Let $p,q \in k[x,y]$, $k \in \{\mathbb{R}, \mathbb{C}\}$, be a Jacobian pair, namely, $p_xq_y-p_yq_x \in k^*$.
A proof of a special case of the two-dimensional Jacobian Conjecture in a respected article from about 20 years ago says that such a map $(x,y) \mapsto (p,q)$ (= an endomorphism having an invertible Jacobian) is injective.
However, according to 1 2 3, if such a map was injective, then it was bijective, and then the two-dimensional Jacobian Conjecture would be proved.
Therefore, there are two options: Either I misunderstood the implication 'injective implies bijective' (the above questions talk about a polynomial map $k^n \to k^n$, and I apply them for an endomorphism of $k[x,y]$-- but isn't an endomorphism define a polynomial map?), or there is an error in the proof (which ruins a nice result).
Any comments are welcome!
Edit: I have now noticed that although the article's claim: "Since $\phi$ is injective (this is ensured by the Jacobian condition)" seems wrong, but fortunately, for the subsequent arguments there is no need for injectivity, and only the invertibility of the Jacobian is used for showing algebraic independence of some elements. Thanks for the commenters for trying to help.