The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on
(
−
1
,
1
)
( - 1,1)
. We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on
(
−
1
,
1
)
( - 1,1)
. Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. We also obtain such a representation for the functions
f
(
z
)
f(z)
analytic in the unit disk, normalized and satisfying
Re
f
′
(
z
)
>
α
\operatorname {Re} f’(z) > \alpha
for
α
>
1
\alpha > 1
. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfying
Re
f
′
(
z
)
>
α
\operatorname {Re} f’(z) > \alpha
.