The Schwarzian derivative for harmonic mappings
The Schwarzian derivative \(S(f)\) of an analytic function \(f\) is defined by \[ S(f) =(f''/f')'-(1/2) (f''/f') ^2 \,. \] It has many natural properties in geometric function theory. For instance, one can give sufficient conditions for univalence in terms of the Schwarzian. Also \(S(T\circ f) =S(f)\) if \(T\) is a Möbius transformation. For an analytic function \(\varphi\), the general analytic function \(f\) with the Schwarzian \(S(f) =\varphi\) has the form \(f=w_1/w_2\) where \(w_1, w_2\) are arbitrary linearly independent solutions of the differential equation \(w'' + \varphi w =0 \,.\) It follows, for instance, that if \(S(f) = 0\) then \(f\) is a Möbius transformation. The purpose of this paper is to explore these results in the case of harmonic mappings. A complex-valued function \(f\) that is harmonic in a simply connected domain \(\Omega \subset {\mathbb C}\) has the canonical representation \(f= h+ \overline g\) where \(f\) and \(g\) are analytic in \(\Omega\) and \(g(z_0 ) =0\) at some prescribed point \(z_0 \in \Omega\,.\) The Schwarzian of a harmonic function is defined in terms of a metric of the minimal surface associated with the harmonic function. One of the results the authors prove is that a harmonic function \(f\) with a vanishing Schwarzian is of the form \(f=h +\alpha \overline h\) for some Möbius transformation \(h\) and some complex constant \(\alpha\) with \(|\alpha|<1 \,.\) They also study the question to what extent \(S(f)\) determines \(f \,.\)
scientific article; zbMATH DE number 2103500