If $p(z)$ is a degree $3$ complex polynomial with distinct critical points (ie. zeros of $p'$), then the corresponding critical values must be distinct.
It is pretty easy to prove this using the level curves of $p$:
PROOF: If the critical values were the same, then both critical points must lie in the same level curve $\Gamma$, so $\Gamma$ must have three faces, each of which must contain at least one zero of $p$ by the max. mod. theorem.
Let $D$ denote the one face of $\Gamma$ which has both of the critical points in its boundary. Then since $p$ takes the same value twice on $\partial D$, the chance in the argument of $p(z)$ about $\partial D$ must be at least $2\pi$, and so $D$ must contain at least two zeros of $p$. Thus $p$ has at least four zeros, a contradiction since $\deg(p)=3$.
My question is: Can someone give an analytic proof (perhaps using Cauchy Integral Formula or something) for this fact?
