I am trying to understand how bound of the p-th moment is coming from the concentration inequality. In general, we would have, say, $S=\sum_{i=1}^na_ix_i$, where $x_i$ are random variables and $a_i$ are in $R$. Then, the concentration inequality can be written:
$$ E(S-ES)^p\leq Cp^p\|a\|_2. $$
In order to bound the $p$-th moment, one would write: $$ E(|S|^p)^{1/p}\leq E|S| +Cp^p\|a\|_2. $$
I cannot understand which inequality used here to get this last line? Would it be Minkovsky?