On the cardinality of a reduced unique range set
UDC 517.5Two meromorphic functions are said to share a set $S\subset \mathbb{C}\cup\{\infty\}$ ignoring multiplicities (IM) if $S$ has the same pre-images under both functions. If any two nonconstant meromorphic functions, sharing a set IM, are identical, then the set is called a “reduced unique ran...
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| author | Chakraborty, B. Chakraborty, Bikash Chakraborty, B. |
| author_facet | Chakraborty, B. Chakraborty, Bikash Chakraborty, B. |
| author_sort | Chakraborty, B. |
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| description | UDC 517.5Two meromorphic functions are said to share a set $S\subset \mathbb{C}\cup\{\infty\}$ ignoring multiplicities (IM) if $S$ has the same pre-images under both functions. If any two nonconstant meromorphic functions, sharing a set IM, are identical, then the set is called a “reduced unique range set for meromorphic functions'' (in short, RURSM or URSM-IM).
From the existing literature, it is known that there exists a RURSM with seventeen elements. In this article, we reduced the cardinality of an existing RURSM and established that there exists a RURSM with fifteen elements. Our result gives an affirmative answer to the question of L. Z. Yang (Int. Soc. Anal., Appl., and Comput., 7, 551–564 (2000)). |
| doi_str_mv | 10.37863/umzh.v72i11.594 |
| first_indexed | 2026-03-24T02:03:13Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i11.594
UDC 517.5
B. Chakraborty (Ramakrishna Mission Vivekananda Centenary College, West Bengal, India)
ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET*
ПРО ПОТУЖНIСТЬ РЕДУКОВАНОЇ МНОЖИНИ УНIКАЛЬНОСТI
Two meromorphic functions are said to share a set S \subset \BbbC \cup \{ \infty \} ignoring multiplicities (IM) if S has the same pre-images
under both functions. If any two nonconstant meromorphic functions, sharing a set IM, are identical, then the set is called
a “reduced unique range set for meromorphic functions” (in short, RURSM or URSM-IM).
From the existing literature, it is known that there exists a RURSM with seventeen elements. In this article, we reduced
the cardinality of an existing RURSM and established that there exists a RURSM with fifteen elements. Our result gives
an affirmative answer to the question of L. Z. Yang (Int. Soc. Anal., Appl., and Comput., 7, 551 – 564 (2000)).
Двi мероморфнi функцiї подiляють мiж собою множину S \subset \BbbC \cup \{ \infty \} , не враховуючи кратнiсть, якщо S має
однаковi прообрази вiдносно обох цих функцiй. Якщо для деякої множини будь-якi двi мероморфнi функцiї, що
не є сталими та подiляють мiж собою цю множину, не враховуючи кратнiсть, обов’язково є тотожними, то така
множина називається редукованою множиною унiкальностi для мероморфних функцiй.
З наявних робiт вiдомо, що iснує редукована множина унiкальностi для мероморфних функцiй, яка складається
з 17 елементiв. У цiй роботi ми скорочуємо вказане число та доводимо, що iснує редукована множина унiкальностi
для мероморфних функцiй, що складається з 15 елементiв. Наш результат дає ствердну вiдповiдь на питання,
поставлене L. Z. Yang (Int. Soc. Anal., Appl., and Comput., 7, 551 – 564 (2000)).
1. Introduction. Suppose that f and g are two nonconstant meromorphic functions and a \in \BbbC . We
say that f and g share the value a-CM (counting multiplicities), if f - a and g - a have the same set
of zeros with the same multiplicities. Similarly, we say that f and g share the value a-IM (ignoring
multiplicities), provided that f - a and g - a have the same set of zeros, where the multiplicities are
not taken into account.
Moreover, we say that f and g share \infty -CM (resp., IM), if 1/f and 1/g share 0-CM (resp.,
IM).
In course of studying the factorization of meromorphic functions, in 1976, F. Gross [6] first
generalized the idea of value sharing by introducing the concept of set sharing. Before going to the
details of this paper, we first recall the definition of set sharing:
Definition 1.1. Let f be a nonconstant meromorphic function and, let S \subset \BbbC \cup \{ \infty \} . The set
Ef (S) =
\bigcup
a\in S
\{ (z,m) \in \BbbC \times \BbbN | f(z) - a = 0\} ,
where a zero of f(z) - a with multiplicity m counts m times in Ef (S), is called the pre-image of S
under f, which is also denoted by f - 1(S). Also, we define
Ef (S) =
\bigcup
a\in S
\{ (z, 1) \in \BbbC \times \BbbN | f(z) - a = 0\} ,
i.e., Ef (S) denotes the set of distinct elements in Ef (S).
