Difference of two sub additive function

Question

A real valued function $ f : \mathbb{R} \to \mathbb{R} $ is said to be sub additive if $ f( x+y ) \leq f( x ) + f( y ), \quad x,y \in \mathbb{R}. $ There are some nice properties related to sub additive function.

1. A function $g$ on $ [0, \infty) $ is sub additive if the function $\frac{g(x)}{x}$ is non increasing.

2. Sum of two sub additive function is a sub additive function.

3. The maximum of two sub additive function is again a sub additive function.

4. If $ f $ and $ g $ are sub additive and $ g $ is non decreasing then the composition function $ h(x) = g( f(x))$ is a sub additive function.

5. If $ f $ and $ g $ are sub additive, $ g $ is positive non decreasing and $ f $ is non positive then the product function $h(x) = f(x) g(x) $ is a sub additive function.

$\textbf{ Q. } $What about the difference of two sub additive function? Can we impose some conditions on the functions so the the difference of two sub additive function becomes a sub additive function?

$\textbf{ Example:}$ Consider $ f(x) = x ,\; x\in [0, \infty) $ and $ g(x) = \frac{x}{1+x} \;, x\in [0, \infty ) .$ Note that both of $f$ and $g$ are sub additive. But $ h(x) = f(x)- g(x) = \frac{x^2}{1+x}$ is not sub additive because the function $\frac{h(x)}{x} = \frac{x}{1+x}$ is an increasing function. Therefore,

$ \frac{h(x+y)}{x+y} \geq \frac{h(x)}{x}$ and $ \frac{h(x+y)}{x+y} \geq \frac{h(y)}{y}.$ Thus, $ h(x) + h(y) \leq \frac{x h(x+y)}{x+y} + \frac{y h(x+y)}{x+y} = h(x+y).$

If $h$ is sub additive then we would have $ h(x) + h(y) \leq h(x+y) \leq h(x) + h(y), \; i.e., \; h(x) + h(y) = h(x+y)$ which is not true.

Any idea or suggestion is appreciated. Thanks in advance.