On the extreme points of some sets of analytic functions

Abstract

Let A A denote the set of analytic functions defined on the open unit disc. The extreme points of F = { f ϵ A : f ( 0 ) = 0 F = \{ f \epsilon A:f(0) = 0 and | Re ⁡ f ( z ) | > π / 2 |\operatorname {Re} f(z)| > \pi /2 are determined. Also a partial characterization is given for the extreme points of G α = { f ϵ A : f ( 0 ) = 1 {G_\alpha } = \{ f\epsilon A:f(0) = 1 and | arg ⁡ f ( z ) | > α π / 2 } , 0 > α > 1 |\arg f(z)| > \alpha \pi /2\} ,0 > \alpha > 1 .