* This paper was supported by the Department of Higher Education, Science and Technology & Biotechnology, Govt.
of West Bengal under the sanction order no. 216(sanc) /ST/P/S&T/16G-14/2018 dated 19/02/2019.
c\bigcirc B. CHAKRABORTY, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11 1553
1554 B. CHAKRABORTY
Definition 1.2. Two meromorphic functions f and g are said to share a set S CM (resp., IM),
if Ef (S) = Eg(S) (resp., Ef (S) = Eg(S)).
Thus, if S is a singleton set, then it coincides with the usual definition of the value sharing
notation.
In 1976, F. Gross [6] proposed the following question which has later became popular as ”Gross’
question”. The question was as follows:
Question 1.1. Does there exist a finite set S such that any two nonconstant entire functions f
and g sharing the set S must be f \equiv g?
In 1982, to give an affirmative answer to the above question, F. Gross and C. C. Yang [7]
introduced the terminology of unique range set for entire function (in short, URSE) as follows:
Definition 1.3. A set S \subset \BbbC is said to be a unique range set for entire functions (in short,
URSE), if for any two nonconstant entire functions f and g, the condition Ef (S) = Eg(S) implies
f \equiv g.
In the same paper [7], they proved the following result.
Theorem A [7]. Let S = \{ z \in \BbbC : ez + z = 0\} . If two entire functions f, g satisfy Ef (S) =
= Eg(S), then f \equiv g.
It is to be observed that the set S given in Theorem A is an unique range set but contains
infinitely many elements. Thus it can not answer to Question 1.1.
Analogue to Definition 1.3, the definition of unique range sets for meromorphic functions was
also introduced in the literature.
Definition 1.4. A set S \subset \BbbC is called a unique range set for meromorphic functions (in short,
URSM), if for any two nonconstant meromorphic functions f and g, the condition Ef (S) = Eg(S)
implies f \equiv g.
Later on, many authors (see, e.g., [3, 4, 10, 11, 15]) gave the existence of such finite sets for
entire functions as well as meromorphic functions to confirm Question 1.1.
The prime concern of the researchers is to find new unique range sets or to make the cardinalities
of the existing range sets as small as possible. To see the remarkable progress in this regard, one can
go through the research monograph of C. C. Yang and H. X. Yi [13].
To carry on the research on unique range sets, in 1997, H. X. Yi [16] introduced the concept of
reduced unique range sets.
Definition 1.5 [16]. A set S \subset \BbbC \cup \{ \infty \} is said to be a unique range set for meromorphic (resp.,
entire) functions in ignoring multiplicity, in short URSM-IM (resp., URSE-IM) or a reduced unique
range set for meromorphic (resp., entire) functions, in short RURSM (resp., RURSE) if Ef (S) =
= Eg(S) implies f \equiv g for any pair of nonconstant meromorphic (resp., entire) functions.
So, the following question is natural.
Question 1.2 [16]. Is there any finite set S such that for any two nonconstant meromorphic
(resp., entire) functions f and g, the condition Ef (S) = Eg(S) implies f \equiv g?
In 1997, H. X. Yi [16] gave an answer to the above question.
Theorem B [16]. Let n and m be two integers with n > 2m+ 14 and m \geq 2, and let a and b
be two non-zero constants such that the algebraic equation zn + azm + b = 0 has no multiple roots.
If n and m are co prime, then
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET 1555
S = \{ z | zn + azm + b = 0\}
is a URSM-IM.
The above theorem gives the existence of a URSM-IM with 19 elements. In 1998, H. X. Yi [17]
further improved the above result as:
Theorem C [17]. Let n(\geq 17) be an integer. Let
S = \{ z | azn - n(n - 1)z2 + 2n(n - 2)bz - (n - 1)(n - 2)b2 = 0\} ,
where a and b be two non-zero constants such that abn - 2 \not = 2. Then the set S is a URSM-IM.
Thus Theorem C gives the existence of a URSM-IM with 17 elements. In this direction, in 1997,
M. Reinders [12] has shown that there exist URSM-IM with 16 elements. But unfortunately, the
proof of a lemma which is necessary in the proof of Reinders’ proof [12] has some gaps [9, p. 204].
In 1998, M. L. Fang and H. Guo [2] gave another example of URSM-IM with 17 elements using
the technique of G. Frank and M. Reinders [3].
For a positive integers n(\geq 3) and a complex number c( \not = 0, 1), we shall denote by P (z) [3]
the following polynomial:
P (z) =
(n - 1)(n - 2)
2
zn - n(n - 2)zn - 1 +
n(n - 1)
2
zn - 2 - c. (1.1)
Clearly, the restrictions on c implies that P (z) has only simple zeros.
Theorem D [2]. Let S = \{ z | P (z) = 0\} , where P (z) is defined by (1.1). If n \geq 17, then the
set S is a URSM-IM.
