Rational functions with positive Schwarzian derivative

This paper examines the sign of the Schwarzian derivative of real valued rational functions of degree 2 and higher. We prove that rational functions whose two polynomials have real, simple, and interlaced zeros have positive Schwarzian. We show the converse holds for rational functions of degree 2 and 3 but fails for higher degrees, as they may have non-real roots. Nonetheless, we prove the real zeros are still interlaced and simple, and we show that the two polynomials have the same number of non-real roots. These results settle two conjectures in the literature.