Let p, q be continuously differentiable functions defined in a vertical strip \(\Sigma\) of the plane and assume that p is a polynomial of degree 1 in the second variable. If the Jacobian determinant of the map (p, q) never vanishes, then (p, q) is globally injective provided that q is a polynomial in the second variable. Moreover, assuming that the Jacobian determinant of (p, q) is bounded away from zero, then (p, q) is injective without any further assumption on q. As a consequence, we generalize results on global injectivity recently given by Sabatini.
