Squared-influence logarithmic lower bound (${\sum }_i{\beta }_i^2\ge t^2(1-t{)}^2(\log n{)}^2/n$)

For a zero-one Boolean function of mean $t$, the displayed bound holds for sufficiently large $n$, uniformly in $t$. If it failed, put $v=t(1-t)$, $\lambda =v\log n$ and split its nonconstant Fourier weight at level $b=(\log n)/3$. The total influence bound puts at most $3v/4$ above that level. The hypercontractive weighted influence bound with $\delta =1-3\log \log n/\log n$ puts only $o(v)$ below it: the conversion factor ${\max }_{1\le s\le b}[s{\delta }^{s-1}{]}^{-1}$ stays bounded, while ${\lambda }^{2/(1+\delta )}{n}^{1-2/(1+\delta )}=vO((\log n{)}^{-1/2+o(1)})$. This contradicts total nonconstant weight $v$.