Second Derivative of Univalent Function

Question

While doing some self-study today, I was reading through some proofs that the derivative of a Univalent Function vanishes nowhere. However, I couldn't find mention of the second derivative of a univalent function.

I thus presumed that there must be trivial counterexamples for second derivatives never vanishing; however, after testing a few simple, univalent maps I've failed to find a counterexample.

My question is thus as follows:

Can second derivatives of Univalent functions vanish somewhere?

Answer 1

After some research, I will update my answer. It turns out that any function entire and injective on $\mathbb{C}$ must be of the form $z\mapsto az+b$, so that if $f$ is univalent on $\mathbb{C}$ then the second derivative must be zero everywhere.

For a proof of this, see here:

entire 1-1 function