P-adic meromorphic functions f'P'(f), g'P'(g) sharing a small function
Résumé
Let $\KK$ be a complete algebraically closed p-adic field of characteristic zero. Let $f,\ g$ be two transcendental meromorphic functions in the whole field $\KK$ or meromorphic functions in an open disk that are not quotients of bounded analytic functions. Let $P$ be a polynomial of uniqueness for meromorphic functions in $\KK$ or in an open disk and let $\alpha$ be a small meromorphic function with regards to $f$ and $g$. If $f'P'(f)$ and $g'P'(g)$ share $\alpha$ counting multiplicity, then we show that $f=g$ provided that the multiplicity order of zeroes of $P'$ satisfy certain inequalities. If $\alpha$ is a Moebius function or a non-zero constant, we can obtain more general results on $P$.