On the cardinality of unique range sets with weight one
UDC 517.9 Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\},$ if $E_{f}(S,l)=E_{g}(S,l),$ where $$$E_{f}(S,l)=\bigcup_{a \in S} \big \{(z,t) \in \mathbb{C}\times\mathbb{N} \bigm| f(z)=a \; \tex...
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2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512221602250752 |
|---|---|
| author | Chakraborty, B. Chakraborty, S. Chakraborty, B. Chakraborty, S. |
| author_facet | Chakraborty, B. Chakraborty, S. Chakraborty, B. Chakraborty, S. |
| author_sort | Chakraborty, B. |
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UDC 517.9
Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\},$ if $E_{f}(S,l)=E_{g}(S,l),$ where $$$E_{f}(S,l)=\bigcup_{a \in S} \big \{(z,t) \in \mathbb{C}\times\mathbb{N} \bigm| f(z)=a \; \text{with multiplicity} \;p \big \},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p>l.$
In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang [Indian J.  Pure and Appl.  Math., 31, No~4, 431–440 (2000)] by showing that there exist a finite set $S$ with 13 elements such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g.$ |
| doi_str_mv | 10.37863/umzh.v72i7.6022 |
| first_indexed | 2026-03-24T03:25:21Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i7.6022
UDC 517.9
B. Chakraborty (Ramakrishna Mission Vivekananda Centenary College, West Bengal, India),
S. Chakraborty (Jadavpur Univ., West Bengal, India)
ON THE CARDINALITY OF UNIQUE RANGE SETS WITH WEIGHT ONE
ПРО ПОТУЖНIСТЬ УНIКАЛЬНОГО НАБОРУ МНОЖИН
З ОДИНИЧНОЮ ВАГОЮ
Two meromorphic functions f and g are said to share the set S \subset \BbbC \cup \{ \infty \} with weight l \in \BbbN \cup \{ 0\} \cup \{ \infty \} , if
Ef (S, l) = Eg(S, l), where
Ef (S, l) =
\bigcup
a\in S
\bigl\{
(z, t) \in \BbbC \times \BbbN
\bigm| \bigm| f(z) = a with multiplicity p
\bigr\}
,
where t = p if p \leq l and t = p+ 1 if p > l.
In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang [Indian J. Pure and Appl. Math., 31,
№ 4, 431 – 440 (2000)] by showing that there exist a finite set S with 13 elements such that Ef (S, 1) = Eg(S, 1) implies
f \equiv g.
Двi мероморфнi функцiї f i g сумiсно мають множину S \subset \BbbC \cup \{ \infty \} з одиничною вагою l \in \BbbN \cup \{ 0\} \cup \{ \infty \} , якщо
Ef (S, l) = Eg(S, l), де
Ef (S, l) =
\bigcup
a\in S
\bigl\{
(z, t) \in \BbbC \times \BbbN
\bigm| \bigm| f(z) = a з кратнiстю p
\bigr\}
,
за умови, що t = p, якщо p \leq l, i t = p+ 1, якщо p > l.
У цiй роботi ми вдосконалюємо та доповнюємо результат L. W. Liao i C. C. Yang [Indian J. Pure and Appl.
Math., 31, № 4, 431 – 440 (2000)], пропонуючи доведення того, що iснує скiнченна множина S, що складається з 13
елементiв, така, що з Ef (S, 1) = Eg(S, 1) випливає f \equiv g.
1. Introduction and definitions. By \BbbC and \BbbN , we mean the set of complex numbers and set of
natural numbers, respectively. By meromorphic function, we mean an analytic function defined on
\BbbC except possibly at isolated singularities, each of which is a pole.
The tool we used in this paper is Nevanlinna theory. For the standard notations of the Nevanlinna
theory, one can go through the Hayman’s monograph [8].
It will be convenient to let that E be denote any set of positive real numbers of finite linear
measure, not necessarily the same at each occurrence. For any nonconstant meromorphic function
h(z), we denote by S(r, h), any quantity satisfying
S(r, h) = o(T (r, h)), r - \rightarrow \infty , r \not \in E.
