On the zeros of the derivative of a rational function and coinvariant subspaces for the shift operator on the Bergman space

If all $n$ $(n>1)$ zeros of a rational function $r$ with simple poles are in a half-plane, then the derivative of $r$ has at least one zero in the same half-plane. This result is used to prove that the number of zeros of a linear combination of $n$ Bergman kernels in the unit disc may range from 0 to $2n-3$.