In 1999, S. Bartels [1] gave another proof of Theorem D. Thus it is observed from the existing
literature that the smallest available reduced unique range set must contains at least 17 elements (see,
e.g., [1, 2, 17]). Let
\lambda M = \mathrm{i}\mathrm{n}\mathrm{f}\{ \sharp (S) : S is a URSM-IM\} ,
where \sharp (S) is the cardinality of S. It is clear from the above discussion that \sharp (S) \leq 17. Also,
examples show that \sharp (S) \geq 6 [13, p. 527]. Combining the above results, L.-Z. Yang pose the
following open question [14, p. 557].
Question 1.3 [14]. What exactly the number \lambda M is?
The main purpose of this paper is to reduce the cardinality of the URSM-IM in Theorem D. As
a result, our paper partially answers to Question 1.3.
2. Main result. The following theorem is the main result of this paper.
Theorem 2.1. Let S = \{ z : P (z) = 0\} , where P (z) is the polynomial of degree n, defined in
(1.1). If n \geq 15, then S is a URSM-IM.
Remark 2.1. In Theorem 2.1, if n \geq 9, then S is a URSE-IM.
3. Notations. We assumed that the readers are familiar with the classical Nevanlinna theory
[8, 13]. Before going to the proof of the main theorem, we explain some well known definitions and
notations.
Definition 3.1. Let f be a meromorphic function. Also, let a \in \BbbC \cup \{ \infty \} and m \in \BbbN .
(i) We denote by N(r, a; f | = 1), the counting function of simple a-points of f.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1556 B. CHAKRABORTY
(ii) We denote by N(r, a; f | \leq m) (resp., N(r, a; f | \geq m)), the counting function of those
a-points of f whose multiplicities are not greater (resp., not less) than m where each a-point is
counted according to its multiplicity.
Let N(r, a; f | \leq m) and N(r, a; f | \geq m) denote the reduced counting function of N(r, a; f | \leq
m) and N(r, a; f | \geq m), respectively.
Similar to the above counting functions, N(r, a; f | < m), N(r, a; f | > m), N(r, a; f | < m)
and N(r, a; f | > m) are defined.
Definition 3.2. For a \in \BbbC \cup \{ \infty \} and p \in \BbbN , we denote, by Np(r, a; f), the sum
N(r, a; f) +N(r, a; f | \geq 2) + . . .+N(r, a; f | \geq p).
Thus, clearly N1(r, a; f) = N(r, a; f).
Definition 3.3. Let f and g be two nonconstant meromorphic functions such that f and g share
a IM, where a \in \BbbC \cup \{ \infty \} . Let z0 be an a-point of f with multiplicity p and z0 be an a-point of g
with multiplicity q.
(i) We denote by N
1)
E (r, a; f), the counting function of those a-points of f and g where
p = q = 1. Thus N
1)
E (r, a; f) = N
1)
E (r, a; g).
(ii) We denote by N
(2
E (r, a; f), the reduced counting function of those a-points of f and g where
p = q \geq 2. So, N
(2
E (r, a; f) = N
(2
E (r, a; g).
(iii) We denote by NL(r, a; f), the reduced counting function of those a-points of f and g where
p > q and by NL(r, a; g), we denote the reduced counting function of those a-points of f and g
where q > p. Thus NL(r, a; f) \not = NL(r, a; g).
We denote by N\ast (r, a; f, g), the reduced counting function of those a-points of f whose multi-
plicities differ from the multiplicities of the corresponding a-points of g. Thus,
N\ast (r, a; f, g) \equiv N\ast (r, a; g, f) and N\ast (r, a; f, g) = NL(r, a; f) +NL(r, a; g).
Definition 3.4. Let a, b \in \BbbC \cup \{ \infty \} . We denote by N(r, a; f | g \not = b), the counting function of
those a-points of f, counted according to multiplicity, which are not the b-points of g.
Definition 3.5. A polynomial \wp (z) over \BbbC , is called a uniqueness polynomial for meromorphic
(resp., entire) functions, if for any two nonconstant meromorphic (resp., entire) functions f and g,
\wp (f) \equiv \wp (g) implies f \equiv g.
In 2000, H. Fujimoto [4] first discovered a special property of a polynomial, which was later
termed as critical injection property.
Definition 3.6 [4]. A polynomial \wp (z) is said to satisfy critical injection property if \wp (\alpha ) \not =
\not = \wp (\beta ), where \alpha and \beta are any two distinct zeros of \wp \prime (z).
