Functionals of higher derivative type

Abstract

Functionals of higher derivative type are linear combinations of functionals of the form f→f(n)(ζ), where n ≥ 2 and 0 < |ζ| < 1. The paper shows that, if L is a functional of higher derivative type and f is a function in the class S of univalent functions that maximises Re{L} over S, then L(f) ≠ 0. In addition, if the function f is a rational function, then it must be a rotation of the Koebe function k(z) = z(1 - z)-2. These results are applied to establish several cases of the two-functional conjecture for functionals of higher derivative type.