Degeneracy of holomorphic mappings into or avoiding Fermat type hypersurfaces

We prove that if $f\text{\hspace{0.17em}⁣}:{\mathbb{C}}^p\to {\mathbb{P}}^n(\mathbb{C})$ is a holomorphic mapping of maximal rank whose image lies in the Fermat hypersurface of degree $d>(n+1)\max \{n-p,1\}$, then its image is contained in a linear subspace of dimension at most $\left[{\displaystyle \frac{n-1}{2}}\right]$. Analog in the logarithmic case is also given. Our result strengthens a classical result of Green and provides a Nevanlinna theoretic proof for a recent result due to Etesse.