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Volume 30, Issue 1
Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function

A. Escassut, K. Boussaf & J. Ojeda

Anal. Theory Appl., 30 (2014), pp. 51-81.

Published online: 2014-03

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  • Abstract

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  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.

  • AMS Subject Headings

12J25, 30D35, 30G06

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COPYRIGHT: © Global Science Press

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@Article{ATA-30-51, author = {}, 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 $\mathbb{K}$ 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  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} }
TY - JOUR T1 - Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function 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 -

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  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.

A. Escassut, K. Boussaf & J. Ojeda. (1970). Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function. Analysis in Theory and Applications. 30 (1). 51-81. doi:10.4208/ata.2014.v30.n1.4
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