SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES*

doi:10.1007/s10473-023-0218-0

Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (2): 821-838.doi: 10.1007/s10473-023-0218-0

Previous Articles Next Articles

SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*

Manli Liu¹, Lingyun Gao^2,†, Shaomei Fang¹

1. Department of Mathematics, South China Agricultural University, Guangzhou 510642, China;
2. Department of Mathematics, Jinan University, Guangzhou 510632, China

Received:2021-10-06 Revised:2022-03-01 Online:2023-03-25 Published:2023-04-12
Contact: †Lingyun GAO, E-mail: tgaoly@jnu.edu.cn.
About author:Manli Liu, E-mail: lml6641@163.com; Shaomei Fang, E-mail: dz90@scau.edu.cn
Supported by:
This work was partially supported by the NSFC (11271227, 11271161), the PCSIRT (IRT1264) and the Fundamental Research Funds of Shandong University (2017JC019).

Abstract

Abstract: In this paper, we mainly investigate the value distribution of meromorphic functions in $\mathbb{C}^m$ with its partial differential and uniqueness problem on meromorphic functions in $\mathbb{C}^m$ and with its $k$-th total derivative sharing small functions. As an application of the value distribution result, we study the defect relation of a nonconstant solution to the partial differential equation. In particular, we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.

Key words: meromorphic function in several variables, Nevanlinna theory, partial differential equation, total derivative

Manli Liu, Lingyun Gao, Shaomei Fang. SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*[J].Acta mathematica scientia,Series B, 2023, 43(2): 821-838.

Trendmd

References

[1] Brück R.On entire functions which share one value CM with their derivates. Result in Math, 1996, 30: 21-24
[2] Gundersen G G, Yang L Z.Entire functions that share one value with one or two of their derivates. J Math Anal Appl, 1998, 223: 245-260
[3] Hayman W K. Meromorphic Functions.Oxford: Clarendon Press, 1964
[4] Hu P C, Li P, Yang C C.Unicity of Meromorphic Mappings. Berlin: Springer Science and Bussiness Media, 2013
[5] Jin L.Theorems of Picard type for entire functions of several complex variables. Kodai Math J, 2003, 26: 221-229
[6] Jin L.A unicity theorem for entire functions of several complex variables. Chin Ann Math, 2004, 25B: 483-492
[7] Li B Q.On Picard's theorem. J Math Anal Appl, 2018, 460: 561-564
[8] Li B Q, Yang L. On Picard type theorems and entire solutions of differential equations. arXiv:1809.05553v1
[9] Lü F.Theorems of Picard type for meromorphic function of several complex variables. Complex Variables and Elliptic Equations, 2013, 58: 1085-1092
[10] Ru M.Nevanlinna Theory and Its Relation to Diophantine Approxmation. Singapore: World Sci Publishing, 2001
[11] Vitter A.The lemma of the logarithmic derivative in several complex variables. Duke Math J, 1977, 44: 89-104
[12] Yi H X.A question of C.C. Yang on the uniqueness of entire functions. Kodai Math J, 1990, 13: 39-46
[13] Yi H X.Uniqueness theorems for meromorphic functions whose N-th derivatives share the same 1-points. Complex Variables and Elliptic Equations, 1997, 34: 421-436
[14] Yang L Z.Further results on entire functions that share one value with their derivates. J Math Anal Appl, 1997, 212: 529-536
[15] Ye Z.A sharp form of Nevanlinna's second main theorem of several complex variables. Math Z, 1996, 222: 81-95
[16] Yu K W. On entire and meromorphic functions that share small functions with thire derivatives. J Inequal Pure Appl Math, 2003, 4(1): Art 21

SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Temur JANGVELADZE, Zurab KIGURADZE, Mikheil GAGOSHIDZE. ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM [J]. Acta mathematica scientia,Series B, 2019, 39(4): 971-988.
[2]	Raluca M BALAN, Lluís QUER-SARDANYONS, Jian SONG. HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE [J]. Acta mathematica scientia,Series B, 2019, 39(3): 717-730.
[3]	Lamine SYLLA. REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATION WITH JUMPS AND VISCOSITY SOLUTION OF SECOND ORDER INTEGRO-DIFFERENTIAL EQUATION WITHOUT MONOTONICITY CONDITION: CASE WITH THE MEASURE OF LÉVY INFINITE [J]. Acta mathematica scientia,Series B, 2019, 39(3): 819-844.
[4]	Jie XIONG, Xu YANG. UNIQUENESS PROBLEM FOR SPDES FROM POPULATION MODELS [J]. Acta mathematica scientia,Series B, 2019, 39(3): 845-856.
[5]	Jinniao QIU, Wenning WEI. UNIQUENESS OF VISCOSITY SOLUTIONS OF STOCHASTIC HAMILTON-JACOBI EQUATIONS [J]. Acta mathematica scientia,Series B, 2019, 39(3): 857-873.
[6]	Yaozhong HU. SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS [J]. Acta mathematica scientia,Series B, 2019, 39(3): 874-914.
[7]	Huifang LIU, Zhiqiang MAO. ON ENTIRE SOLUTIONS OF SOME TYPE OF NONLINEAR DIFFERENCE EQUATIONS [J]. Acta mathematica scientia,Series B, 2018, 38(3): 819-828.
[8]	Junfeng LIU, Ciprian A. TUDOR. STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY [J]. Acta mathematica scientia,Series B, 2017, 37(6): 1545-1566.
[9]	Nguyen Van THIN. NORMAL FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE [J]. Acta mathematica scientia,Series B, 2017, 37(3): 623-656.
[10]	Yang TAN. SEVERAL UNIQUENESS THEOREMS OF ALGEBROID FUNCTIONS ON ANNULI [J]. Acta mathematica scientia,Series B, 2016, 36(1): 295-316.
[11]	Hua LIU. ANALYTIC BOUNDARY VALUE PROBLEMS ON CLASSICAL DOMAINS [J]. Acta mathematica scientia,Series B, 2015, 35(5): 1037-1045.
[12]	LI Hai Chou,GAO Ling Yun. MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS [J]. Acta mathematica scientia,Series B, 2015, 35(1): 195-206.
[13]	LI Xiao-Min, YI Hong-Xun, HU Hai-Yan. UNIQUENESS RESULTS OF MEROMORPHIC FUNCTIONS WHOSE DERIVATIVES SHARE FOUR SMALL FUNCTIONS [J]. Acta mathematica scientia,Series B, 2012, 32(4): 1593-1606.
[14]	LI Sheng, GAO Zong-Sheng. RESULTS ON A QUESTION OF ZHANG AND YANG [J]. Acta mathematica scientia,Series B, 2012, 32(2): 717-723.
[15]	Yanni Zeng. LARGE TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR VISCOELASTIC MODEL WITH FADING MEMORY [J]. Acta mathematica scientia,Series B, 2012, 32(1): 219-236.