In the paper (http://www.sciencedirect.com/science/article/pii/S0022247X97953439), Webster obtained a unique solution of the functional equation $f(x+1)=g(x)f(x)$ (where $f,g:\mathbb{R}^+\rightarrow \mathbb{R}^+$) under some conditions one of which is $\lim_\limits{x\to \infty}\frac{g(x+w)}{g(x)}=1$ for all $w>0$.

Now, I'm looking for a function $g:\mathbb{R}^+\rightarrow \mathbb{R}^+$ such that the sequence
$\frac{g(n+1)}{g(n)}\rightarrow 1$, $\lim_\limits{x\to \infty}\frac{g(x+w_0)}{g(x)}$ does not exist, for some
$w_0>0$, and having a solution $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ for the functional equation $f(x+1)=g(x)f(x)$ (**Gamma-type functional equation**) with the following properties?:

(a) $f$ is eventually $\log$-convex (i.e., it is $\log$-convex from a number on);

(b) $f(1)=1$.

Thanks in advance.