Displacement Convexity of the Logarithmic Energy on the Real Line

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, in dimension $d=1$, with the identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. The set $\mathcal{D}_{\log}$ and the logarithmic energy $\mathcal{E}_{\log}$, and the set $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ of measures of finite free Fisher information with the free score $\Xi_{\mu}\in T_{\mu}\subseteq L^{2}(\mu;\mathbb{R})$, are those of those definitions; $-\Xi_{\mu}$ is its negative in the vector space $L^{2}(\mu;\mathbb{R})$. The displacement pairing $\mathcal{J}$ and optimal couplings are those of The Intrinsic Calculus on the Wasserstein Space: Standing Notation (i) and Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i).

(Displacement convexity)(i) Let $\mu\in\mathcal{D}_{\log}\cap\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, let $\nu\in\mathcal{D}_{\log}$, and let $\pi\in\Pi(\mu,\nu)$ be an optimal coupling. Then

$$\mathcal{E}_{\log}(\mu)+\mathcal{J}(-\Xi_{\mu},\pi)\le\mathcal{E}_{\log}(\nu).$$