Suppose f and g be two nonconstant meromorphic functions and a \in \BbbC . We say that f and g share
the value a-CM (counting multiplicities), provided that f - a and g - a have the same 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 the course of studying the factorization of meromorphic functions, F. Gross [6] first generalized
the idea of value sharing by introducing the concept of a unique range set. Before going to the details
of the paper, we first recall the definition of set sharing.
c\bigcirc B. CHAKRABORTY, S. CHAKRABORTY, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7 997
998 B. CHAKRABORTY AND S. CHAKRABORTY
Definition 1.1 [15]. For a nonconstant meromorphic function f and any set S \subset \BbbC \cup \{ \infty \} , we
define
Ef (S) =
\bigcup
a\in S
\{ (z, p) \in \BbbC \times \BbbN | f(z) = a with multiplicity p\} ,
Ef (S) =
\bigcup
a\in S
\{ (z, 1) \in \BbbC \times \BbbN | f(z) = a with multiplicity p\} .
If Ef (S) = Eg(S)
\bigl(
resp., Ef (S) = Eg(S)
\bigr)
, then it is said that f and g share the set S counting
multiplicities or in short CM (resp., ignoring multiplicities or in short IM).
Thus, if S is singleton, then it coincides with the usual definition of value sharing.
In 1977, F. Gross [6] proposed the following problem which has later became popular as ”Gross
problem”. The problem was as follows:
Question A. 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, F. Gross and C. C. Yang [7] gave the affirmative answer to the above question as follows:
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.
In paper [7], they first introduced the terminology unique range set for entire function (in short
URSE). Later the analogous definition for meromorphic functions was also introduced in literature.
Definition 1.2 [15]. Let S \subset \BbbC \cup \{ \infty \} and f and g be two nonconstant meromorphic (resp.,
entire) functions. If Ef (S) = Eg(S) implies f \equiv g, then S is called a unique range set for
meromorphic (resp., entire) functions or in brief URSM (resp., URSE).
In 1997, H. X. Yi [18] introduced the analogous definition for reduced unique range sets.
Definition 1.3 [18]. 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.
During the last few years the notion of unique range sets as well as reduced unique range sets
have been generating an increasing interest among the researchers. For the literature, one can also
go through the research monograph written by C. C. Yang and H. X. Yi [15].
Next we recall the following notations:
\lambda M = \mathrm{i}\mathrm{n}\mathrm{f}\{ \sharp (S) | S is an URSM\} and \lambda E = \mathrm{i}\mathrm{n}\mathrm{f}\{ \sharp (S) | S is an URSE\} ,
where \sharp (S) is the cardinality of the set S.
In 1996, P. Li [11] showed that \lambda E \geq 5 whereas C. C. Yang and H. X Yi [15] established that
\lambda M \geq 6. Also, G. Frank and M. Reinders [3] estimated that \lambda M \leq 11. And for entire functions
corresponding estimation is \lambda E \leq 7. Till date these estimations are the best.
In course of time, researchers are also paying their attention to find the lowest cardinality of
URSM-IM as well as URSE-IM. In 1997, H. X. Yi [18] gave the existence of URSM-IM with
19 elements. After one year, in 1998, H. X. Yi [19] further improved his result [8] and obtained
URSM-IM of 17 elements. Also in 1998, M. L. Fang and H. Guo [2] and in 1999, S. Bartels [1]
independently gave the existence of URSM-IM with 17 elements. In connection to our discussions,
the following question is natural:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE CARDINALITY OF UNIQUE RANGE SETS WITH WEIGHT ONE 999
Question 1.1. Can one further reduced the lower bound of the unique range sets by relaxing the
sharing notations?
