= Scalar estimate for Talagrand product induction
For $a\in[0,1]$,
$$
\inf_{0\leq t\leq1}e^{t^2/4}a^{-(1-t)}\leq2-a,
$$
where the endpoint $t=1$ is used directly if $a=0$. Put $L=-\log a$. For $L\leq1/2$, choose $t=2L$ and <set> $h(L)=\log(2-e^{-L})-L+L^2$. Then $h(0)=h'(0)=0$ and $h''(L)=2-2e^L/(2e^L-1)^2\geq0$, proving the bound. For $L\geq1/2$, take $t=1$; the result follows from the already proved boundary case and monotonicity of $2-a$. This estimate closes the section-by-section induction for the exponential <Talagrand convex distance> moment.
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