I would like to prove the following statement:
Let $X_1, \ldots, X_n$ be independent centered random variables such that
$$ \mathbb{E}(|X_\ell|^m) \le m! K^{m-2} \frac{\sigma_\ell}{2} $$
for all $\ell = 1, \ldots, n$, $ m \in \mathbb{N}, m \ge 2 $ and some positive constants $K, \sigma_\ell$. Then for all $t > 0$,
$$ \mathbb{P}(|\sum_{\ell=1}^n X_\ell| > t) \le 2\exp\left(-\frac{\frac{t^2}{2}}{\sigma^2 + Kt}\right) $$
for $\sigma^2 := \sum_{\ell = 1}^n \sigma_\ell^2$.
So far I tried to use Markov's inequality with a parameter and the exponential function, but then I'm stuck. There I tried to expand the exponential to its power series and applying the bound on the moments