  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.