As an attempt to reduce the cardinalities of unique range sets, L. W. Liao and C. C. Yang [13]
introduced the following notation:
Definition 1.4. Let f be a nonconstant meromorphic function and S \subset \BbbC \cup \{ \infty \} . We define
E1)(S, f) =
\bigcup
a\in S
E1)(a, f),
where E1)(a, f) is the set of all simple a-points of f.
For a positive integers n \geq 3 and c \not = 0, 1, we shall denote by P (z), the Frank – Reinders
polynomial [3] as
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. Using the methods of Frank –
Reinders [3], in connection to the Question 1.1, L. W. Liao and C. C. Yang [13] proved following
theorem.
Theorem B [13]. Suppose that n(\geq 1) be a positive integer. Further suppose that S = \{ z :
P (z) = 0\} , where the polynomial P (z) of degree n defined by (1.1). Let f and g be two nonconstant
meromorphic functions satisfying E1)(S, f) = E1)(S, g). If n \geq 15, then f \equiv g.
The motivation of this paper is to improve and supplement Theorem B utilizing the method of
Frank – Reinders [3]. Before going to state our main result, we recall some well-known definitions
which will be useful for the proof of the main result of this paper.
Definition 1.5 [10]. For a nonconstant meromorphic function f and any set S \subset \BbbC \cup \{ \infty \} ,
l \in \BbbN \cup \{ 0\} \cup \{ \infty \} , we define
Ef (S, l) =
\bigcup
a\in S
\{ (z, t) \in \BbbC \times \BbbN | f(z) = a with multiplicity p\} ,
where t = p if p \leq l and t = p+ 1 if p > l.
Two meromorphic functions f and g are said to share the set S with weight l, if Ef (S, l) =
= Eg(S, l).
Clearly Ef (S) = Ef (S,\infty ) and Ef (S) = Ef (S, 0).
Definition 1.6. Suppose 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.
ii) By N(r, a; f | \leq m) (resp.,N(r, a; f | \geq m), we denote the counting function of those a-
points of f whose multiplicities are not greater (resp., less) than m where each a-point is counted
according to its multiplicity.
Similarly, one can define N(r, a; f | \leq m) and N(r, a; f | \geq m) as the reduced counting function
of N(r, a; f | \leq m) and N(r, a; f | \geq m), respectively.
Analogously, N(r, a; f | < m), N(r, a; f | > m), N(r, a; f | < m) and N(r, a; f | > m) are
defined.
Definition 1.7. Suppose that f and g be two nonconstant meromorphic functions such that f
and g share (a, 0). Further, suppose that z0 be an a-point of f with multiplicity p, an a-point of g
with multiplicity q.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
1000 B. CHAKRABORTY AND S. CHAKRABORTY
i) We denote by NL(r, a; f), the reduced counting function of those a-points of f and g where
p > q.
ii) By N
1)
E (r, a; f), the counting function of those a-points of f and g where p = q = 1.
iii) By N
(2
E (r, a; f), the reduced counting function of those a-points of f and g where p = q \geq 2.
Similarly, we can define NL(r, a; g), N
1)
E (r, a; g), N
(2
E (r, a; g).
If f and g share (a,m), m \geq 1, then N
1)
E (r, a; f) = N(r, a; f | = 1).
Definition 1.8. We denote by N(r, a; f | = k), the reduced counting function of those a-points
of f whose multiplicities is exactly k, where k \geq 2 is an integer.
Definition 1.9 [10]. Let f, g share a value a IM. We denote by N\ast (r, a; f, g), the reduced
counting function of those a-points of f whose multiplicities differ from the multiplicities of the
corresponding a-points of g. Clearly
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).
2. Main results.
Theorem 2.1. Suppose that n(\geq 1) be a positive integer. Further suppose that S = \{ z :
P (z) = 0\} where the polynomial P (z) of degree n is defined by (1.1). Let f and g be two
nonconstant meromorphic functions satisfying Ef (S, 1) = Eg(S, 1). If n \geq 13, then f \equiv g.
