THE TUMURA–CLUNIE THEOREM IN SEVERAL COMPLEX VARIABLES

2014 ◽  
Vol 90 (3) ◽  
pp. 444-456 ◽  
Author(s):  
PEI-CHU HU ◽  
CHUNG-CHUN YANG
Keyword(s):  
Meromorphic Function ◽  
Complex Variable ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Several Variables

AbstractIt is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

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.

scholarly journals Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables

2021 ◽  
Vol 6 (11) ◽  
pp. 11796-11814
Author(s):  
Hong Li ◽  
◽  
Keyu Zhang ◽  
Hongyan Xu ◽  
◽  
...  
Keyword(s):  
Finite Order ◽  
Difference Equations ◽  
Complex Variable ◽  
Nevanlinna Theory ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Systems Of Equations ◽  
Entire Solutions ◽  
Partial Differential ◽  
Single Complex

<abstract><p>By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate some properties of the transcendental entire solutions for several systems of partial differential difference equations of Fermat type, and obtain some results about the existence and the forms of transcendental entire solutions of the above systems, which improve and generalize the previous results given by Cao, Gao, Liu <sup>[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b24">24</xref>,<xref ref-type="bibr" rid="b39">39</xref>]</sup>. Some examples are given show that there exist some significant differences in the forms of transcendental entire solutions with finite order of the systems of equations with between several complex variables and a single complex variable.</p></abstract>

scholarly journals Theq-Difference Theorems for Meromorphic Functions of Several Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zhi-Tao Wen
Keyword(s):  
Meromorphic Function ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Zero Order ◽  
Logarithmic Density ◽  
Several Variables

We investigateq-shift analogue of the lemma on logarithmic derivative of several variables. Letfbe a meromorphic function inℂnof zero order such thatf(0)≠0,∞, and letq∈ℂn\{0}. Then we havem(r,f(qz)/f(z))=o(T(r,f))on a set of logarithmic density 1. Theq-shift analogue of the first and the second main theorems of Nevanlinna theory of several variables and their applications is also shown.

Remark on Irreducible Factorizations of Polynomials in Several Variables*

1981 ◽  
Vol 24 (4) ◽  
pp. 493-496
Author(s):  
Leiba Rodman
Keyword(s):  
Complex Variables ◽  
Several Complex Variables ◽  
Analytic Set ◽  
Several Variables ◽  
Set Of Points

AbstractIt is proved that a polynomial on several complex variables, whose coefficients depend analytically on a parameter ε, admits a factorization which is irreducible for every value of the parameter, with the possible exception of an analytic set of points. Moreover, the coefficients of the irreducible factors can be chosen to depend analytically on ε in a neighborhood of every point not belonging to this analytic set.

scholarly journals Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1707
Author(s):  
Renata Długosz ◽  
Piotr Liczberski
Keyword(s):  
Power Series ◽  
Power Series Expansion ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Homogeneous Polynomials ◽  
Positive Real ◽  
Sharp Estimates ◽  
Several Variables

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families.

scholarly journals The star function for meromorphic functions of several complex variables

2017 ◽  
Vol 64 (1) ◽  
pp. 26-39
Author(s):  
Faruk Abi-Khuzam ◽  
Florian Bertrand ◽  
Giuseppe Della Sala
Keyword(s):  
Meromorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables
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