@Article{ATA-30-51,
author = {A. Escassut and K. Boussaf and J. Ojeda},
title = {Complex and p-Adic Meromorphic Functions f'P'(f), g'P'(g) Sharing a Small Function},
journal = {Analysis in Theory and Applications},
year = {2014},
volume = {30},
number = {1},
pages = {51--81},
abstract = { Let $\KK$ be a complete algebraically closed $p$-adic field of characteristic zero. We apply results in algebraic geometry and a new Nevanlinna theorem for $p$-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on $\KK$ and on $\CC$. Let $P$ be a polynomial of uniqueness for meromorphic functions in $\KK$ or $\CC$ or in an open disk. Let $f$, $g$ be two transcendental meromorphic functions in the whole field $\KK$ or in $\CC$ or meromorphic functions in an open disk of $\KK$ that are not quotients of bounded analytic functions. We show that if $f'P'(f)$ and $g'P'(g)$ share a small function $\alpha$ counting multiplicity, then $f=g$, provided that the multiplicity order of zeros of $P'$ satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities $n\geq k+2$ or $n\geq k+3$ used in previous papers by Hypothesis (G). In the $p$-adic context, another consists of giving a lower bound for a sum of $q$ counting functions of zeros with $(q-1)$ times the characteristic function of the considered meromorphic function.},
issn = {1573-8175},
doi = {https://doi.org/10.4208/ata.2014.v30.n1.4},
url = {http://global-sci.org/intro/article_detail/ata/4473.html}
}