Corollary 2.1. Suppose that n(\geq 1) be a positive integer. Further suppose that S = \{ z : P (z) =
= 0\} where the polynomial P (z) of degree n is defined by (1.1). Let f and g be two nonconstant
entire functions satisfying Ef (S, 1) = Eg(S, 1). If n \geq 8, then f \equiv g.
3. Lemmas. We define for any two nonconstant meromorphic functions f and g
Q(z) =
P (z) + c
c
, F = Q(f), G = Q(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)
.
Lemma 3.1 [14]. Let f be a nonconstant meromorphic function and
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)) = dT (r, f) + S(r, f),
where d = \mathrm{m}\mathrm{a}\mathrm{x}\{ n,m\} .
Lemma 3.2 [15]. For a nonconstant meromorphic function f,
T
\biggl(
r,
1
f
\biggr)
= T (r, f) +O(1),
where O(1) is a bounded quantity.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE CARDINALITY OF UNIQUE RANGE SETS WITH WEIGHT ONE 1001
Lemma 3.3 [15]. For a nonconstant meromorphic function f and for a complex number a \in
\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.
Lemma 3.4 [15, p. 23]. Suppose that f is a nonconstant meromorphic function in the complex
plane and a1, a2, . . . , aq are q(\geq 3) distinct values in \BbbC \cup \{ \infty \} . Then
(q - 2)T (r, f) <
q\sum
j=1
N(r, aj ; f) + S(r, f),
where S(r, f) is a quantity such that
S(r, f)
T (r, f)
\rightarrow 0 as r \rightarrow +\infty out side of a set E in (0,\infty ) with
finite linear measure.
A polynomial P (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, P (f) \equiv P (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. A polynomial P (z) is said to satisfy critical injection property
if P (\alpha ) \not = P (\beta ) for any two distinct zeros \alpha , \beta of the derivative P \prime (z).
Let P (z) be a monic polynomial without multiple zero whose derivatives has mutually distinct
k-zeros given by d1, d2, . . . , dk with multiplicities q1, q2, . . . , qk respectively. The following theorem
of Fujimoto helps us to find many uniqueness polynomials.
Lemma 3.5 [5]. Suppose that P (z) satisfy critical injection property. Then P (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. If k = 3 and \mathrm{m}\mathrm{a}\mathrm{x}\{ q1, q2, q3\} \geq
\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.
4. Proof of Theorem 2.1. By the assumption, it is clear that F and G shares (1, 1). Now we
consider two cases.
Case 1. First we assume that H \not \equiv 0. Then by simple calculations, we have
N(r,\infty ;H) \leq N(r, 0;F | \geq 2) +N(r, 0;G| \geq 2) +N(r,\infty ;F )+
+N(r,\infty ;G) +N\ast (r, 1;F,G) +N0(r, 0;F
\prime ) +N0(r, 0;G
\prime ),
where N0(r, 0;F
\prime ) is the reduced counting function of zeros of F \prime which is not zeros of F (F - 1).
Similarly N(r, 0;G\prime ) is defined. Thus,
N(r,\infty ;H) \leq N(r, 0; f) +N(r, 0; g) +N(r, 1; f)+
+N(r, 1; g) +N(r,\infty ; f) +N(r,\infty ; g)+
+N\ast (r, 1;F,G) +N \star (r, 0; f
\prime ) +N \star (r, 0; g
\prime ), (4.1)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
1002 B. CHAKRABORTY AND S. CHAKRABORTY
where N \star (r, 0; f
\prime ) is the reduced counting function of zeros of f \prime which is not zeros of f(f - 1)
and (F - 1) . We denote by N\ast (r, 1;F,G) the reduced counting function of those 1-points of F
whose multiplicities differ from the multiplicities of the corresponding 1-points of G. Clearly
N(r, 1;F | = 1) = N(r, 1;G| = 1) \leq N(r,\infty ;H), (4.2)
where N(r, 1;F | = 1) is the counting function of those simple 1-points of F which are also simple
1-points of G.
