[proofplan] The proof is a direct optimization of the HWI inequality. The HWI bound estimates the entropy by a concave quadratic expression in the Wasserstein distance $W_2(\mu,\nu)$, with coefficient determined by the positive curvature parameter $\lambda$. Since this quadratic has maximum $I_\nu(\mu)/(2\lambda)$ on $[0,\infty)$, evaluating it at the actual distance gives the logarithmic Sobolev inequality. [/proofplan]

[step:Apply the HWI inequality with positive curvature parameter]Define the entropy value $E\in[0,\infty)$, the Fisher information value $J\in[0,\infty)$, and the Wasserstein distance value $W\in[0,\infty)$ by \begin{align*} E=\operatorname{Ent}_\nu(\mu) \end{align*} \begin{align*} J=I_\nu(\mu) \end{align*} \begin{align*} W=W_2(\mu,\nu). \end{align*} The hypotheses on $\nu=e^{-V}\,d\mathcal L^n$, the lower Hessian bound $J(\nabla V)_x\ge\lambda I$ with $\lambda>0$, and the stated regularity, moment, entropy, Fisher-information, and decay assumptions on $\mu=f\nu$ are exactly the hypotheses required to apply the [Otto-Villani HWI Inequality][citetheorem:9570]. Hence \begin{align*} E\le W\sqrt{J}-\frac{\lambda}{2}W^2. \end{align*}[/step]

[guided]We introduce three scalar quantities because the rest of the proof is purely one-dimensional. Let \begin{align*} E=\operatorname{Ent}_\nu(\mu) \end{align*} be the relative entropy, let \begin{align*} J=I_\nu(\mu) \end{align*} be the relative Fisher information, and let \begin{align*} W=W_2(\mu,\nu) \end{align*} be the quadratic Wasserstein distance between $\mu$ and $\nu$. The theorem assumes that $J<\infty$ and that $\mu$ lies in the smooth finite-entropy and finite-Fisher-information class covered by the HWI theorem, so all three quantities are finite nonnegative real numbers in the present argument. Now apply the [Otto-Villani HWI Inequality][citetheorem:9570]. Its hypotheses require precisely the curvature lower bound $J(\nabla V)_x\ge\lambda I$, the probability reference measure $\nu=e^{-V}\,d\mathcal L^n$, and the smoothness, positivity, finite-moment, finite-entropy, finite-Fisher-information, and integration-by-parts assumptions on $\mu=f\nu$. Therefore it gives \begin{align*} \operatorname{Ent}_\nu(\mu)\le W_2(\mu,\nu)\sqrt{I_\nu(\mu)}-\frac{\lambda}{2}W_2(\mu,\nu)^2. \end{align*} With the abbreviations above, this is exactly \begin{align*} E\le W\sqrt{J}-\frac{\lambda}{2}W^2. \end{align*}[/guided]

[step:Maximize the resulting quadratic upper bound] For every $w\in[0,\infty)$, completing the square gives \begin{align*} w\sqrt{J}-\frac{\lambda}{2}w^2 = \frac{J}{2\lambda}-\frac{\lambda}{2}\left(w-\frac{\sqrt{J}}{\lambda}\right)^2. \end{align*} Since $\lambda>0$, the second term on the right-hand side is nonpositive. Therefore \begin{align*} w\sqrt{J}-\frac{\lambda}{2}w^2\le \frac{J}{2\lambda} \end{align*} for every $w\ge0$. Applying this with $w=W$ yields \begin{align*} E\le \frac{J}{2\lambda}. \end{align*} [/step]

[step:Translate the scalar inequality back to entropy and Fisher information] Substituting back the definitions of $E$ and $J$, we obtain \begin{align*} \operatorname{Ent}_\nu(\mu)\le \frac{1}{2\lambda}I_\nu(\mu). \end{align*} This is the claimed logarithmic Sobolev inequality for every $\mu=f\nu$ in the smooth finite-entropy and finite-Fisher-information class covered by the HWI theorem. [/step]