Delta (\alpha ,\beta ) = \{ z:0 > \left | z \right | > \infty ,\alpha > \arg z > \beta \}
for which
lim
_
r
→
∞
|
f
(
r
e
i
θ
)
|
>
0
{\underline {{\operatorname {lim}}} _{r \to \infty }}\left | {f(r{e^{i\theta }})} \right | > 0
, for all
θ
∈
S
2
\theta \in {S_2}
, then there exists a sector
Δ
(
α
′
,
β
′
)
⊆
Δ
(
α
,
β
)
\Delta (\alpha ’,\beta ’) \subseteq \Delta (\alpha ,\beta )
such that
(
α
′
,
β
′
)
⊆
S
¯
2
,
S
2
(\alpha ’,\beta ’) \subseteq {\bar S_2},{S_2}
is second category in
(
α
′
,
β
′
)
(\alpha ’,\beta ’)
, and
f
(
z
)
f(z)
has no zero in
Δ
(
α
′
,
β
′
)
\Delta (\alpha ’,\beta ’)
. Based on this property, we prove several uniqueness theorems.