Some inequalities for the distributions of sums of independent random variables

Let $X_i$, $i=\overline{1,n}$ be independent random variables,
$$ S_n=\sum_1^nX_i,\ F_i(x)=\mathbf P(X_i<x),\ \overline{\alpha}_k=\int_0^\infty x^t\,dF_k(x). $$

Upper estimates are given for $\mathbf P(S_n\ge x)$ in terms of the sum
$$ \sum_{1\le i_1\le\dots\le i_p\le n}\overline{\alpha}_{i_1}\dots\overline{\alpha}_{i_p}. $$

Upper and lower estimates are obtained for $\mathbf M|S_n|^t$, $t>2$.