4. Auxiliary lemmas.
Lemma 4.1 (First fundamental theorem of Nevanlinna, [13]). For a nonconstant meromorphic
function f and for a complex number a \in \BbbC \cup \{ \infty \} ,
T
\biggl(
r,
1
f - a
\biggr)
= T (r, f) +O(1),
where O(1) is a bounded quantity depending on a.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET 1557
Lemma 4.2 (Second fundamental theorem of Nevanlinna, [13, p. 15]). Suppose that f is a non-
constant meromorphic function in the complex plane and a1, a2, . . . , aq are q(\geq 2) distinct values
in \BbbC . Then
(q - 1)T (r, f) \leq N(r,\infty ; f) +
q\sum
j=1
N(r, aj ; f) - Nram(r, f) + S(r, f), (4.1)
where
Nram(r, f) = 2N(r,\infty ; f) - N(r,\infty ; f \prime ) +N(r, 0; f \prime )
and S(r, f) is a quantity such that
S(r, f)
T (r, f)
\rightarrow 0 as r \rightarrow +\infty outside of a set E(\subset (0,\infty )) with
finite linear measure.
Remark 4.1. Clearly, (4.1) can be written as
(q - 1)T (r, f) \leq N(r,\infty ; f) +
q\sum
j=1
N(r, aj ; f) - N\circ (r, 0; f
\prime ) + S(r, f),
where N\circ (r, 0; f
\prime ) is the counting function of those zeros of f \prime which is not zeros of
\prod q
j=1
(f - aj).
Lemma 4.3 [13, p. 28]. Let f be a nonconstant meromorphic function and let
R(f) =
\sum n
k=0
akf
k\sum m
j=0
bjf
j
be an irreducible rational function in f with constant coefficients \{ ak\} and \{ bj\} , where an \not = 0 and
bm \not = 0. Then
T (r,R(f)) = d \cdot T (r, f) + S(r, f),
where d = \mathrm{m}\mathrm{a}\mathrm{x}\{ n,m\} .
Lemma 4.4 [5]. Let \wp (z) be a monic polynomial without multiple zero whose derivative has
mutually k-distinct zeros, given by d1, d2, . . . , dk with multiplicities q1, q2, . . . , qk, respectively.
Suppose that \wp (z) satisfy the “critical injection property”. Then \wp (z) will be a uniqueness
polynomial if and only if \sum
1\leq l<m\leq k
q
l
qm >
k\sum
l=1
q
l
.
In particular, the above inequality is always satisfied whenever k \geq 4. When k = 3 and
\mathrm{m}\mathrm{a}\mathrm{x}\{ q1, q2, q3\} \geq 2 or k = 2, \mathrm{m}\mathrm{i}\mathrm{n}\{ q1, q2\} \geq 2 and q1 + q2 \geq 5, then also the above inequality
holds.
Lemma 4.5 [13, p. 376]. Let \scrF and \scrG be two non constant meromorphic functions sharing
1 CM. If
N2(r, 0;\scrF ) +N2(r, 0;\scrG ) +N2(r,\infty ;\scrF ) +N2(r,\infty ;\scrG ) < (\mu + o(1))T (r),
where \mu < 1, r \in I, T (r) = \mathrm{m}\mathrm{a}\mathrm{x}\{ T (r,\scrF ), T (r,\scrG )\} , I is a set of infinite linear measure of
r \in (0,\infty ). Then one of the following holds:
i) \scrF \equiv \scrG ,
ii) \scrF \scrG \equiv 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1558 B. CHAKRABORTY
Lemma 4.6. Let \scrF and \scrG be two non constant meromorphic functions sharing 1 IM. Then
N(r, 1;\scrF ) +N(r, 1;\scrG ) - N
1)
E (r, 1;\scrF ) +N\ast (r, 1;\scrF ,\scrG ) \leq
\leq 1
2
\{ N(r, 1;\scrF ) +N(r, 1;\scrG )\} +N(r, 1;\scrF | \geq 2) +N(r, 1;\scrG | \geq 2).
Proof. Given \scrF and \scrG share 1 IM. Let z0 be an 1-point of \scrF of multiplicity p and let z0 be
an 1-point of \scrG of multiplicity q. Now, we consider following cases:
Case 1. Assume p = q.
If p = q = 1, then z0 is counted (1 + 1 - 1 + 0) = 1 times in the left-hand side of the above
inequality whereas it is counted
1
2
(1 + 1) + 0 + 0 = 1 times in the right-hand side of the same.
If p = q \geq 2, then z0 is counted (1 + 1 - 0 + 0) = 2 times in the left-hand side of the above
inequality whereas it is counted
1
2
(p+ p) + p+ p = 3p times in the right-hand side of the same.
Case 2. Assume p > q.
If p > q and q = 1, then p \geq 2 and z0 is counted (1 + 1 - 0 + 1) = 3 times in the left-hand
side of the above inequality whereas it is counted
1
2
(p+ 1) + p+ 0 =
3p
2
+
1
2
\biggl(
\geq 3 +
1
2
\biggr)
times in
the right-hand side of the same.
If p > q and q \geq 2, then z0 is counted (1 + 1 - 0 + 1) = 3 times in the left-hand side of the
above inequality whereas it is counted
1
2
(p+ q) + p+ q =
3
2
(p + q)(> 3q) times in the right-hand
side of the same.
