The value distribution theory was introduced by R. Nevanlinna who was the famous Finnish mathematician, Since then, the value distribution theory has not only led to a new field of mathematics, but has also been applied in various fields of mathematics and has made many advances. The value distribution theory of Nevanlinna has played an important role in the study of the growth characteristics of functions, uniqueness, and type of functions. The uniqueness of complex meromorohic functions is a new and original version of the uniqueness of holomorphic functions, which is the core part of the theory of value distributions. Therefore uniqueness of complex meromorphic functions is an outstanding problem in Nevanlinna value distribution theory. There is a lot of research on the uniqueness of two meromorphic functions sharing four values on annuli. In this paper, we have showed uniqueness of functions that are meromorphic on an annulus. For detail, we show uniqueness of two functions that are transcendental meromorphic on an annulus, share for small different functions and satisfy additional condition for characteristic functions, which is an improvement and extension of the results obtained by N. Wu, Q. Ge in 2015 and by D. W. Meng, S. Y. Liu and N. Lu in 2020.
| Published in | Science Discovery Mathematics (Volume 1, Issue 1) |
| DOI | 10.11648/j.sdmath.20260101.15 |
| Page(s) | 43-47 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2026. Published by Science Publishing Group |
Uniqueness, Meromorphic Function, Sharing Value, Small Function
| [1] | T. B. Cao, H. X. Yi, Uniqueness theorems of meromorphic functions sharing sets IM on annuli, Acta Math. Sinica, 54 (2011), 623–632. |
| [2] | T. B. Cao, H. X. Yi and H. Y. Xu, On the multiple values and uniqueness of meromorphic functions on annuli, Comput. Math. Appl., 58 (2009), 1457–1465 |
| [3] | A. Y. Khrystiyanyn, A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. I, Mat. Stud., 23 (2005), 19-30. |
| [4] | A. Y. Khrystiyanyn, A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. II, Mat. Stud., 24 (2005), 57-68. |
| [5] | A. A. Kondratyuk, I. Laine, Meromorphic functions in multiply connected domains, Fourier series methods in complex analysis, Univ. Joensuu, 2006. |
| [6] | R. Korhonen, Nevanlinna theory in an annulus, value distribution theory and related topics, Adv. Complex Anal. Appl., 3 (2004), 167-179. |
| [7] | H. F. Liu, Z. Q. Mao, Meromorphic functions in the unit disc that share slowly growing functions in an angular domain, Comput. Math. Appl., 62 (2011), 4539-4546 |
| [8] | D. W. Meng, S. Y. Liu and N. Lu, On the uniqueness of meromorphic functions that share small functions on annuli, AIMS Mathematics, 5(4), 3223-3230. |
| [9] | R. Nevanlinna, Eindentig keitss¨atze in der theorie der meromorphen funktionen, Acta. Math., 48(1926), 367-391. |
| [10] | N. Wu, Q. Ge, On uniqueness of meromorphic functions sharing five small functions on annuli, Bull. Iranian Math. Soc., 41 (2015), 713-722. |
| [11] | H. Y. Xu, Z. J. Wu, The shared set and uniqueness of meromorphic functions on annuli, Abstr. Appl. Anal., 2013 (2013), 1-10. |
| [12] | H. Y. Xu, Z. X. Xuan, The uniqueness of analytic functions on annuli sharing some values, Abstr. Appl. Anal., 2012 (2012), 309-323. |
| [13] | J. H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad J. Math., 47 (2004), 152-160. |
| [14] | Si, D. Q., Unicity of meromorphic functions sharing some small function. Int. J. Math. 23(9) (2012) |
| [15] | Si D. Q, Tran A. H., Nguyen T. T. H., Ha H. G., Meromorphic functions having the same inverse images of four values on annuli, Bull. Iran. Math. Soc. 44 (2018), 19-41 |
APA Style
Pak, D. Y., Choe, P. (2026). Uniqueness of Meromorphic Functions for Four Small Functions on Annuli. Science Discovery Mathematics, 1(1), 43-47. https://doi.org/10.11648/j.sdmath.20260101.15
ACS Style
