Convex hulls and extreme points of families of symmetric univalent functions

Earlier in 1935[12], M. S. Robertson introduced the class of quadrant preserving functions. More precisely, let Q be the class of all functions f(z) analytic in the unit disk $D = {z : $\mid$z$\mid$ < 1}$ such that f(0) = 0, f'(0) = 1, and the range f(z) is in the j-th quadrant whenever z is in the j-th quadrant of D, j = 1,2,3,4. This class Q contains the subclass of normalized, odd univalent functions which have real coefficients. On the other hand, this class Q is contained in the class T of odd typically real functions which was introduced by W. Rogosinski [13]. Clearly, if $f \in Q$, then f(z) is real when z is real and therefore the coefficients of f are all real. Recently, it was observed by Y. Abu-Muhanna and T. H. MacGregor [1] that any function $f \in Q$ is odd. Instead of functions "preserving quadrants", the authors [1] have introduced the notion of "preserving sectors".