Inequality containing finite sum. For what value of k the following inequality holds?
$\sum_{i=1}^{n}a_{i}^3<k|\sqrt{\sum_{i=1}^{n}a_{i}}|$
I don't have any idea to solve this.

There is no constant $k$ that will work for all $a_i$. Take for example $a_i=c\gt 0$ for all $i$. Then the left-hand side is $nc^3$, while the right-hand side is $k\sqrt{nc}$. Whatever number $k$ we pick, by choosing $c$ large enough, we can make $nc^3$ bigger than $k\sqrt{nc}$.
