Q:

Relation between weighted sum of squares and weighted sum

Is there a relation between $$X = \sum_{i=1}^n p_i x_i\quad  \text{and} \quad X' = \sum_{i=1}^n p_i x_i ^2?$$ 
We can assume that $\sum_{i=1}^n p_i =1$ and $\forall i\in[1,n], p_i \geq 0$ (basically, the $p_i$'s in my case are normalized weights).
I know when $p_1 = p_2 = \dots = p_n = p$, we have a special case of the Cauchy-Schwarz inequality and get the relation: 
$$\sum_{i=1}^n x_i ^2 \geq \frac{(\sum_{i=1}^n x_i)^2}{n}.$$
My question is: Is there any known lower bound or relation linking the two sums when the weights are not equal?
Thanks!

A:

Since the function $f(x) = x^2$ is convex, you can apply Jensen's inequality to get:
$$
\left(\sum_{i=1}^n p_i x_i\right)^2 \le \sum_{i=1}^n p_i x_i^2
$$

