Consider the following operation
\begin{align*} a=\left|\underset{1\leq j\leq n}{\max}\frac{1}{n-j+1}\left(\sum_{i=j}^{n}r_{i}\right)^{2}-\underset{1\leq j\leq n}{\max}\frac{1}{n-j+1}\left(\sum_{i=j}^{n}s_{i}\right)^{2}\right|, \end{align*} where $n$ , $r_{i}$ and $s_{i}$ are given scalars. Is it possible to bound $a$ by a function of the following norms $|r-s|_1, |r-s|_2, |r-s|_\infty$ please? By definition, we have
\begin{align*} |r-s|_{1}= & \sum_{i=1}^{n}|r_{i}-s_{i}|,\\ |r-s|_{2}= & \sqrt{\sum_{i=1}^{n}|r_{i}-s_{i}|^{2}},\\ |r-s|_{\infty}= & \underset{1\leq j\leq n}{\max}|r_{i}-s_{i}|. \end{align*} Thanks.