Pak, D. Y.; Choe, P. Uniqueness of Meromorphic Functions for Four Small Functions on Annuli. Sci. Discov. Math. 2026, 1(1), 43-47. doi: 10.11648/j.sdmath.20260101.15
AMA Style
Pak DY, Choe P. Uniqueness of Meromorphic Functions for Four Small Functions on Annuli. Sci Discov Math. 2026;1(1):43-47. doi: 10.11648/j.sdmath.20260101.15
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.sdmath.20260101.15,
author = {Du Yong Pak and Pyongil Choe},
title = {Uniqueness of Meromorphic Functions for Four Small Functions on Annuli},
journal = {Science Discovery Mathematics},
volume = {1},
number = {1},
pages = {43-47},
doi = {10.11648/j.sdmath.20260101.15},
url = {https://doi.org/10.11648/j.sdmath.20260101.15},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sdmath.20260101.15},
abstract = {The value distribution theory was introduced by R. Nevanlinna who was the famous Finnish mathematician, Since then, the value distribution theory has not only led to a new field of mathematics, but has also been applied in various fields of mathematics and has made many advances. The value distribution theory of Nevanlinna has played an important role in the study of the growth characteristics of functions, uniqueness, and type of functions. The uniqueness of complex meromorohic functions is a new and original version of the uniqueness of holomorphic functions, which is the core part of the theory of value distributions. Therefore uniqueness of complex meromorphic functions is an outstanding problem in Nevanlinna value distribution theory. There is a lot of research on the uniqueness of two meromorphic functions sharing four values on annuli. In this paper, we have showed uniqueness of functions that are meromorphic on an annulus. For detail, we show uniqueness of two functions that are transcendental meromorphic on an annulus, share for small different functions and satisfy additional condition for characteristic functions, which is an improvement and extension of the results obtained by N. Wu, Q. Ge in 2015 and by D. W. Meng, S. Y. Liu and N. Lu in 2020.},
year = {2026}
}
TY - JOUR T1 - Uniqueness of Meromorphic Functions for Four Small Functions on Annuli AU - Du Yong Pak AU - Pyongil Choe Y1 - 2026/03/26 PY - 2026 N1 - https://doi.org/10.11648/j.sdmath.20260101.15 DO - 10.11648/j.sdmath.20260101.15 T2 - Science Discovery Mathematics JF - Science Discovery Mathematics JO - Science Discovery Mathematics SP - 43 EP - 47 PB - Science Publishing Group UR - https://doi.org/10.11648/j.sdmath.20260101.15 AB - The value distribution theory was introduced by R. Nevanlinna who was the famous Finnish mathematician, Since then, the value distribution theory has not only led to a new field of mathematics, but has also been applied in various fields of mathematics and has made many advances. The value distribution theory of Nevanlinna has played an important role in the study of the growth characteristics of functions, uniqueness, and type of functions. The uniqueness of complex meromorohic functions is a new and original version of the uniqueness of holomorphic functions, which is the core part of the theory of value distributions. Therefore uniqueness of complex meromorphic functions is an outstanding problem in Nevanlinna value distribution theory. There is a lot of research on the uniqueness of two meromorphic functions sharing four values on annuli. In this paper, we have showed uniqueness of functions that are meromorphic on an annulus. For detail, we show uniqueness of two functions that are transcendental meromorphic on an annulus, share for small different functions and satisfy additional condition for characteristic functions, which is an improvement and extension of the results obtained by N. Wu, Q. Ge in 2015 and by D. W. Meng, S. Y. Liu and N. Lu in 2020. VL - 1 IS - 1 ER -
Department of Mathematics, University of Science, Pyongyang, DPR Korea
Information