For independent Rademacher random variables $\epsilon_i, i=1,2, \cdots, n$, i.e. $P(\epsilon_i=-1)=P(\epsilon_i=1)=\frac{1}{2}$, do we have

$$max_{0\le a_i\le1, i=1,2,\cdots,n}P(|\sum_{i=1}^n\epsilon_ia_i^2|>x)\le max_{0\le a_i\le1, i=1,2,\cdots,n}P(|\sum_{i=1}^n\epsilon_ia_i|>x), \forall x>0.$$

For independent Rademacher random variables $\epsilon_i, i=1,2, \cdots, n$, i.e. $P(\epsilon_i=-1)=P(\epsilon_i=1)=\frac{1}{2}$, do we have

$$max_{0\le a_i\le b, i=1,2,\cdots,n}P(|\sum_{i=1}^n\epsilon_ia_i^2|>x)\le max_{0\le a_i\le b, i=1,2,\cdots,n}P(|\sum_{i=1}^n\epsilon_ia_i|>x), \forall x>0,$$ where $0\le b<1$.

For independent Rademacher random variables $\epsilon_i, i=1,2, \cdots, n$, i.e. $P(\epsilon_i=-1)=P(\epsilon_i=1)=\frac{1}{2}$, do we have

$$P(|\sum_{i=1}^n\epsilon_ia_i^2|>x)\le P(|\sum_{i=1}^n\epsilon_ia_i|>x), \forall x>0$$ provided $0\le a_i\le1, i=1,2,\cdots,n$.

For independent Rademacher random variables $\epsilon_i, i=1,2, \cdots, n$, i.e. $P(\epsilon_i=-1)=P(\epsilon_i=1)=\frac{1}{2}$, do we have

$$max_{0\le a_i\le1, i=1,2,\cdots,n}P(|\sum_{i=1}^n\epsilon_ia_i^2|>x)\le max_{0\le a_i\le1, i=1,2,\cdots,n}P(|\sum_{i=1}^n\epsilon_ia_i|>x), \forall x>0.$$