Q:

An upper-bound problem of sum of positive numbers

I came across the following problem of inequality.
If $ \ \ \sum_{i=1}^{n}x_i^3\leq S$ 
then find the value of $K$ such that  
$\sum_{i=1}^{n}x_i\leq K$. 
It is given that $x_i>0,\forall i\in \{1,2,3,..,n\}$ 
I guess it can be solved using Jensen's theorem but I can't think of the function I have to use. I think $K$ should be as a function of $n$ and $S$.

A:

You can use power-mean inequality.
$$\sqrt[3]{\frac{a_{1}^{3}+a_{2}^{3}+\cdots +a_{n}^{3}}{n}}\ge \frac{a_{1}+a_{2}+\cdots +a_{n}}{n}$$

