Do functions $f:(0,\infty) \to [0,\infty)$ which satisfy the property that $x \to f(e^x)$ is convex have a standard name in the mathematical literature?
Equivalently, define $g:\mathbb R \to [0,\infty)$ by setting $f(x)=g(\log x)$. So, our property is the convexity of $g$.
Here we precompose with $\log x$; this is not to be confused with the notion of a "Logarithmically convex function", where we first apply $f$, then take the logarithm.
I wonder whether there is a standard name for such functions $f$.
Comment:
This property does not imply the convexity of $f$, or vice versa:
Indeed, consider $f(x)= (\log x)^2$, and $f(x)=(x-1)^2$.
I am not aware of a standard name, but you may perhaps try convex up to an exponential, if it makes sense in English (I am not an English native speaker...). You should of course precisely define this notion before using it.