Higher order extremal problem and proper holomorphic mapping (Q2771483)

It is well-known that the Riemann mapping function of a simply connected domain \(\Omega\) in the plane maximizes uniquely the value \(h'(a)> 0\) where \(h\) maps holomorphic \(\Omega\) into the closed unit disc. That means NEWLINE\[NEWLINEf_a'(a)= \max h'(a).NEWLINE\]NEWLINE The author shows that the extremal solution to NEWLINE\[NEWLINEf^{(m)}(a)= \max h^{(m)}(a)NEWLINE\]NEWLINE for all holomorphic functions \(h\) from \(\Omega\) into the closed unit disc with NEWLINE\[NEWLINEh(a)= h'(a)=\cdots= h^{(n- 1)}(a)= 0NEWLINE\]NEWLINE and \(h^{(m)}(a)> 0\) is a suited holomorphic mapping of \(\Omega\) into the unit disc. In case of a simply connected domain \(f= f^m_a\). It is lined out that the solution of this problem is connected with Ahlfors' generalization of the Riemann mapping. This article has a nice introduction (here called motivation) which contains relations of this problem to several kernel functions (Szegő kernel, Cauchy kernel, Garabedian kernel, Bergman kernel).