TY - JOUR
T1 - Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function
AU - A. Escassut, K. Boussaf & J. Ojeda
JO - Analysis in Theory and Applications
VL - 1
SP - 51
EP - 81
PY - 2014
DA - 2014/03
SN - 30
DO - http://doi.org/10.4208/ata.2014.v30.n1.4
UR - https://global-sci.org/intro/article_detail/ata/4473.html
KW - Meromorphic, nevanlinna, sharing value, unicity, distribution of values.
AB - <p style="text-align: justify;">Let $\mathbb{K}$ 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 $\mathbb{K}$ and on $\mathbb{C}$. Let $P$ be a polynomial of uniqueness for meromorphic functions in $\mathbb{K}$ or $\mathbb{C}$ or in an open disk. Let $f$, $g$ be two transcendental meromorphic functions in the whole field $\mathbb{K}$ or in $\mathbb{C}$ or meromorphic functions in an open disk of $\mathbb{K}$ that are not quotients of bounded&nbsp; analytic functions. We show that if $f&#39;P&#39;(f)$ and $g&#39;P&#39;(g)$ share a small function $\alpha$ counting multiplicity, then $f=g$, provided that the multiplicity order of zeros of $P&#39;$ 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.</p>