MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS

doi:10.1007/s10473-025-0415-0

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1514-1528.doi: 10.1007/s10473-025-0415-0

MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS

Xiaotang NIE¹, Jiaxing HUANG^1,*, Yuefei WANG^1,2,*, Chengfa WU^1,3

1. School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, China;
2. Institute of Mathematics, Chinese Academia Sinica, Beijing 100080, China;
3. Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China

收稿日期:2024-05-21 修回日期:2024-09-19 出版日期:2025-10-10 发布日期:2025-10-10

MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS

Xiaotang NIE¹, Jiaxing HUANG^1,*, Yuefei WANG^1,2,*, Chengfa WU^1,3

1. School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, China;
2. Institute of Mathematics, Chinese Academia Sinica, Beijing 100080, China;
3. Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China

Received:2024-05-21 Revised:2024-09-19 Online:2025-10-10 Published:2025-10-10
Contact: ^*Jiaxing HUANG, E-mail: hjxmath@szu.edu.cn; Yuefei WANG, E-mail: wangyf@math.ac.cn
About author:Xiaotang NIE, E-mail: nxt0126@163.com; Chengfa WU, E-mail:cfwu@szu.edu.cn
Supported by:
National Natural Science Foundation of China (12201420, 12231013, 11701382 and 11971288) and the Guangdong Basic and Applied Basic Research Foundation, China (2021A1515010054) and NCAMS.

摘要/Abstract

摘要： In this paper, we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients $$f(z+1)-f(z-1)+a(z)S(f,z)=\frac{P(z, f(z))}{Q(z, f(z))}.$$ We obtain the necessary conditions on the degree of $R(z,f)$ for these delay differential equations and give a classification of the delay Schwarzian differential equations according to the multiplicities of the root of $Q(z, f)$ on $f$. Finally, we provide some examples to illustrate that all cases occur.

关键词: meromorphic functions, delay differential equations, Nevanlinna theory, Schwarzian derivatives, minimal hyper-type

Abstract: In this paper, we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients $$f(z+1)-f(z-1)+a(z)S(f,z)=\frac{P(z, f(z))}{Q(z, f(z))}.$$ We obtain the necessary conditions on the degree of $R(z,f)$ for these delay differential equations and give a classification of the delay Schwarzian differential equations according to the multiplicities of the root of $Q(z, f)$ on $f$. Finally, we provide some examples to illustrate that all cases occur.

Key words: meromorphic functions, delay differential equations, Nevanlinna theory, Schwarzian derivatives, minimal hyper-type

Xiaotang NIE, Jiaxing HUANG, Yuefei WANG, Chengfa WU. MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2025, 45(4): 1514-1528.

Xiaotang NIE, Jiaxing HUANG, Yuefei WANG, Chengfa WU. MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1514-1528.

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MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS

MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS

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