Let
A
A
be the set of functions regular in the unit disc
U
\mathcal {U}
and
A
0
{A_0}
the set of all functions
f
∈
A
f \in A
which satisfy
f
(
0
)
=
1
f(0) = 1
. For
V
⊂
A
0
V \subset {A_0}
define the dual set
V
∗
=
{
f
∈
A
0
|
f
∗
g
≠
0
for all
g
∈
V
,
z
∈
U
}
,
V
∗
∗
=
(
V
∗
)
∗
{V^ \ast } = \{ f \in {A_0}|f \ast g \ne 0{\text { for all }}g \in V,z \in \mathcal {U}\} ,{V^{ \ast \ast }} = {({V^ \ast })^ \ast }
. Here
f
∗
g
f \ast g
denotes the Hadamard product. THEOREM. Let
V
⊂
A
0
V \subset {A_0}
have the following properties: (i)
V
V
is compact, (ii)
f
∈
V
f \in V
implies
f
(
x
z
)
∈
V
f(xz) \in V
for all
|
x
|
⩽
1
|x| \leqslant 1
. Then
λ
(
V
)
=
λ
(
V
∗
∗
)
\lambda (V) = \lambda ({V^{ \ast \ast }})
for all continuous linear functionals
λ
\lambda
on
A
A
. This theorem has many applications to functions in
A
A
which are defined by properties like bounded real part, close-to-convexity, univalence etc.