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
