Assume $x_1, \dots, x_k$ are distributed according to a multinomial distribution with parameters $n, p_1, \dots, p_k$. How can I compute the following expectation or a non-trivial upper bound for it?
$$ \mathbb{E} \Big[ \frac{1}{\sqrt{x_i}}\; \Big| \; x_i > 0 \Big]$$