Local $C^r$-right equivalence of $C^{r+1}$ functions

Abstract

Let $f,g:(\mathbb{R}^n,0)\rightarrow (\mathbb{R},0)$ be $C^{r+1}$ functions, $r\in \mathbb{N}$. We will show that if $\nabla f(0)=0$ and there exist a neigbourhood $U$ of $0\in \mathbb{R}^n$ and a constant $C>0$ such that $$ \left|\partial^m(g-f)(x)\right|\leq C \left|\nabla f(x)\right|^{r+2-|m|}, \quad x\in U, $$ for any $m\in \mathbb{N}_0^n$ such that $|m|\leq r$, then there exists a $C^r$ diffeomorphism $\varphi:(\mathbb{R}^n,0)\rightarrow (\mathbb{R}^n,0)$ such that $f=g\circ \varphi$ in a neighbourhood of $0$.