Suppose $f$ is a continuous function with bounded derivatives (if need be, we can also assume $f$ is holomorphic). Can I get bounds for the norm of $n$th derivative of $f$? $f$ is a function of several variables to several variables.
I know if $f$ is from single variable to single variable and is holomorphic, I can use Cauchy's Estimates to bound the nth derivative as:
Suppose $f$ is holomorphic on a neighborhood of the closed ball ${B}(z^*, R)$, and suppose that \begin{align} m := \max \{ |f(z)| : |z - z^*| = r \}, \quad (m < \infty). \end{align} Then \begin{align} \left|f^{(n)}(z^*) \right| \leq \frac{n! \, m}{r^n}. \end{align}
I am looking for a similar result if $f$ is from several variables to several variables.
Edit 1: I should add more details when I say bound on higher-order derivatives. For instance, $f^{(2)}(\mathbf{x})$ would be multiple hessian-like matrices, one for each component of $f$. When I say I want $||f^{(2)}(\mathbf{x})|| \leq M$, I want $M$ that is the supremum of all the upper bounds to the hessian-like matrices.