Now using the Second Fundamental Theorem, equations (4.1) and (4.2), we get
(n+ 1)(T (r, f) + T (r, g)) \leq
\leq N(r,\infty ; f) +N(r,\infty ; g) +N(r, 0; f) +N(r, 0; g)+
+N(r, 1; f) +N(r, 1; 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) \leq
\leq 2\{ N(r,\infty ; f) +N(r,\infty ; g) +N(r, 0; f) +N(r, 0; g)+
+N(r, 1; f) +N(r, 1; g)\} + \{ N(r, 1;F ) +N(r, 1;G) -
- N(r, 1;F | = 1)\} +N\ast (r, 1;F,G) + S(r, f) + S(r, g). (4.3)
Again
N(r, 1;F ) +N(r, 1;G) - N(r, 1;F | = 1) +N\ast (r, 1;F,G) \leq
\leq 1
2
\{ N(r, 1;F ) +N(r, 1;G)\} + 1
2
\{ N(r, 1;F | \geq 2) +N(r, 1; , G| \geq 2)\} \leq
\leq n
2
(T (r, f) + T (r, g)) +
1
2
\{ N(r, 0; f \prime | f \not = 0) +N(r, 0; g\prime | g \not = 0)\} . (4.4)
Using (4.3) and (4.4), we get \Bigl( n
2
- 3
\Bigr)
(T (r, f) + T (r, g)) \leq
\leq 2\{ N(r,\infty ; f) +N(r,\infty ; g)\} + 1
2
\biggl\{
N
\biggl(
r, 0;
f \prime
f
\biggr)
+N
\biggl(
r, 0;
g\prime
g
\biggr) \biggr\}
+
+S(r, f) + S(r, g) \leq
\leq 2\{ N(r,\infty ; f) +N(r,\infty ; g)\} + 1
2
\biggl\{
N
\biggl(
r,\infty ;
f \prime
f
\biggr)
+N
\biggl(
r,\infty ;
g\prime
g
\biggr) \biggr\}
+
+S(r, f) + S(r, g) \leq
\leq 5
2
\{ N(r,\infty ; f) +N(r,\infty ; g)\} + 1
2
\{ N(r, 0; f) +N(r, 0; g)\} + S(r, f) + S(r, g).
That is
(n - 7)(T (r, f) + T (r, g)) \leq 5\{ N(r,\infty ; f) +N(r,\infty ; g)\} + S(r, f) + S(r, g),
which is a contradiction if n \geq 13 (resp., 8) for meromorphic (resp., entire) case.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE CARDINALITY OF UNIQUE RANGE SETS WITH WEIGHT ONE 1003
Case 2. Next we assume that H \equiv 0. Now on integration two times, we have
F \equiv AG+B
CG+D
, (4.5)
where A,B,C,D are constant satisfying AD - BC \not = 0.
Thus, applying Lemma 3.1 in equation (4.5), we get
T (r, f) = T (r, g) +O(1). (4.6)
Next we consider the following two cases.
Subcase 2.1. First we assume that AC \not = 0. Then equation (4.5) can be written as
F - A
C
=
BC - AD
C(CG+D)
.
Thus,
N
\biggl(
r,
A
C
;F
\biggr)
= N(r,\infty ;G).
Now applying the Second Fundamental Theorem and equation (4.6), we obtain
nT (r, f) +O(1) = T (r, F ) \leq
\leq N(r,\infty ;F ) +N(r, 0;F ) +N
\biggl(
r,
A
C
;F
\biggr)
+ S(r, F ) \leq
\leq N(r,\infty ; f) +N(r, 0; f) + 2T (r, f) +N(r,\infty ; g) + S(r, f) \leq 5T (r, f) + S(r, f)
which is impossible as n > 5.
Subcase 2.2. Thus we consider AC = 0. Since AD - BC \not = 0, so A = C = 0 never occur.
