Elementary proof of a theorem on conformal rigidity

Abstract

Let f(z) be single valued and analytic in the annulus A: a < I zI <b, f(A) CA, and suppose that f maps every closed curve with winding number +1 about z =0 onto a curve with the same property. Then [1], [2], [3], [4] f if a rotation. Stimulated by a discussion with Professor A. Marden we shall give a short proof of this result with a minimum of topological notions. By hypothesis, the integral of f'(z) z f(Z) around any closed curve in A vanishes. Hence there exists a branch, F(z), of log [z/f(z) ] single valued in A. Let us define u(rei0) = Re F(rei0) = log r -log I f(rei0) , a < r < b, f2 I(r) = (2Xr)-l u(rei0) do.