Is there any function $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ satisfying the following conditions?:
(1) the sequence $f(n)$ is convergent;
(2) $\log f$ is eventually monotone (i.e., it is monotone from a number on);
(3) $\log f$ is not eventually convex or concave.
If yes, is there a (big) class of such functions (with various properties)?
There are lots of such functions. Let g be a positive increasing function on $\mathbb R^{+}$ such that g(n)=1-1/n for each n and such that g does not have a left derivative at some point in (k,k+1) for each k. Let $f=e^{g}$. Then $log f$ is not concave or convex eventually because convex and concave functions have left derivatives at every point. Of course, one can use right derivatives instead of left derivatives.