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Fessler and Gutierrez \cite{Fe,Gu} proved that if a non-singular planar maphas Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective.We prove that the same holds replacing $(0,+\infty)$ with any unbounded curvedisconnecting the upper (lower) complex half-plane. Additionally we prove thata Jacobian map $(P,Q)$ is injective if $P_x + Q_y$ is not a surjectivefunction.

Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space