Bennett's Inequality is stated with a rather unintuitive function,
$$ h(u) = (1+u) \log(1+u) - u $$
See here. I have seen in multiple places that Bernstein's Inequality, while slightly weaker, can be obtained by bounding $h(u)$ from below,
$$ h(u) \ge \frac{ u^2 }{ 2 + \frac{2}{3} u} $$
and plugging it back into Bennett's Inequality. However, I can't see where this expression comes from. Could someone point me in the right direction?