= Solution
By the <Cauchy-Schwarz inequality>, $\sum_i\beta_i\leq\sqrt{n\sum_i\beta_i^2}<\lambda$. The <total influence> identity in part (i) therefore gives $\boxed{\sum_{S\ne\varnothing}|S|\alpha_S^2<\lambda/4}$.
The <Holder inequality> gives $\sum_i\beta_i^{2/p}\leq n^{1-1/p}(\sum_i\beta_i^2)^{1/p}<\lambda^{2/p}n^{1-2/p}$. Substitute this into the <hypercontractive weighted influence bound> and multiply by $\delta$ to obtain
$$
\sum_{S\ne\varnothing}|S|\delta^{|S|}\alpha_S^2<\frac\delta4\lambda^{2/p}n^{1-2/p}\leq\boxed{\frac14\lambda^{2/p}n^{1-2/p}},
$$
as required. The version with $\delta^{|S|-1}$ is slightly stronger and will also be useful below.
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