Prove that:
$ \rightarrow\sum_{k=1}^n f(\frac{k}{n})\sum_{k=1}^n k{f(\frac{k}{n})}^2\le\sum_{k=1}^n kf(\frac{k}{n})\sum_{k=1}^n{f(\frac{k}{n})}^2 $
Given $f(x)$ is a positive function and also monotonic decreasing function.
Prove that:
$ \rightarrow\sum_{k=1}^n f(\frac{k}{n})\sum_{k=1}^n k{f(\frac{k}{n})}^2\le\sum_{k=1}^n kf(\frac{k}{n})\sum_{k=1}^n{f(\frac{k}{n})}^2 $
Given $f(x)$ is a positive function and also monotonic decreasing function.
Denote $a_k=f\left(\frac{k}{n}\right).$
(Proof by induction). Let $P(n)$ be the statement $\sum\limits_{k=1}^n f\left(\frac{k}{n}\right)\sum\limits_{k=1}^n k\left(f\left(\frac{k}{n}\right)\right)^2\leqslant\sum\limits_{k=1}^n kf\left(\frac{k}{n}\right)\sum\limits_{k=1}^n\left(f\left(\frac{k}{n}\right)\right)^2$ or, in shorter form $\sum\limits_{k=1}^n a_k\sum\limits_{k=1}^n ka_k^2\leqslant\sum\limits_{k=1}^n ka_k\sum\limits_{k=1}^na_k^2.$