Families of real and symmetric analytic functions

Abstract

Abstract. We introduce families of functions analytic in the unit disk and having rotational symmetries. The families include the /c-fold symmetric univalent functions which have real coefficients. We relate the families to special classes of functions with a positive real part and then determine their extreme points. The case k = 2 corresponds to the odd functions which "preserve quadrants" and the extreme points of this set are characterized by having a radial limit which is real or imaginary almost everywhere. We also find estimates on the initial coefficients of functions in the families.