# subadditive function with special growth

Related to one of my previous question (for which I have received an answer, thanks) I have the following new one. Maybe I am describing the empty set but not being a specialist at all of the domain I prefer to ask the community. I’m looking for a real valued function $$g(x)$$ with the following behaviour:

$$g(x)$$ defined $$[-\infty,0]$$ with first derivative on $$(-\infty,0)$$

$$g(x)$$ sub-additive on $$(-\infty,+\infty)$$

$$e^{x} \cdot g(x)$$ convex

$$g(0) < +\infty$$

$$\frac{g(x)}{x} \rightarrow -\infty$$ when $$x \rightarrow -\infty$$

Hope I’m clear enough.

Thanks for any suggestion