The Entropy of the Push-Forward of a Measure by the Gradient of a Twice Continuously Differentiable Function with Pinched Hessian

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment, and with finite entropy, the entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ as in that definition. Let $\Phi:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$, which is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, with Hessian matrix $D^{2}\Phi(x)$ and gradient map $\nabla\Phi:x\mapsto D\Phi(x)$ as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, and let $\varepsilon$ and $L$ be positive real numbers with $\varepsilon I_{d}\preceq D^{2}\Phi(x)\preceq L\,I_{d}$ for every $x\in\mathbb{R}^{d}$. The map $\nabla\Phi$ is Borel by The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse §bijection, The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse §inverse and Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §regularity, and $0<\det D^{2}\Phi(x)$ for every $x$ by Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §pinching and Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §positive, with $\det$ the determinant and $\log$ the natural logarithm. Let $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$.

1. (Second moment)¶ $(\nabla\Phi)_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

2. (Entropy)¶ The function $x\mapsto\log\det D^{2}\Phi(x)$ is Borel and bounded, $(\nabla\Phi)_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, and