A best function that satisfies required conditions

Question

I am looking for a smooth and Lipschitz function that satisfies the following condition :

1) It is a two-variable function f(x,y) that must have a direct relation with the variable x, and inverse relation with the variable y. In other words, increasing x lead to increase f(x,y) and increasing y lead to decrease f(x,y).

2) It must have a lower and upper bound.

I was wondering if anyone can help me.

Thank you

Answer 1

Suppose you have a bounded monotonically increasing function $g(x).$ Let the lower bound of $g(x)$ be $L$. Then $g(x) + 1 -L \ge 1$ and $$\frac{1}{g(x) + 1 -L}$$ is bounded and monotonically decreasing. Finally you can define the function you need as $$ f(x,y) = \frac{g(x)}{g(y) + 1 -L}. $$

One example of such $g(x)$ is the sigmoid function, i.e. $$ g(x)=\frac{1}{1+e^{-x}} $$ with $L=0$.