Case 3. Assume q > p.
The explanations are similar to Case 2. Hence, the proof is completed.
5. Proof of Theorem 2.1. Given that f and g share the set S IM. Now, we define
F := Q(f), G := Q(g),
where
Q(z) :=
P (z) + c
c
=
=
(n - 1)(n - 2)
2c
zn - 2
\biggl(
z2 - 2n
n - 1
z +
n
n - 2
\biggr)
and P (z) is defined in (1.1), c \in \BbbC \setminus \{ 0, 1\} .
Thus F and G share the value 1 IM, and, hence, EF (\{ 1\} ) = EG(\{ 1\} ). Now we consider two
cases:
Case 1. First, we assume that F and G are linearly dependent. Then there exist a non zero
constant k such that
F \equiv k G.
Thus, using Lemma 4.3, we obtain
T (r, f) = T (r, g) + S(r, g).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET 1559
Subcase 1.1. If EF (\{ 1\} )\cap EG(\{ 1\} ) \not = \phi , then there exist a z0 \in \BbbC such that F (z0) = G(z0) =
= 1. Thus, k = 1, i.e.,
F \equiv G, i.e., P (f) \equiv P (g).
Since P \prime (z) =
n(n - 1)(n - 2)
2
zn - 3(z - 1)2 and P (0) \not = P (1). So, P (z) satisfies ”critical injection
property”. Thus, in view of Lemma 4.4, P (z) is a uniqueness polynomial, i.e., f \equiv g.
Subcase 1.2. If EF (\{ 1\} )\cap EG(\{ 1\} ) = \phi , then EF (\{ 1\} ) = EG(\{ 1\} ) = \phi . Thus, we can assume
that F and G share 1 CM.
First, we show that under the given conditions,
FG \not \equiv 1,
because, otherwise if FG \equiv 1, then
fn - 2
2\prod
i=1
(f - \gamma i) g
n - 2
2\prod
i=1
(g - \gamma i) \equiv
4c2
(n - 1)2(n - 2)2
,
where \gamma i, i = 1, 2, are the roots of the equation z2 - 2n
n - 1
z +
n
n - 2
= 0.
Let z0 be a \gamma i-point of f of order p. Then z0 must be a pole of g (say, of order q). Then
p = nq \geq n. So,
N(r, \gamma i; f) \leq
1
n
N(r, \gamma i; f) \leq
1
n
T (r, f) +O(1).
Again, let z0 be a zero of f of order t. Then z0 must be a pole of g (say, of order s). Then
(n - 2)t = ns. Thus t > s. Now, 2s = (n - 2)(t - s) \geq (n - 2). Thus (n - 2)t = ns gives t \geq n
2
.
So,
N(r, 0; f) \leq 2
n
N(r, 0; f) \leq 2
n
T (r, f) +O(1).
Similar calculations are valid for g also. Thus, applying the second fundamental theorem, we get
T (r, f) \leq N(r, 0; f) +
2\sum
i=1
N(r, \gamma i; f) + S(r, f) \leq
\leq 2
n
T (r, f) +
2
n
T (r, f) + S(r, f),
which is impossible as n \geq 5. Thus, FG \not \equiv 1.
Now, our claim is F \equiv G. Since
N2(r, 0;F ) +N2(r, 0;G) +N2(r,\infty ;F ) +N2(r,\infty ;G) \leq
\leq 2N(r, 0; f) + 2T (r, f) + 2N(r, 0; g) + 2T (r, g) + 2N(r,\infty ; f) + 2N(r,\infty ; g)+
+S(r, f) + S(r, g) \leq
\leq 4T (r, f) + 4T (r, g) + 2N(r,\infty ; f) + 2N(r,\infty ; g) + S(r, f) + S(r, g) <
< (1 + o(1))T (r) (as n \geq 9 (resp., n \geq 15) for URSE-IM (resp., URSM-IM)),
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1560 B. CHAKRABORTY
where T (r) = \mathrm{m}\mathrm{a}\mathrm{x}\{ T (r, F ), T (r,G)\} and S(r) = o(T (r)), so, in view of Lemma 4.5, we obtain
F \equiv G. (5.1)
Thus, from (5.1), we have
P (f) \equiv P (g).
Since P \prime (z) =
n(n - 1)(n - 2)
2
zn - 3(z - 1)2 and P (0) \not = P (1). So, P (z) satisfies “critical injection
property”. Thus, in view of Lemma 4.4, P (z) is a uniqueness polynomial, i.e.,
f \equiv g.
