 Scalar estimate for Talagrand product induction

For $a\in [0,1]$,where the endpoint $t=1$ is used directly if $a=0$. Put $L=-\log a$. For $L\le 1/2$, choose $t=2L$ and set $h(L)=\log (2-e^{-L})-L+L^2$. Then $h(0)=h^{\prime }(0)=0$ and $h^{\prime \prime }(L)=2-2e^L/(2e^L-1{)}^2\ge 0$, proving the bound. For $L\ge 1/2$, 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.