Proof of Kuhn's polynomial conjectures (Q791702)

Let \(f\) be a polynomial of exact degree n mapping the interval \(I=[-1,1]\) into itself and having all roots in \(I\). The rational function \(R_ f(x) = \frac{1-f^ 2(x)}{(1-x^ 2)(f'(x))^ 2}\) is positive on those subintervals of \(I\) where it is regular, and its positive squareroot \(\sqrt{R_ f}\) satisfies \(\left(\sqrt{R_ f}\right)''>0\) there, unless either \(f\) or \(-f\) equals the Chebyshev polynomial \(T_ n\) (in these cases \(R_ f\) is constant and minimal). There is a similar result for trigonometric polynomials.