Level sets, a Gauss-Fourier conjecture, and a counter-example to a conjecture of Borcea and Shapiro

We use level sets to give a counter-example to a conjecture of J. Borcea and B. Shapiro. In particular we show that there are real polynomials P such that there is a chord of the level set Im P'(z)=P(z) = 0 that does not have a non-real zero of P lying on it. In addition, we use level sets to show that there are no bounded components of the set {z: Im z > 0 and ImQ(z) > 0} where Q = z -f/f" and f is a real entire function.