If $(a_k)$ and $(b_k)$ are positive sequences of the same length, and $$0<m\le \frac{a_k}{b_k} \le M<\infty$$ $$A=\frac{m+M}{2},\ \ G=\sqrt{mM}$$ then $$(\Sigma{a_k}^2)(\Sigma{b_k}^2) \le (\frac{A}{G}\Sigma{a_kb_k})^2=\frac{A^2}{G^2}(\Sigma{a_kb_k})^2$$ This is a reverse of Cauchy-Schwarz which follows from the trivial inequality $(M-\frac{a_k}{b_k})(\frac{a_k}{b_k}-m) \ge 0$ and the arithmetic-geometric mean inequality.
Applying this for $\sqrt{a_k}, \frac{1}{\sqrt{a_k}}$, we recover the inequality given in the answer by kodlu.
