Nevanlinna Theory of Meromorphic Functions

2014 ◽  
pp. 1-24
Author(s):  
Junjiro Noguchi ◽  
Jörg Winkelmann
Keyword(s):  
Nevanlinna Theory ◽  
Meromorphic Functions

scholarly journals Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hong Yan Xu ◽  
Da Wei Meng ◽  
Sanyang Liu ◽  
Hua Wang
Keyword(s):  
Finite Order ◽  
Difference Equations ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Complex Variables ◽  
Second Order ◽  
Several Complex Variables ◽  
Entire Solutions ◽  
Partial Differential

AbstractThis paper is concerned with description of the existence and the forms of entire solutions of several second-order partial differential-difference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.

Nevanlinna theory of meromorphic functions on annuli

2010 ◽  
Vol 53 (3) ◽  
pp. 547-554 ◽  
Author(s):  
Mark Lund ◽  
Zhuan Ye
Keyword(s):  
Nevanlinna Theory ◽  
Meromorphic Functions

Some results for complex partial q-difference equations in ℂ n \mathbb{C}^{n}

2019 ◽  
Vol 26 (3) ◽  
pp. 471-481
Author(s):  
Yue Wang
Keyword(s):  
Difference Equations ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Zero Order ◽  
Value Distribution ◽  
Meromorphic Solutions ◽  
Difference Polynomials

Abstract Using the Nevanlinna theory of the value distribution of meromorphic functions, the value distribution of complex partial q-difference polynomials of meromorphic functions of zero order is investigated. The existence of meromorphic solutions of some types of systems of complex partial q-difference equations in {\mathbb{C}^{n}} is also investigated. Improvements and extensions of some results in the literature are presented. Some examples show that our results are, in a certain sense, the best possible.

scholarly journals URS AND URSIMS FOR P-ADIC MEROMORPHIC FUNCTIONS INSIDE A DISC

2001 ◽  
Vol 44 (3) ◽  
pp. 485-504 ◽  
Author(s):  
Abdelbaki Boutabaa ◽  
Alain Escassut
Keyword(s):  
Characteristic Function ◽  
Analytic Functions ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Closed Field ◽  
Open Disc ◽  
Absolute Value ◽  
Unique Range Set ◽  
Bounded Characteristic ◽  
Primary 12H25

AbstractLet $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

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