Let $b=(b_1,...,b_n), b_i\in \mathbb R,$ for $i=1,..,n$. Let $\epsilon=(\epsilon_1,..,\epsilon_n)$ be a Rademacher sequence, i.e. $Prob(\epsilon_i=1)=Prob(\epsilon_i=-1)=\frac 12$. It is known that for all $p\geq 2$,
$\left(E|\sum_{i=1}^n\epsilon_ib_i|^p\right)^{1/p}\leq Cp^{1/2} ||b||_2$.
Show similar inequality if in addition:
1) $\sum_{i=1}^n\epsilon_i=0$
2) $\sum_{i=1}^n\epsilon_i=1$
3) $\sum_{i=1}^n\epsilon_i=-1$