Convex hulls and extreme points of some families of univalent functions

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 Re f’(z) > α for α > 1 α > 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 Re f’(z) > α .