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