I've encountered a paper that given $a_1, a_2, \ldots, a_m \in \mathbb N_+$ and $k \in \mathbb N_+, k \le m$ assumes that the following inequality holds without further comment:
$\hspace{20pt}\mathcal O\left(\sum\limits_{i = 1}^{m - k + 1} \left(a_i + a_{i + 1} + \ldots + a_{i + k - 1}\right)^{2k - 1}\right) \le \mathcal O\left(\left(a_1 + a_2 + \ldots + a_m\right)^{2k - 1}\right)$
I can't seem to come up with a proof.
Edit: Added Big O notation.