Case 2. Assume that F and G are linearly independent. Then F \not \equiv G. Henceforth we shall
denote by H the following function:
H :=
\Biggl(
F
\prime \prime
F \prime - 2F
\prime
F - 1
\Biggr)
-
\Biggl(
G
\prime \prime
G\prime - 2G
\prime
G - 1
\Biggr)
. (5.2)
Again we consider two subcases:
Subcase 2.1. Assume H \equiv 0. Then on integration, we get from (5.2) that
1
G - 1
\equiv A
F - 1
+B,
where A(\not = 0), B are constants. Thus F and G share the value 1 CM. Since,
N2(r, 0;F ) +N2(r, 0;G) +N2(r,\infty ;F ) +N2(r,\infty ;G) \leq
\leq 2N(r, 0; f) + 2T (r, f) + 2N(r, 0; g) + 2T (r, g) + 2N(r,\infty ; f) + 2N(r,\infty ; g)+
+S(r, f) + S(r, g) \leq
\leq \lambda
n
T (r) + S(r) (\lambda = 8, or 12 according to f and g both are entire,
or meromorphic functions, respectively) <
< (1 + o(1))T (r) (as n \geq 9 (resp., n \geq 15) for URSE-IM (resp., URSM-IM)),
where T (r) = \mathrm{m}\mathrm{a}\mathrm{x}\{ T (r, F ), T (r,G)\} and S(r) = o(T (r)), thus, in view of Lemma 4.5, we obtain
FG \equiv 1 or F \equiv G.
But, already we have seen that, if n \geq 5, then FG \not \equiv 1. Thus, F \equiv G, which contradicts the fact
that F and G are linearly independent.
Subcase 2.2. We assume that H \not \equiv 0. Then by simple calculations, we have
N(r,\infty ;H) \leq N(r,\infty ;F ) +N(r,\infty ;G) +N(r, 0;F | \geq 2) +N(r, 0;G| \geq 2)+
+N\ast (r, 1;F,G) +N0(r, 0;F
\prime ) +N0(r, 0;G
\prime ),
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ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET 1561
where N0(r, 0;F
\prime ) is the reduced counting function of those zeros of F \prime which are not zeros of
F (F - 1), similarly, N0(r, 0;G
\prime ) is defined.
Since
cF =
(n - 1)(n - 2)
2
fn - 2
\prod 2
i=1
(f - \gamma i)
and
cF \prime =
n(n - 1)(n - 2)
2
fn - 3(f - 1)2f \prime ,
where \gamma i, i = 1, 2, are the roots of the equation z2 - 2n
n - 1
z +
n
n - 2
= 0. Thus,
N(r,\infty ;H) \leq N(r,\infty ; f) +N(r,\infty ; g) +N(r, 0; f) +N(r, 0; g) +N\ast (r, 1;F,G)+
+N(r, 1; f) +N(r, 1; g) +N \star (r, 0; f
\prime ) +N \star (r, 0; g
\prime ) \leq
\leq N(r,\infty ; f) +N(r,\infty ; g) + 2\{ T (r, f) + T (r, g)\} +N\ast (r, 1;F,G)+
+N \star (r, 0; f
\prime ) +N \star (r, 0; g
\prime ), (5.3)
where N \star (r, 0; f
\prime ) is the reduced counting function of those zeros of f \prime which are not zeros of
f(f - 1) and (F - 1), N \star (r, 0; g
\prime ) denotes similarly according to g. Again
N
1)
E (r, 1;F ) = N
1)
E (r, 1;G) \leq N(r,\infty ;H) + S(r, f) + S(r, g), (5.4)
where N
1)
E (r, 1;F ) is the counting function of those simple 1-points of F which are also simple
1-points of G.
Thus, using (5.3), (5.4) and Lemma 4.6 and first fundamental theorem, we have
N(r, 1;F ) +N(r, 1;G) \leq
\leq N(r, 1;F ) +N(r, 1;G) - N
1)
E (r, 1;F ) +N(r,\infty ;H) + S(r, f) + S(r, g) \leq
\leq N(r,\infty ; f) +N(r,\infty ; g) + 2\{ T (r, f) + T (r, g)\} +N\ast (r, 1;F,G)+
+N(r, 1;F ) +N(r, 1;G) - N
1)
E (r, 1;F ) +N \star (r, 0; f
\prime ) +N \star (r, 0; g
\prime )+
+S(r, f) + S(r, g) \leq
\leq N(r,\infty ; f) +N(r,\infty ; g) + 2\{ T (r, f) + T (r, g)\} +
+
1
2
\{ N(r, 1;F ) +N(r, 1;G)\} +N(r, 1;F | \geq 2) +N(r, 1;G | \geq 2)+
+N \star (r, 0; f
\prime ) +N \star (r, 0; g
\prime ) + S(r, f) + S(r, g) \leq
\leq N(r,\infty ; f) +N(r,\infty ; g) +
\Bigl(
2 +
n
2
\Bigr)
\{ T (r, f) + T (r, g)\} +
+N(r, 1;F | \geq 2) +N(r, 1;G | \geq 2) +N \star (r, 0; f
\prime ) +N \star (r, 0; g
\prime )+
+S(r, f) + S(r, g), (5.5)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1562 B. CHAKRABORTY
where N \star (r, 0; f
\prime ) is the counting function of those zeros of f \prime which are not zeros of f(f - 1) and
(F - 1), N \star (r, 0; g
\prime ) denotes similarly according to g.
