Lorian Saceanu's Inequality with Many Variables

Solution

For the first case, define $A=1\ge a_k\ge a\gt 0$ and $B=1\ge b_k\ge b\gt 0.$ In the second case, $A=1\le a_k\le a$ and $B=1\le b_k\le b.$ Then the Polya-Szego inequality gives

$\displaystyle \frac{\displaystyle \left(\sum_{k=1}^na_k^2\right)\left(\sum_{k=1}^nb_k^2\right)}{\displaystyle \left(\sum_{k=1}^na_kb_k\right)^2}\le\frac{1}{4}\left(\sqrt{\frac{AB}{ab}}+\sqrt{\frac{ab}{AB}}\right)^2=\frac{(ab+AB)^2}{4abAB}$

which is equivalent to the required inequality.

Acknowledgment

Lorian Saceanu has kindly communicated to me his problem and solution via my facebook account.