Massart eq. (4): the modified log-Sobolev summand, single fibre

Massart eq (4) — modified logarithmic Sobolev summand (single-fiber, ψ-form). For a probability measure $\mu$ (one conditional fiber of the entropy tensorization), a measurable $Z$ with $e^{\lambda Z}$ and $\lambda Z e^{\lambda Z}$ integrable, and a constant reference exponent $c$ (the leave-one-out value $Z_k$, which is $\sigma(\text{coords}\neq k)$-measurable hence constant on the $k$-fiber), the entropy functional of $e^{\lambda Z}$ is bounded by the ψ-summand:

This is the per-coordinate conditional modified-LSI brick (BLM Theorem 6.6 / Massart's lemma). Summed over the coordinates $k$ against the sub-additivity (Han) tensorization of entropy applied to $e^{\lambda Z}$, it yields the full Massart eq (4) modified log-Sobolev inequality, which — with Bousquet's eq (6) per-summand bound and condition (3) — gives the differential inequality $G'' \le v\,e^{\lambda}$ integrated by the Herbst step to the sub-gamma cgf bound. Proof: the constant-reference variational (dual) bound of entropy $\mathrm{Ent}_\mu(Y)\le\int(Y\log Y - Y\log u-(Y-u))$ at $Y=e^{\lambda Z}$, $u=e^{\lambda c}>0$, followed by the pointwise identity $e^{a}(a-b)-(e^{a}-e^{b})=e^{a}(e^{b-a}-(b-a)-1)=e^{a}\psi(a-b)$. Here $\psi$ is written inline as $e^{-x}-1+x$ (= BLM's $\varphi(-x)$, $\varphi(x)=e^{x}-x-1$).