If \alpha 1, \alpha 2, . . . , \alpha n be the n-distinct zeros of P (z) = 0, then
c(F - 1) = P (f) =
(n - 1)(n - 2)
2
n\prod
i=1
(f - \alpha i)
and
c(G - 1) = P (g) =
(n - 1)(n - 2)
2
n\prod
i=1
(g - \alpha i).
Thus, applying the second fundamental theorem for n+ 2 distinct values 0, 1, \alpha 1, \alpha 2, . . . , \alpha n, we
obtain
(n+ 1) (T (r, f) + T (r, g)) \leq
\leq N(r,\infty ; f) +N(r, 0; f) +N(r, 1; f) +
n\sum
i=1
N(r, \alpha i; f) - N \star (r, 0, f
\prime )+
+N(r,\infty ; g) +N(r, 0; g) +N(r, 1; g) +
n\sum
i=1
N(r, \alpha i; g) - N \star (r, 0, g
\prime )+
+S(r, f) + S(r, g) \leq
\leq N(r,\infty ; f) +N(r,\infty ; g) + 2(T (r, f) + T (r, g)) +N(r, 1;F ) +N(r, 1;G) -
- N \star (r, 0, f
\prime ) - N \star (r, 0, g
\prime ) + S(r, f) + S(r, g). (5.6)
By using inequalities (5.5) and (5.6), we get\Bigl( n
2
- 3
\Bigr)
(T (r, f) + T (r, g)) \leq
\leq 2(N(r,\infty ; f) +N(r,\infty ; g)) + \{ N(r, 1;F | \geq 2) +N(r, 1;G| \geq 2)\} +
+S(r, f) + S(r, g). (5.7)
Also, using first fundamental theorem and elementary calculations, we have
N
\biggl(
r, 0;
f \prime
f
\biggr)
\leq T
\biggl(
r,
f \prime
f
\biggr)
+O(1) = N
\biggl(
r,\infty ;
f \prime
f
\biggr)
+ S(r, f) \leq
\leq N(r, 0; f) +N(r,\infty ; f) + S(r, f) \leq T (r, f) +N(r,\infty ; f) + S(r, f).
Thus on simplifying (5.7), we obtain\Bigl( n
2
- 3
\Bigr)
(T (r, f) + T (r, g)) \leq
\leq 2\{ N(r,\infty ; f) +N(r,\infty ; g)\} +N(r, 1;F | \geq 2) +N(r, 1;G| \geq 2)+
+S(r, f) + S(r, g) \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET 1563
\leq 2\{ N(r,\infty ; f) +N(r,\infty ; g)\} +N(r, 0; f \prime | f \not = 0) +N(r, 0; g\prime | g \not = 0)+
+S(r, f) + S(r, g) \leq
\leq 2\{ N(r,\infty ; f) +N(r,\infty ; g)\} +N
\biggl(
r, 0;
f \prime
f
\biggr)
+N
\biggl(
r, 0;
g\prime
g
\biggr)
+
+S(r, f) + S(r, g) \leq
\leq 3\{ N(r,\infty ; f) +N(r,\infty ; g)\} + T (r, f) + T (r, g) + S(r, f) + S(r, g).
That is,
(n - 8)(T (r, f) + T (r, g)) \leq
\leq 6\{ N(r,\infty ; f) +N(r,\infty ; g)\} + S(r, f) + S(r, g)
which is impossible as n \geq 15 (resp., 9) for URSM-IM (resp., URSE-IM) case.
References
1. S. Bartels, Meromorphic functions sharing a set with 17 elements ignoring multiplicities, Complex Variables, Theory
and Appl., 39, 85 – 92 (1999).
2. M. L. Fang, H. Guo, On unique range sets for meromorphic or entire functions, Acta Math. Sin. (New Ser.), 14,
№ 4, 569 – 576 (1998).
3. G. Frank, M. Reinders, A unique range set for meromorphic function with 11 elements, Complex Variables, Theory
and Appl., 37, 185 – 193 (1998).
4. H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122, № 6, 1175 – 1203
(2000).
5. H. Fujimoto, On uniqueness polynomials for meromorphic functions, Nagoya Math. J., 170, 33 – 46 (2003).
6. F. Gross, Factorization of meromorphic functions and some open problems, Proc. Conf. Univ. Kentucky, Leixngton,
Ky (1976); Lect. Notes Math., 599, 51 – 69 (1977).
