Let $Y_1,\cdots,Y_n$ are independent and identically distributed random variables. Let $X = \sum_{i=1}^n Y_i$. What is upper and lower bounds of $\mathbf{E}[X^2]$ in terms of first and second moment of $Y_i$'s?
One upper bound is
$$ \mathbf{E}\left[X^2\right] \leq \sum_{i=1}^n \mathbf{E}\left[Y_i^2\right] + \left(\sum_{i=1}^n \mathbf{E}[Y_i]\right)^2. $$