Support points and double poles

Abstract

This paper gives some sufficient conditions for support points of the class S of univalent functions to be rotations of the Koebe function k ( z ) = z ( 1 − z ) − 2 k(z) = z{(1 - z)^{ - 2}} . If f is a support point associated with a continuous linear functional L and if the function Φ ( w ) = L ( f 2 / ( f − w ) ) \Phi (w) = L({f^2}/(f - w)) does not have a double pole, then under some mild additional assumptions, a rational support point f must be a rotation of the Koebe function. The situation is more complicated when Φ \Phi has a double pole. However, we are able to prove the two-functional conjecture for derivative functionals, where Φ \Phi has a double pole.