7. F. Gross, C. C. Yang, On preimage and range sets of meromorphic functions, Proc. Japan Acad., 58, 17 – 20 (1982).
8. W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford (1964).
9. P. C. Hu, P. Li, C. C. Yang, Unicity of meromorphic mappings, Springer, DOI 10.1007/978-1-4757-3775-2.
10. P. Li, C. C. Yang, Some further results on the unique range set of meromorphic functions, Kodai Math. J., 18,
437 – 450 (1995).
11. P. Li, C. C. Yang, On the unique range set for meromorphic functions, Proc. Amer. Math. Soc., 124, 177 – 185 (1996).
12. M. Reinders, Unique range sets ignoring multiplicities, Bull. Hong Kong Math. Soc., 1, 339 – 341 (1997).
13. C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Kluwer Acad. Publ. (2003).
14. L. Z. Yang, Some recent progress in the uniqueness theory of meromorphic functions, Proc. Second ISAAC Congr.,
Int. Soc. Anal., Appl. and Comput., 7, 551 – 564 (2000).
15. H. X. Yi, Unicity theorems for meromorphic and entire functions III, Bull. Austral. Math. Soc., 53, 71 – 82 (1996).
16. H. X. Yi, The reduced unique range sets for entire or meromorphic functions, Complex Variables, Theory and Appl.,
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Received 17.12.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
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| id | umjimathkievua-article-594 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:13Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0e/4fdffb6bd2ce636a5afdc30462f7080e.pdf |
| spelling | umjimathkievua-article-5942025-03-31T08:49:35Z On the cardinality of a reduced unique range set ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET On the cardinality of a reduced unique range set Chakraborty, B. Chakraborty, Bikash Chakraborty, B. Meromorphic function URSM Set Sharing Ignoring Multiplicities Meromorphic function URSM Set Sharing Ignoring Multiplicities UDC 517.5Two meromorphic functions are said to share a set $S\subset \mathbb{C}\cup\{\infty\}$ ignoring multiplicities (IM) if $S$ has the same pre-images under both functions. If any two nonconstant meromorphic functions, sharing a set IM, are identical, then the set is called a “reduced unique range set for meromorphic functions'' (in short, RURSM or URSM-IM). From the existing literature, it is known that there exists a RURSM with seventeen elements. In this article, we reduced the cardinality of an existing RURSM and established that there exists a RURSM with fifteen elements. Our result gives an affirmative answer to the question of L. Z. Yang (Int. Soc. Anal., Appl., and Comput., 7, 551–564 (2000)). УДК 517.5 Про потужність редукованої множини унікальності Дві мероморфні функції поділяють між собою множину $S\subset \mathbb{C}\cup\{\infty\},$ не враховуючи кратність, якщо $S$ має однакові прообрази відносно обох цих функцій. Якщо для деякої множини будь-які дві мероморфні функції, що не є сталими та поділяють між собою цю множину, не враховуючи кратність, обов'язково є тотожними, то така множина називається редукованою множиною унікальності для мероморфних функцій. З наявних робіт відомо, що існує редукована множина унікальності для мероморфних функцій, яка складається з 17 елементів. У цій роботі ми скорочуємо вказане число та доводимо, що існує редукована множина унікальності для мероморфних функцій, що складається з 15 елементів. Наш результат дає ствердну відповідь на питання, поставлене L. Z. Yang (Int. Soc. Anal., Appl., and Comput., 7, 551–564 (2000)). Institute of Mathematics, NAS of Ukraine 2020-11-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/594 10.37863/umzh.v72i11.594 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 11 (2020); 1553-1563 Український математичний журнал; Том 72 № 11 (2020); 1553-1563 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/594/8784 Copyright (c) 2020 B. Chakraborty |
| spellingShingle | Chakraborty, B. Chakraborty, Bikash Chakraborty, B. On the cardinality of a reduced unique range set |
| title | On the cardinality of a reduced unique range set |
| title_alt | ON THE CARDINALITY OF A REDUCED UNIQUE RANGE SET On the cardinality of a reduced unique range set |
| title_full | On the cardinality of a reduced unique range set |
| title_fullStr | On the cardinality of a reduced unique range set |
| title_full_unstemmed | On the cardinality of a reduced unique range set |
| title_short | On the cardinality of a reduced unique range set |
| title_sort | on the cardinality of a reduced unique range set |
| topic_facet | Meromorphic function URSM Set Sharing Ignoring Multiplicities Meromorphic function URSM Set Sharing Ignoring Multiplicities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/594 |
| work_keys_str_mv | AT chakrabortyb onthecardinalityofareduceduniquerangeset AT chakrabortybikash onthecardinalityofareduceduniquerangeset AT chakrabortyb onthecardinalityofareduceduniquerangeset |