Given a closed subset of the family
S
∗
(
α
)
{S^\ast }(\alpha )
of functions starlike of order
α
\alpha
of a particular form, a continuous Fréchet differentiable functional, J, is constructed with this collection as the solution set to the extremal problem
max
Re
J
(
f
)
\max \operatorname {Re} J(f)
over
S
∗
(
α
)
{S^\ast }(\alpha )
. Similar results are proved for families which can be put into one-to-one correspondence with
S
∗
(
α
)
{S^\ast }(\alpha )
. The support points of
S
∗
(
α
)
{S^\ast }(\alpha )
and
K
(
α
)
K(\alpha )
, the functions convex of order
α
\alpha
, are completely characterized and shown to coincide with the extreme points of their respective convex hulls. Given any finite collection of support points of
S
∗
(
α
)
{S^\ast }(\alpha )
(or
K
(
α
)
K(\alpha )
), a continuous linear functional, J, is constructed with this collection as the solution set to the extremal problem
max
Re
J
(
f
)
\max \operatorname {Re} J(f)
over
S
∗
(
α
)
{S^\ast }(\alpha )
(or
K
(
α
)
K(\alpha )
).