This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into ...

Critical Point: A point in the complex plane where the derivative of a polynomial vanishes, revealing important information about the function's behaviour.

Arthur B. Brown, Relations Between the Critical Points of a Real Analytic Functions of N Independent Variables, American Journal of Mathematics, Vol. 52, No. 2 (Apr., 1930), pp. 251-270 ...