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
and 
defined in a region 
, it is said that two functions 
is equivalent to 
, and is denoted by 
if 
. 
and 
be meromorphic functions defined in a region, and let 
be a complex number or small function for 
and 
. We denote the set of points that 
by 
( 
) counting (ignoring) multiplicities. 
( 
) means that 
and 
share 
considering multiplicities (or ignoring multiplicities). 
( 
) be an annulus in complex plane and 
be a interger value function whose support is discrete subset of 
, and 
are meromorphic functions on the annlus. Then for a positive interger, we define the counting function of 
as follows; 
. 
if 
. In particular, we call 
the counting function of 
when 
is number of zeros of 
in 
. And we call 
, and call 
, where 
. 
that is nonconstant meromorphic on 
, it is said that is transcendental on 
provide with 
. 
on 
, we have that 
. 
is said that it is small for the meromorphic function 
on 
, which is defined as follows: 
, except for set 
such that 
; 
, except for set 
such that 
. 
) of 
and 
on 
by 
. 
and 
share IM if 
, or if 
. 
instead of notions 
and so on. 
and 
in the complex plane 
, can’t have the same inverse images of five different values in 
. This is just the first result for uniqueness of meromorphic functions. 
( 
) 
and 
be two functions that are transcendental meromorphic on an annulus 
( 
), and 
be five different complex values of 
. 
for 
, then it implies that 
. 
as follows: 
and 
be two functions that are transcendental meromorphic on an annulus 
( 
), and let 
be five different small functions for 
and 
. 
for 
, then 
. 
and 
be two functions that are transcendental meromorphic on an annulus 
( 
), and let 
be five different small functions for 
and 
. 
for 
and 
, where 
is the counting function of the common zeros of 
and 
(ignoring multiplicities) on 
and 
is reduced one of 
. 
and 
be two functions that are transcendental meromorphic on an annulus 
( 
), and let 
be five different small functions for 
and 
. 
for 
and 
is the counting function of the different zeros of 
and 
(ignoring multiplicities) on 
and 
is reduced one of 
. 
and 
be two functions that are transcendental meromorphic on an annulus 
( 
), and let 
be five different small functions for 
and 
. 
for 
and 
, 
. 
by 
yields the same result. 
and 
be two functions that are transcendental meromorphic on an annulus 
( 
), and let 
be five different small functions for 
and 
. 
for 
and 
. 
be a nonconstant function that is meromorphic on an annulus 
( 
). Then 
. 
be a nonconstant function that is meromorphic on an annulus 
( 
) and 
. Then, for 
, 
. 
be a nonconstant function that is meromorphic on an annulus 
( 
), and 
be different values of 
. Then 
. 
and 
be two nonconstant functions that is meromorphic on an annulus 
( 
) and share 
IM. Then we have 
. 
, we get 
with 
we obtain 
. 
( 
), it implies that 
. 
. 
and 
be transcendental meromorphic functions on annulus 
( 
) and let 
be five different small functions for 
and 
. And let 
and 
share 
IM. 
, from Lemma 2.4 we have 
, which is contradiction to the supposition that 
is a transcendental meromorphic function. Thus we get 
. Therefore, from Theorem C, we obtain 
. (End). 
only come from of zeros of 
and poles of 
. We distinguish the following distinct cases. 
be a zero of 
and not a zero of 
. 
. 
be a 
-order zero of 
and 
-order zero of 
. 
. 
be a pole of 
and not a zero of 
. 
. 
is not greater than 
. 
. 
, we have 
. (End). | [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
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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
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