Support points of families of analytic functions described by subordination

Abstract

We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the typically-real functions, and the functions which are subordinate to a given majorant. If the majorant F F is univalent then each support point has the form F ∘ ϕ F \circ \;\phi , where ϕ \phi is a finite Blaschke product and ϕ ( 0 ) = 0 \phi (0) = 0 . This completely characterizes the set of support points when F F is convex. The set of support points is found for some specific majorants, including F ( z ) = ( ( 1 + z ) / ( 1 − z ) ) p F(z) = {((1 + z)/(1 - z))^p} where p > 1 p > 1 . Let K K and St {\text {St}} denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families K ∗ {K^{\ast } } and St ∗ {\text {St}}^{\ast } defined by the property of being subordinate to some member of K K or St {\text {St}} , respectively.