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 ...

A.J. VAN DER POORTEN, A NOTE ON HADAMARD ROOTS OF RATIONAL FUNCTIONS, The Rocky Mountain Journal of Mathematics, Vol. 26, No. 3 (SUMMER 1996), pp. 1183-1197 ...