Further properties of Kuhn's ratio (Q581573)

Let f be a polynomial mapping the interval \(I=[-1,1]\) into itself. We consider the rational function \[ R_ f(x)=\frac{1-f^ 2(x)}{(1-x^ 2)(f'(x))^ 2} \] on I and show that \(\sqrt{R_ f}\) is convex where defined if one of the two additional assumptions on f is fulfilled: Either \((f')^{-1}(0)\subset I\) or \(f^{-1}(0)\subset {\mathbb{R}}.\) This theorem extends an earlier result where the assumption was \(f^{- 1}(0)\subset I\). Several more possible extensions are mentioned but not seriously believed in.