Thus the following two subcases are obvious:
2.2.1. First we assume that A = 0 and C \not = 0. Then obviously B \not = 0. Hence, equation (4.5)
can be written as
F \equiv 1
\gamma G+ \delta
, (4.7)
where \gamma =
C
B
and \delta =
D
B
. Now we assume that there exist no z0 such that F (z0) = 1. Then
applying the Second Fundamental Theorem and using the equation (4.6), we have
T (r, F ) \leq N(r,\infty ;F ) +N(r, 0;F ) +N(r, 1;F ) + S(r, F ) \leq
\leq N(r,\infty ; f) +N(r, 0; f) + 2T (r, f) + S(r, f) \leq 4
n
T (r, F ) + S(r, F )
which is impossible as n > 4.
Thus, there exist atleast one z0 such that F (z0) = 1. Since F and G share 1, hence, \gamma + \delta = 1
with \gamma \not = 0. Thus, equation (4.7) becomes
F \equiv 1
\gamma G+ 1 - \gamma
,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
1004 B. CHAKRABORTY AND S. CHAKRABORTY
and
N
\biggl(
r, 0;G+
1 - \gamma
\gamma
\biggr)
= N(r,\infty ;F ).
If \gamma \not = 1, then the Second Fundamental Theorem and equation (4.6), imply
T (r,G) \leq N(r,\infty ;G) +N(r, 0;G) +N
\biggl(
r, 0;G+
1 - \gamma
\gamma
\biggr)
+ S(r,G) \leq
\leq N(r,\infty ; g) +N(r, 0; g) + 2T (r, g) +N(r,\infty ; f) + S(r, g) \leq 5
n
T (r, F ) + S(r, F ),
which is again impossible as n > 5. Hence, \gamma = 1, i.e., 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
, (4.8)
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).
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). Consequently (n - 2)t = ns gives
t \geq n
2
. So,
N(r, 0; f) \leq 2
n
N(r, 0; f).
Similar calculations are valid for g also. Again
N(r,\infty ; f) \leq N(r, 0; g) +
2\sum
i=0
N(r, \gamma i; g) \leq
\leq 2
n
N(r, 0; g) +
1
n
2\sum
i=0
N(r, \gamma i; g) \leq
4
n
T (r, g) +O(1).
Next we apply the Second Fundamental Theorem for the identity (4.8) and we get
2T (r, f) \leq N(r,\infty ; f) +N(r, 0; f) +
2\sum
i=0
N(r, \gamma i; f) + S(r, f) \leq
\leq 4
n
T (r, f) +
2
n
T (r, f) +
2
n
T (r, f) + S(r, f)
which is a contradiction as n \geq 5.
2.2.2. Thus we consider that A \not = 0 and C = 0. Then obviously D \not = 0 and the equation (4.5)
can be written as
F \equiv \lambda G+ \mu ,
where \lambda =
A
D
and \mu =
B
D
.
Then obviously there exist atleast one z0 such that F (z0) = 1, otherwise we arrived at a contra-
diction like previous subcase. Thus, \lambda + \mu = 1 with \lambda \not = 0. Hence,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE CARDINALITY OF UNIQUE RANGE SETS WITH WEIGHT ONE 1005
N
\biggl(
r, 0;G+
1 - \lambda
\lambda
\biggr)
= N(r, 0;F ).
If \lambda \not = 1, then using the Second Fundamental Theorem and equation (4.6), we obtain
T (r,G) \leq N(r,\infty ;G) +N(r, 0;G) +N
\biggl(
r, 0;G+
1 - \lambda
\lambda
\biggr)
+ S(r,G) \leq
\leq N(r,\infty ; g) +N(r, 0; g) + 2T (r, g) +N(r, 0; f) + 2T (r, f) + S(r, g) \leq 7
n
T (r,G) + S(r,G)
which is a contradiction as n > 7. Thus, \lambda = 1, i.e., F \equiv G, that is, 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) satisfy critical injection property. Thus, in view of Lemma 3.5, P (z) is
a uniqueness polynomial. Hence, f \equiv g.
Theorem 2.1 is proved.
