The Mean Value Theorem (inequality) for complex-valued functions of several real variables

Question

I am having trouble finding any good reference to the Mean Value Theorem for functions $f:\mathbb{R}^n \to \mathbb{C}$. By Mean Value Theorem here I actually mean the inequality, that is, I want something like $$ |f(b)-f(a)| \leq C|b-a|, $$ where $C$ is some supremum over the first order partial derivatives of $f$.

I need the reference to be a book, or at least an article since I would like to refer to it in a paper.

Answer 1

By identifying ${\bf{C}}$ as ${\bf{R}}^{2}$, we let $u=f(x)-f(a)$, then we have something like the Mean Value Theorem that \begin{align*} u\cdot(f(x)-f(a))=u\cdot Df(c)(x-a), \end{align*} where the $\cdot$ here is the usual dot product in ${\bf{R}}^{2}$ but not the complex multiplication, then \begin{align*} \|f(x)-f(a)\|^{2}=(f(x)-f(a))\cdot Df(c)(x-a)\leq\|f(x)-f(a)\|\|Df(c)\|\|x-a\|, \end{align*} so \begin{align*} \|f(x)-f(a)\|\leq\|Df(c)\|\|x-a\|. \end{align*}

Answer 2

Take the mean value theorem for functions $f:\>{\mathbb R}^n\to{\mathbb R}^m$ as found in any textbook, and let $m=2$. (Since you explicitly want a citation I don't quote the theorem here.)