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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 $x\in \mathbb{R}^2$. 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 $\displaystyle O\Big(x\ln x(\ln \frac{\ln x}{\ln\ln x})^2\Big)^{-1}$ by the half-Reeb component method. This improves the theorems of Gutiérrez ^{[<span class="xref"><a href="#b16">16</a></span>]} and Rabanal ^{[<span class="xref"><a href="#b27">27</a></span>]}. The $W$-condition is optimal for the half-Reeb component method in this paper setting. This work is related to the Jacobian conjecture.