Schwarzian derívate of a polynomial with different real roots

Question

I’m working on the exercise 7.5.2 of the book Brin and Stuck introduction to dynamical systems.

It says: Show that any polynomial with distinct real roots has negative schwarzian derívative.

The schwarzian derivative being defined as $Sf=\frac{f’’’}{f’}-\frac{3}{2}\left(\frac{f’’}{f’}\right)^2$

I was thinking about this for a while but had no ideas on how to solve it.

Answer 1

If $f$ has distinct real roots $z_i$, then so has $f'$ distinct real roots $w_i$. So compute the logarithmic derivative $$ \frac{f''(x)}{f'(x)}=\sum\frac1{x-w_i} $$ and the derivative of that $$ \frac{f'''(x)}{f'(x)}-\left(\frac{f''(x)}{f'(x)}\right)^2=-\sum\frac1{(x-w_i)^2} $$ This is negative and becomes even more so if you subtract one-half of the square again.