= Modified logarithmic Sobolev inequality
A <modified logarithmic Sobolev inequality> controls an <entropy functional> by an energy adapted to exponentials. For <independent random variables> $X_i$, let $f_i(X^{(i)})$ be any <measurable> coordinate-deleted <functions>, and put $Z=f(X)$. Where the displayed <expected values> are finite, <tensorization of entropy> gives
$$
\operatorname{Ent}(e^{\lambda Z})\leq
\mathbb E\left[e^{\lambda Z}\sum_i\phi\bigl(-\lambda(Z-f_i)\bigr)\right],\qquad
\phi(u)=e^u-u-1.
$$
Indeed, the one-coordinate <entropy functional> obeys $\operatorname{Ent}_i(Y)\leq\mathbb E_i[Y(\log Y-\log a)-(Y-a)]$ for every positive comparison value $a$ <independent> of that coordinate. Taking $Y=e^{\lambda f}$ and $a=e^{\lambda f_i}$ produces the displayed integrand, and then <tensorization of entropy> sums it.
If the sum of $\phi$ terms is bounded by $\lambda^2v/2$ pointwise, the result becomes $\operatorname{Ent}(e^{\lambda Z})\leq(\lambda^2v/2)\mathbb Ee^{\lambda Z}$. The <Herbst argument> integrates this to a <moment-generating function> bound. The energy bound must retain its dependence on $\lambda$; a bare statement that an unspecified energy is at most $v$ does not determine the normalization.
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