Support points and double poles

Goh, Say Song

doi:10.1090/S0002-9939-1994-1197537-4

Support points and double poles
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Proc. Amer. Math. Soc. 122 (1994), 463-468 Request permission

Abstract:

This paper gives some sufficient conditions for support points of the class S of univalent functions to be rotations of the Koebe function $k(z) = z{(1 - z)^{ - 2}}$. If f is a support point associated with a continuous linear functional L and if the function $\Phi (w) = L({f^2}/(f - w))$ 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 $\Phi$ has a double pole. However, we are able to prove the two-functional conjecture for derivative functionals, where $\Phi$ has a double pole.

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