Global injectivity of differentiable maps via W-condition in R2

In this paper, we study the intrinsic relations between the global injectivity of the differentiable local homeomorphism map F and the rate of the Spec(F) tending to zero, where Spec(F) denotes the set of all (complex) eigenvalues of Jacobian matrix JF(x), for all x is an element of R-2. They depend deeply on the W-condition which extends the *-condition and the B-condition. The W-condition reveals the rate that tends to zero of the real eigenvalues of JF, which can not exceed O(x ln x(ln lnx/ln ln x)(2))(-1) by the half-Reeb component method. This improves the theorems of Gutierrez [16] and Rabanal [27]. The W-condition is optimal for the half-Reeb component method in this paper setting. This work is related to the Jacobian conjecture.