I have come across this equation (Lemma 1) in this paper:
$$P\left(\left\lVert \frac{1}{m}\sum_{i=1}^{m}(Z_i-E[Z_i])\right\rVert>\epsilon\right) = 2 \exp\left(-\frac{m\epsilon}{2M}\log\left(1+\frac{mM\epsilon}{\sigma^2}\right)\right)$$
where $Z_i$ is a random variable taking values in a Hilbert space, and we have $\left\lVert Z_i\right\rVert≤ M≤\infty$ and $\sigma^2 = E\left[\left\lVert Z_i\right\rVert^2\right]$.
Unfortunately, it comes without proof. It says it can be derived from Bennett inequality and using the following elementary inequality:
$$t\log(1+t) ≥ 2t - 2\log(1+t).$$
The additional reference given in the paper is not helpful for me. I can't make sense of it. I have looked at the derivation of Bennett's inequality, but I am unable to derive the inequality above. Any help would be greatly appreciated.