Let $f : \mathbb{R} \mapsto \mathbb{R}$ be some decreasing, differentiable function. Then, the function $g : \mathbb{R} \mapsto \mathbb{R}$ defined as, $$g(x) = \dfrac{1 + 2f(x)}{1 - f(x)}$$ is also decreasing, as can easily be checked by the quotient rule for derivatives.
But if $f : \mathbb{N} \mapsto \mathbb{Q}$ be a decreasing function, can we say that the function $g(n) = \dfrac{1 + 2f(n)}{1 - f(n)}$ is also decreasing? How can this be proven?
I am actually working with $f(n) = \displaystyle\prod_{r = 2}^{r = n} \dfrac{r^3 - 1}{r^3 + 1}$. I tried finding the first few values of $g(n)$ and they are indeed decreasing. But I don't know how to prove that $g(n)$ is decreasing.
Thanks for any hints or ideas about this.