About the hypothesis on two functionals

Abstract

We consider linear continuous functionals L andN which differ from constant ones. We assume that no one of them can be reduced to another one by multiplication by a real constant. We consider the problem about the form of a schlicht and analytic function which maximizes the real parts of these functionals. For this problem theDuren hypothesis on two functionals is well known. This hypothesis assumes that if some function f ∈ S provides a global maximum ofReL andReN , then f is one of the Koebe functions: kθ(z) = z (1−zeiθ)2 . Further we consider the functionals which depend on a finite number of coefficients of the function f :