Let $y=f(x)$ be a real-analytic function defined on $ U\subset \mathbb{R}^n$, and $B(0.r)\subset U$, where $r>0$ is fixed, and
$$ B(0,r)=\{x\in \mathbb{R}^n: |x|<r\}. $$
Assume $f(0)=0$, we can write
$$ f(x)=\sum_{\alpha\in \mathbb{N}^n}a_\alpha x^\alpha. $$
I want to know that, if $|f|\leq M$ on $B(0,r)$, then the upper bounds
$$ |a_\alpha|\leq \frac{M}{r^{|\alpha|}} $$
holds for all multi-index $\alpha$ ?
Maybe $f(x)=\sin x$ is a counterexample? In this case $f'(0)=1$ but for any $r\in (0,1)$, $|\sin r|/r<1$.
(How to post this question make sense?)
(I only know in the case of several complex variables it is true and is known as Cauchy's inequality.)
This question is from Lemma 6.1.1 in Krantz's book of the implicit function theorem, Page 118. See https://link.springer.com/book/10.1007/978-1-4614-5981-1
I do not know if it is conflict.
