Given a function $f$ with $f: \mathbb{N}_{\geq 0} \rightarrow \mathbb{R}^+$. I want to show that $f^*$ with $f^*(x) = \log f(x)$ is concave. Judging from the plot this is true, however, $f$ is a quite complex function.
What I know is that for $\lim_{n \rightarrow \infty} \frac{1}{n} f^*(n)$ is zero. Does this suffice, or do I also have to show that $f$ is a (strictly) increasing function? Or is there anything else I forgot?
The second derivative has to be negative, this means: $$y'(x)=\frac{f'(x)}{f(x)}$$ so $$y''=\frac{f''(x)f(x)-(f'(x))^2}{f(x)^2}<0$$