Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space - Sabatini, Marco (Topological Methods in Nonlinear Analysis, 2023)
Fessler and Gutierrez Fe, Gu proved that if a non-singular planar map has Jacobian matrix without eigenvalues in (0,+∞), then it is injective. We prove that the same holds replacing (0,+∞) with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove that a Jacobian map (P,Q) is injective if P/x + Q/y is not a surjective function.