Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set''
Saved in:
| Date: | 2025 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2025
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8063 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512894437818368 |
|---|---|
| author | Chakraborty, B. Chakraborty, B. Chakraborty, B. |
| author_facet | Chakraborty, B. Chakraborty, B. Chakraborty, B. |
| author_sort | Chakraborty, B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-04-16T12:01:19Z |
| doi_str_mv | 10.3842/umzh.v76i9.8063 |
| first_indexed | 2026-03-24T03:36:02Z |
| format | Article |
| fulltext |
DOI: 10.3842/umzh.v76i9.8063
CORRECTION TO THE B. CHAKRABORTY’S ARTICLE
“ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET”
[Ukr. Mat. Zh., 72, № 11, 1553 – 1563 (2020)]
ВИПРАВЛЕННЯ ДО СТАТТI Б. ЧАКРАБОРТI
„ПРО КАРДИНАЛЬНIСТЬ ЗВЕДЕНОЇ УНIКАЛЬНОЇ ДIАПАЗОННОЇ
МНОЖИНИ” [Укр. мат. журн., 72, № 11, 1553 – 1563 (2020)]
Let M(\BbbC ) denote the field of all meromorphic functions in \BbbC . We define M\ast (\BbbC ) = \{ f \in M(\BbbC ) :
N(r,\infty ; f | = 1) = S(r, f)\} .
For a positive integers n(\geq 3) and a complex number c(\not = 0, 1), we shall denote by P (z) [1]
the following polynomial:
P (z) =
(n - 1)(n - 2)
2
zn - n(n - 2)zn - 1 +
n(n - 1)
2
zn - 2 - c. (1)
The statement of Theorem 2.1 on p. 1555 of [2] should be the following.
Theorem 2.1. Let S = \{ z : P (z) = 0\} , where P (z) is the polynomial of degree n, defined in
(1). Let f, g \in M\ast (\BbbC ). If f and g share S IM and n \geq 15, and then f \equiv g.
In p. 1563, the calculations should be as follows:\Bigl( n
2
- 3
\Bigr)
(T (r, f) + T (r, g))
\leq 2
\bigl\{
N(r,\infty ; f) +N(r,\infty ; g)
\bigr\}
+ 2N(r, 0; f \prime | f \not = 0)
+ 2N(r, 0; g\prime | g \not = 0) + S(r, f) + S(r, g)
\leq 2
\bigl\{
N(r,\infty ; f) +N(r,\infty ; g)
\bigr\}
+ 2N
\biggl(
r, 0;
f \prime
f
\biggr)
+ 2N
\biggl(
r, 0;
g\prime
g
\biggr)
+ S(r, f) + S(r, g)
\leq 4
\bigl\{
N(r,\infty ; f) +N(r,\infty ; g)
\bigr\}
+ 2T (r, f) + 2T (r, g) + S(r, f) + S(r, g)
\leq 4
\bigl\{
N(r,\infty ; f | \geq 2) +N(r,\infty ; g| \geq 2)
\bigr\}
+ 2T (r, f) + 2T (r, g) + S(r, f) + S(r, g)
\leq 2
\bigl\{
N(r,\infty ; f) +N(r,\infty ; g)
\bigr\}
+ 2T (r, f) + 2T (r, g) + S(r, f) + S(r, g),
that is,
(n - 10)(T (r, f) + T (r, g)) \leq 4\{ N(r,\infty ; f) +N(r,\infty ; g)\} + S(r, f) + S(r, g), (2)
which is impossible as n \geq 15 (resp., 11) and f, g \in M\ast (\BbbC ) (resp., f, g \in H(\BbbC )).
The statement of Remark 2.1 on page 1555 of [2] should be the following.
Remark 2.1. Let S = \{ z : P (z) = 0\} , where P (z) is the polynomial of degree n, defined in
(1). If n \geq 11, then S is a URSE-IM.
References
1. G. Frank, M.Reinders, A unique range set for meromorphic function with 11 elements, Complex Var. Theory Appl.,
37, 185 – 193 (1998).
2. B. Chakraborty, On the cardinality of a reduced unique-range set, Reprint of Ukr. Mat. Zh., 72, № 11, 1553 – 1563
(2020); English translation: Ukr. Math. J., 72, № 11, 1794 – 1806 (2021).
1424 ISSN 1027-3190. Укр. мат. журн., 2024, т. 76, № 9
|
| id | umjimathkievua-article-8063 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:36:02Z |
| publishDate | 2025 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ce/578345a1c483327b863c7c964e4262ce.pdf |
| spelling | umjimathkievua-article-80632025-04-16T12:01:19Z Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' Chakraborty, B. Chakraborty, B. Chakraborty, B. Unique range set IM sharing Cardinality Institute of Mathematics, NAS of Ukraine 2025-04-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/8063 10.3842/umzh.v76i9.8063 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 9 (2024); 1424 Український математичний журнал; Том 76 № 9 (2024); 1424 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8063/10172 Copyright (c) 2024 Bikash Chakraborty |
| spellingShingle | Chakraborty, B. Chakraborty, B. Chakraborty, B. Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' |
| title | Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' |
| title_alt | Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' |
| title_full | Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' |
| title_fullStr | Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' |
| title_full_unstemmed | Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' |
| title_short | Correction to the B. Chakraborty's article ``On the cardinality of a reduced unique range set'' |
| title_sort | correction to the b. chakraborty's article ``on the cardinality of a reduced unique range set'' |
| topic_facet | Unique range set IM sharing Cardinality |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8063 |
| work_keys_str_mv | AT chakrabortyb correctiontothebchakraborty039sarticleonthecardinalityofareduceduniquerangeset039039 AT chakrabortyb correctiontothebchakraborty039sarticleonthecardinalityofareduceduniquerangeset039039 AT chakrabortyb correctiontothebchakraborty039sarticleonthecardinalityofareduceduniquerangeset039039 |