Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{\frac{x \left(1 + \frac{2}{\log{\left(x + 1 \right)}}\right)}{x + 1} - 2}{\left(x + 1\right) \log{\left(x + 1 \right)}^{2}} = 0$$
Solve this equationSolutions are not found,
maybe, the function has no inflections