The Schwarzian Derivative for Harmonic Mappings

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting its connection with linear differential equations. Osgood and Stowe [8] have unified these various univalence criteria through a general theorem involving the curvature of a metric. The purpose of the present paper is to offer a definition of Schwarzian derivative that applies more generally to complex-valued harmonic functions. The formula is derived in a natural way by passing to the minimal surface associated locally with a given harmonic function. The derivation then appeals to a definition given by Osgood and Stowe [9] for the Schwarzian derivative of a conformal mapping between arbitrary Riemannian manifolds. The resulting expression reduces to standard form when the harmonic function is analytic, and various classical properties of Schwarzian derivatives generalize in appropriate ways, suggesting that the definition we propose is the "right" one. The Schwarzian derivative of a locally univalent analytic function f is defined by S,(f) : (f,,/f,), __ ~(fl n/f,) 2.