Let
H
=
H
(
∗
,
[
+
]
)
H = H{(^ \ast },[ + ])
denote the real linear space of locally schlicht normalized functions in
|
z
|
>
1
|z| > 1
as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If
M
⊂
H
M \subset H
there is a closed nonempty convex set in the
α
β
\alpha \beta
-plane such that for
(
α
,
β
)
(\alpha ,\beta )
in this set
α
∗
f
[
+
]
β
∗
g
∈
C
{\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C
(in K) whenever f,
g
∈
M
g \in M
. This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.