Let the function $g:\mathbb{R^+}\rightarrow\mathbb{R^+}$ have the properties that for each $w>0$, $\displaystyle{\lim_{x\to\infty}}\frac{g(x+w)}{g(x)} = 1$ and $\log g(x)$ is concave. Now, my questions are:
(a) Why if $\log g(x)$ is concave, then it has derivative $ \frac{g^\prime_{-}(x) + g^\prime_{+}(x) }{2g(x)}$ except, where $g^\prime_{+}(x)$ and $g^\prime_{-}(x)$ are right and left derivatives, respectively (I know that if $g$ is concave, then $g^\prime_{+}(x)$ and $g^\prime_{-}(x)$ exist).
(b) Is the following inequality known? how to get it?
$0\leq \gamma_g + \log g(1)\leq \frac{g^\prime_{-}(1) + g^\prime_{+}(1) }{2g(1)}$,
where $\sum_{i=1}^n\frac{g^\prime_{-}(i) + g^\prime_{+}(i) }{2g(i)} - \log g(n)\rightarrow \gamma_g$ as $n\rightarrow\infty$. Please guide me.