I was playing around with a problem and I obtained a certain combinatorial sum. I was wondering if there was a way to simplify or bound it.

I have a real valued function $f$, which satisfies $|f(x)| \ll x^{\alpha}$ and $|\sum_{j=i}^n f(j)| \ll n^{\beta}$, $\alpha, \beta > 0$.

The sum I want to bound is $$ S(n) = \sum_{j_1 + j_2 = n-2} f(j_1)f(j_2) - 2 \sum_{j_1 + j_2 = n-1} f(j_1)f(j_2) + \sum_{j_1 + j_2 = n} f(j_1)f(j_2). $$

I was also wondering if bound $S(n) \ll \max( n^{2 \alpha}, n^{2 \beta} )$ is a reasonable bound to expect or not. (Maybe this is too strong I'm not sure...) I would appreciate any help finding a bound for $S(n)$ or simplifying it somehow. Thank you very much.