Bounds on the degree of APN polynomials The Case of $x^{-1}+g(x)$

We prove that functions $f:\text{f}{2}^m\to \text{f}{2}^m$ of the form $f(x)=x^{-1}+g(x)$ where $g$ is any non-affine polynomial are APN on at most a finite number of fields $\text{f}{2}^m$. Furthermore we prove that when the degree of $g$ is less then 7 such functions are APN only if $m\le 3$ where these functions are equivalent to $x^3$.