References
1. S. Bartels, Meromorphic functions sharing a set with 17 elements ignoring multiplicities, Complex Variables, Theory
and Appl., 39, № 1, 85 – 92 (1999).
2. M. L. Fang, H. Guo, On unique range sets for meromorphic or entire functions, Acta Math. Sinica (N.S.), 14, № 4,
569 – 576 (1998).
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and Appl., 37, № 1, 185 – 193 (1998).
4. H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122, 1175 – 1203 (2000).
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Ky (1976), p. 51 – 67; Lect. Notes Math., 599, Springer, Berlin (1977).
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437 – 450 (1995).
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Received 29.05.17,
after revision — 08.12.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
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| id | umjimathkievua-article-6022 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:21Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/15/6ac874a84732c920dded6df06fd2f815.pdf |
| spelling | umjimathkievua-article-60222022-03-26T11:01:56Z On the cardinality of unique range sets with weight one On the cardinality of unique range sets with weight one On the cardinality of unique range sets with weight one Chakraborty, B. Chakraborty, S. Chakraborty, B. Chakraborty, S. UDC 517.9 Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\},$ if $E_{f}(S,l)=E_{g}(S,l),$ where $$$E_{f}(S,l)=\bigcup_{a \in S} \big \{(z,t) \in \mathbb{C}\times\mathbb{N} \bigm| f(z)=a \; \text{with multiplicity} \;p \big \},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p&gt;l.$ In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang [Indian J. &nbsp;Pure and Appl. &nbsp;Math., 31, No~4, 431–440 (2000)] by showing that there exist a finite set $S$ with 13 elements such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g.$ UDC 517.9 Про потужнiсть унiкального набору множин з одиничною вагою Двi мероморфнi функцiї $f$ i $g$ сумiсно мають множину $S\subset \mathbb{C}\cup\{\infty\}$ з одиничною вагою $l\in\mathbb{N}\cup\{0\}\cup\{\infty\},$ якщо $E_{f}(S,l)=E_{g}(S,l),$ де$$E_{f}(S,l)=\bigcup_{a \in S} \big \{(z,t) \in \mathbb{C}\times\mathbb{N} \bigm| f(z)=a \; \text{з кратністю} \;p \big \},$$ за умови, що $t=p$, якщо $p\leq l$, i $t=p+1$, якщо $p&gt;l.$У цiй роботi ми вдосконалюємо та доповнюємо результат L. W. Liao і C. C. Yang [Indian J. Pure and Appl. &nbsp;Math., 31, No~4, 431–440 (2000)], пропонуючи доведення того, що iснує скiнченна множина $S,$ що складається з 13 елементiв, така, що з $E_{f}(S,1)=E_{g}(S,1)$ випливає $f\equiv g.$ &nbsp; &nbsp; Institute of Mathematics, NAS of Ukraine 2020-07-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6022 10.37863/umzh.v72i7.6022 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 7 (2020); 997-1005 Український математичний журнал; Том 72 № 7 (2020); 997-1005 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6022/8734 |
| spellingShingle | Chakraborty, B. Chakraborty, S. Chakraborty, B. Chakraborty, S. On the cardinality of unique range sets with weight one |
| title | On the cardinality of unique range sets with weight one |
| title_alt | On the cardinality of unique range sets with weight one On the cardinality of unique range sets with weight one |
| title_full | On the cardinality of unique range sets with weight one |
| title_fullStr | On the cardinality of unique range sets with weight one |
| title_full_unstemmed | On the cardinality of unique range sets with weight one |
| title_short | On the cardinality of unique range sets with weight one |
| title_sort | on the cardinality of unique range sets with weight one |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6022 |
| work_keys_str_mv | AT chakrabortyb onthecardinalityofuniquerangesetswithweightone AT chakrabortys onthecardinalityofuniquerangesetswithweightone AT chakrabortyb onthecardinalityofuniquerangesetswithweightone AT chakrabortys onthecardinalityofuniquerangesetswithweightone |