For a polynomial $V(x)$ of degree $r$ defined in $\mathbb{R}^d$, we denote
$$A_V(x)=\sum_{1\le |\alpha|\le r}|\partial^{\alpha}_xV(x)|^{\frac{1}{|\alpha|}}$$
I want to prove the following result: there exists a constant $c>0$ such that for all $x,x_0\in \mathbb{R}^d$
$$
|x-x_0|\le \Bigl(\frac{A_V(x)}{ \log(A_V(x))}\Bigr)^{-1}$$
implies
$$
\left(\frac{\dfrac{A_V(x)}{\log(A_V(x))}}{\dfrac{A_V(x_0)}{ \log(A_V(x_0))}}\right)^{\pm 1}\le c.
$$
Here we consider polynomials $V$ such that $A_V(x)\neq 1$ for all $x\in \mathbb{R}^d$.
Please help me to do so.
Thanks in advance