Let $r_i, i=1, \ldots, n$ be a random variables that takes values $\alpha$ and $-\alpha$ with equal probability. Let $a_i, i=1, \ldots, n$ be real valued numbers.
How to find (or bound from above) the following conditional expectation for $p\geq 2, \quad T\in R$: $$ E\left(\left|\sum_{i=1}^na_ir_i\right|^p \big| \sum_{i=1}^nr_i=T\right)? $$