About the hypothesis on two functionals (Q941244)

Let \(S\) be the class of schlicht analytic functions, \(f(z)=z+ \sum^\infty_{n=2}c_nz^n\), in the circle \(\Delta=\{z\mid| z|< 1\}\). On the set of all analytic functions in \(\Delta\), we introduce the metric \(\rho(f,g)= \max_{| z|=\tfrac12}| f(z)-g(z)|\). The auther considers continuous linear functionals \(L\) and \(N\) which differ from constant ones, and assumes that no one of them can be reduced to the other one by multiplication by a real constant. In this article, the author considers the problem about the form of a schlicht analytic function which maximizes the real parts of these functionals. For this problem, the Duren hypothesis on two functional is well known: if some function \(f\in S\) provides a global maximum of \(\text{Re\,}L\) and \(\text{Re\,}N\), then \(f\) is one of the Koebe functions \(K_\theta(z)=\frac {z}{(1-ze^{i\theta})^2}\).