Multivariate Taylor Expansion with Uniform Second-Order Remainder

Let $n\ge1$ be a natural number, let $W\subseteq\mathbb{R}^n$ be an open subset of Euclidean space, and let $f:W\to\mathbb{R}$ be a $C^1$ map; write $\partial_i f$ for the partial derivative with respect to the $i$-th coordinate, and $\partial_j\partial_i f$ for $\partial_j$ applied to $\partial_i f$.

Let $x,y\in W$ be points whose segment $\{x+\tau(y-x):\tau\in[0,1]\}$ is contained in $W$, and write $h=y-x$ with components $h_1,\dots,h_n$ and $|h|$ for the Euclidean distance between $x$ and $y$.

(i) (Lipschitz bound.) If $M_1$ is a nonnegative real number with $|\partial_i f(z)|\le M_1$ for every point $z$ of the segment and every $i\in\{1,\dots,n\}$, then $\big|f(y)-f(x)\big|\le \sqrt{n}\,M_1\,|h|.$

For parts (ii) and (iii), assume in addition that each $\partial_i f$ ($i\in\{1,\dots,n\}$) is again a $C^1$ map on $W$.

(ii) (First-order remainder.) If $M_2$ is a nonnegative real number with $|\partial_j\partial_i f(z)|\le M_2$ for every point $z$ of the segment and all $i,j\in\{1,\dots,n\}$, then $\Big|f(y)-f(x)-\sum_{i=1}^n\partial_i f(x)\,h_i\Big|\le \tfrac{1}{2}\,n\,M_2\,|h|^2.$

(iii) (Second-order remainder.) If $\bar{\varepsilon}$ is a nonnegative real number with $|\partial_j\partial_i f(z)-\partial_j\partial_i f(x)|\le\bar{\varepsilon}$ for every point $z$ of the segment and all $i,j\in\{1,\dots,n\}$, then $\Big|f(y)-f(x)-\sum_{i=1}^n\partial_i f(x)\,h_i-\tfrac{1}{2}\sum_{i=1}^n\sum_{j=1}^n\partial_j\partial_i f(x)\,h_i h_j\Big|\le \tfrac{1}{2}\,n\,\bar{\varepsilon}\,|h|^2.$