A minimax principle to the injectivity of the Jacobian conjecture

The main result of this paper is to prove some type of Real Jacobian Conjecture. It is proved by the Minimax Principle and asserts if the eigenvalues of $F'(x)$ are bounded from zero and all the eigenvalues of $F'(x)+F'(x)^T$ are strictly same sign, where $ F $ is $ C^1 $ mapping from $ \mathbb{R}^n $ to $ \mathbb{R}^n $, then $ F $ is injective. Moreover $F$ has a $ C^1 $ mapping inverse.