In the proof of Lemma 1.1 of this article, we read
The zeros of $f''(x+iy) = 0$ are critical points of $f(x+iy)$.
This claim is essential to the proof yet appears to be wrong. For example, $f(z) = z^3 + z + 1$ is holomorphic with $f''(z) = 0$ at $z = 0$, yet $f'(z) = 1$.
What (if anything) have I got wrong? If I do have something wrong, can the proof be salvaged?