Geodesic
LOCAL C r -RIGHT EQUIVALENCE OF C r+1 FUNCTIONS
Glasgow mathematical journal, Tome 59 (2017) no. 1, pp. 265-272

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Let f,g:(Rn , 0) → (R, 0) be C r+1 functions, r ∈ N. We will show that if ∇f(0)=0 and there exist a neighbourhood U of 0 ∈ Rn and a constant C > 0 such that $$\begin{equation*}\left|\partial^m(g-f)(x)\right| ≤ C \left|\nabla f(x)\right|^{r+2-|m|} \quad \textrm{ for } x\in U,\end{equation*}$$ and for any m ∈ N0 n such that |m| ≤ r, then there exists a C r diffeomorphism φ:(Rn , 0) → (Rn , 0) such that f = g ° φ in a neighbourhood of 0 ∈ Rn . 
MIGUS, PIOTR. LOCAL C r -RIGHT EQUIVALENCE OF C r+1 FUNCTIONS. Glasgow mathematical journal, Tome 59 (2017) no. 1, pp. 265-272. doi: 10.1017/S0017089516000161
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     title = {LOCAL {C} r {-RIGHT} {EQUIVALENCE} {OF} {C} r+1 {FUNCTIONS}},
     journal = {Glasgow mathematical journal},
     pages = {265--272},
     year = {2017},
     volume = {59},
     number = {1},
     doi = {10.1017/S0017089516000161},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000161/}
}
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