2018 Putnam B5

Let $f = (f_1, f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives

$\frac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that

$$

\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left( \frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0

$$

everywhere. Prove that $f$ is one-to-one.

令 $f = (f_1, f_2)$ 为从 $\mathbb{R}^2$ 到 $\mathbb{R}^2$ 且具有连续偏导数的函数

$\frac{\partial f_i}{\partial x_j}$ 处处为正。假设

$$

\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left( \frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0

$$

无处不在。证明 $f$ 是一对一的。