Diffeomorphisms of the circle and hyperbolic curvature

The trace $Tf$ of a smooth function $f$ of a real or complex variable is defined and shown to be invariant under conjugation by Möbius transformations. We associate with a convex curve of class $C^2$ in the unit disk with the Poincaré metric a diffeomorphism of the circle and show that the trace of the diffeomorphism is twice the reciprocal of the geodesic curvature of the curve. Then applying a theorem of Ghys on Schwarzian derivatives we give a new proof of the four-vertex theorem for closed convex curves in the hyperbolic plane.