PreprintCC BY 4.0arXiv
Meromorphic functions together with their partial derivatives sharing zeros and poles in ℂm
Abstract
In the paper, using Nevanlinna's value distribution theory of meromorphic functions in ℂm, we study the problem: Let f and g be two finite order non-constant meromorphic functions in ℂm such that f and g share $0$ and ∞ CM. If ∂zi(f) and ∂zi(g) share $0$ CM and if for one n>1, ∂nzi(f) and ∂nzi(g) share $0$ CM, where i=1,2,…,m, then f and g have special presentations. Our finding extends the result of Köhler (Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros, Complex Var., 11 (1989), 39-48) into higher dimensions. Moreover, we show by an example that the restriction on the growth of f and g is necessary.
Related
- On a class of meromorphic functions with three singular values
- Meromorphic Solutions of Certain Kind of Non-Linear Differential equation with finite sums of Exponential Functions
- Geometric Characteristics of Meromorphic Starlike Functions Linked with a Novel Image Domain
- Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc
- Partial Derivatives of J and Hopf Problem
BibTeX
@misc{xu2026meromorphic,
title = {{Meromorphic functions together with their partial derivatives sharing zeros and poles in $\mathbbC^m$}},
author = {Hong Yan Xu and Sujoy Majumder and Goutam Haldar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02682},
eprint = {2611.02682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02682},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Meromorphic functions together with their partial derivatives sharing zeros and poles in ℂ^m
AU - Hong Yan Xu
AU - Sujoy Majumder
AU - Goutam Haldar
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02682
UR - https://arxiv.org/abs/2611.02682
AB - In the paper, using Nevanlinna's value distribution theory of meromorphic functions in ℂ^m, we study the problem: Let f and g be two finite order non-constant meromorphic functions in ℂ^m such that f and g share $0$ and ∞ CM. If ∂_z_i(f) and ∂_z_i(g) share $0$ CM and if for one n>1, ∂^n_z_i(f) and ∂^n_z_i(g) share $0$ CM, where i=1,2,…,m, then f and g have special presentations. Our finding extends the result of Köhler (Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros, Complex Var., 11 (1989), 39-48) into higher dimensions. Moreover, we show by an example that the restriction on the growth of f and g is necessary.
ER -