Conformal Mapping and Ellipses

We answer a question raised by M. Chuaqui, P. Duren, and B. Osgood by showing that a conformal mapping of a simply connected domain cannot take two circles onto two proper ellipses. The goal of this note is to prove the following. Proposition 1. Let Ω be a simply connected domain on C = C ∪ {∞} containing circles (on C) C1 and C2, C1 = C2. Let f be a conformal mapping from Ω into C. If f maps each of the circles C1 and C2 onto an ellipse, then f is a Möbius transformation and therefore the ellipses are actually circles. Thus this proposition answers affirmatively a question raised by M. Chuaqui, P. Duren, and B. Osgood in [1] that a conformal mapping of a simply connected domain cannot take two circles onto two ellipses. Precomposing f with a Möbius transformation and postcomposing it with a linear transformation, we may assume that C1 = {z : |z| = 1} and that f maps the unit disc D = {z : |z| < 1} onto the interior of an ellipse L1 with foci ±1 such that f(r) = 1, f(−r) = −1 for some 0 < r < 1. Then it is a standard exercise in complex analysis to verify that w = f(z) is defined by (1) w = 1 2 (ζ + 1/ζ) with ζ = i exp { − πi 2K(r2) ∫ z/r 0 dt √ (1− t2)(1− r4t2) }