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

In the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $d$ be a natural number with $1\le d$, read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background (i) where a real number is required, and 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, with gradient $D\Phi(x)$ and Hessian matrix $D^{2}\Phi(x)\in\mathcal{S}(d)$. Let $\nabla\Phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ denote the gradient map $x\mapsto D\Phi(x)$, whose components $\partial_{i}\Phi$ are of class $C^{1}$ on $\mathbb{R}^{d}$ by clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ. For a map $G:\mathbb{R}^{d}\to\mathbb{R}^{d}$ whose components are of class $C^{1}$ on $\mathbb{R}^{d}$, $DG(y)$ denotes its Jacobian matrix at $y$, with entry $\partial_{j}G_{i}(y)$ in row $i$ and column $j$. Positive definiteness and the inverse matrix are those of those definitions. Let $\varepsilon$ and $L$ be positive real numbers with

$$\varepsilon I_{d}\preceq D^{2}\Phi(x)\preceq L\,I_{d}\qquad\text{for every }x\in\mathbb{R}^{d}.$$

1. (Two-sided Lipschitz bounds)(i) For all $x,y\in\mathbb{R}^{d}$,

$$\varepsilon\lVert x-y\rVert\le\lVert\nabla\Phi(x)-\nabla\Phi(y)\rVert\le d\,L\,\lVert x-y\rVert .$$

2. (Bijection)(ii) $\nabla\Phi$ is a bijection of $\mathbb{R}^{d}$ onto $\mathbb{R}^{d}$.

3. (The inverse)(iii) For every $x\in\mathbb{R}^{d}$ the matrix $D^{2}\Phi(x)$ is positive definite and $D(\nabla\Phi)(x)=D^{2}\Phi(x)$. The inverse map $G=(\nabla\Phi)^{-1}$ has components of class $C^{1}$ on $\mathbb{R}^{d}$, and for every $x\in\mathbb{R}^{d}$ the matrix $DG(\nabla\Phi(x))$ is the inverse matrix of $D^{2}\Phi(x)$.