  We study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=\lambda u(z) with
the boundary conditions that u(z) decays to zero as z tends to infinity along
the rays \arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}, where P(z)=a_1 z^{m-1}+a_2
z^{m-2}+...+a_{m-1} z is a real polynomial and m\geq 2. We prove that if for
some 1\leq j\leq\frac{m}{2}, we have (j-k)a_k\geq 0 for all 1\leq k\leq m-1,
then the eigenvalues are all positive real. We then sharpen this to a slightly
larger class of polynomial potentials.
  In particular, this implies that the eigenvalues are all positive real for
the potentials \alpha iz^3+\beta z^2+\gamma iz when \alpha,\beta and \gamma are
all real with \alpha\not=0 and \alpha \gamma \geq 0, and with the boundary
conditions that u(z) decays to zero as z tends to infinity along the positive
and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin.
