Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field
UDC 517.53 First of all, we indicate a severe error in the analysis of the main results of both  Chakraborty [Ukr. Math. J., 72, No. 11, 1794–1806 (2021)] and Chakraborty–Chakraborty [Ukr. Math. J., 72, No. 7, 1164–1174 (2020)], to show that both these  papers cease to be t...
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| author | Banerjee, A. Maity, S. Banerjee, A. Maity, S. |
| author_facet | Banerjee, A. Maity, S. Banerjee, A. Maity, S. |
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| description | UDC 517.53
First of all, we indicate a severe error in the analysis of the main results of both  Chakraborty [Ukr. Math. J., 72, No. 11, 1794–1806 (2021)] and Chakraborty–Chakraborty [Ukr. Math. J., 72, No. 7, 1164–1174 (2020)], to show that both these  papers cease to be true.  Further, pertinent to the results of these two papers, we  deal with the unique range set of a meromorphic function over a non-Archimedean field with the smallest possible weights 0 and 1 under the aegis of its most  generalized form to improve the existing result. |
| doi_str_mv | 10.37863/umzh.v74i12.6717 |
| first_indexed | 2026-03-24T03:29:55Z |
| format | Article |
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DOI: 10.37863/umzh.v74i12.6717
UDC 517.53
A. Banerjee, S. Maity1 (Univ. Kalyani, West Bengal, India)
SUBSEQUENT INVESTIGATIONS OF THE LEAST CARDINALITIES
OF UNIQUE RANGE SET FOR TWO MINIMUM WEIGHTS
OVER A NON-ARCHIMEDEAN FIELD
ПОДАЛЬШI ДОСЛIДЖЕННЯ НАЙМЕНШИХ ПОТУЖНОСТЕЙ
МНОЖИНИ УНIКАЛЬНОСТI ЗА ДВОМА МIНIМАЛЬНИМИ ВАГАМИ
НАД НЕАРХIМЕДОВИМ ПОЛЕМ
First of all, we indicate a severe error in the analysis of the main results of both Chakraborty [Ukr. Math. J., 72, № 11,
1794 – 1806 (2021)] and Chakraborty – Chakraborty [Ukr. Math. J., 72, № 7, 1164 – 1174 (2020)], to show that both these
papers cease to be true. Further, pertinent to the results of these two papers, we deal with the unique range set of a
meromorphic function over a non-Archimedean field with the smallest possible weights 0 and 1 under the aegis of its most
generalized form to improve the existing result.
Насамперед вказано на грубу помилку в аналiзi основних результатiв, що наведенi в статтях Chakraborty [Ukr.
Math. J., 72, № 11, 1794 – 1806 (2021)] та Chakraborty – Chakraborty [Ukr. Math. J., 72, № 7, 1164 – 1174 (2020)], щоб
показати, що обидвi статтi втрачають силу. Далi, що стосується результатiв цих двох статей, розглянуто множину
унiкальностi мероморфної функцiї над неархiмедовим полем з найменшими можливими вагами 0 i 1 пiд егiдою
його найбiльш загальної форми для того, щоб покращити iснуючий результат.
1. Introduction and motivation. At the outset, we would like to mention that though the whole
paper has been oriented about uniqueness theory over non-Archimedean field but to enlighten some
important facts relevant to the focus of the paper we have to mention some terminologies of value
distribution theory over complex field available in the book [12]. Let z be a zero of f(z) - a = 0,
the multiplicity of z is denoted by w(a, f ; z). Let \scrM (\BbbC ) denotes the collection of all meromorphic
functions on \BbbC . For f \in \scrM (\BbbC ) and a \in \BbbC \cup \{ \infty \} we define
Ef (a) =
\bigl\{
(z, w(a, f ; z)) : z is zero of f(z) - a = 0
\bigr\}
.
In the case of ignoring multiplicities we denote the set by Ef (a). Let f, g \in \scrM (\BbbC ), we say
f and g share the value a CM (counting multiplicity) if Ef (a) = Eg(a) and share the value a IM
(ignoring multiplicity) if Ef (a) = Eg(a). Now, for f \in \scrM (\BbbC ) and S \subset \BbbC \cup \{ \infty \} , define
Ef (S) =
\bigcup
a\in S
\{ (z, w(a, f ; z)) : z is zero of f(z) - a = 0\} .
In the case of ignoring multiplicities we denote the set by Ef (S). Two functions f, g \in \scrM (\BbbC )
are said to share a set S CM (IM), if Ef (S) = Eg(S)(Ef (S) = Eg(S)).
The notion of weighted sharing of sets, introduced in [15], defined as follows:
1 Corresponding author, e-mail: sayantanmaity100@gmail.com.
c\bigcirc A. BANERJEE, S. MAITY, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12 1587
1588 A. BANERJEE, S. MAITY
Let k \in \BbbZ + \cup \{ \infty \} , the set of all a-points of f with multiplicity m is counted m times if
m \leq k and counted k + 1 times if m > k is denoted by Ek
f (a). For two functions f, g \in \scrM (\BbbC )
if Ek
f (a) = Ek
g (a), then we say f, g share the value a with weight k. We say f, g share the set S
with weight k if Ek
f (S) = Ek
g (S) for a set S \subset \BbbC \cup \{ \infty \} . We write f, g share (S, k) to mean that
f, g share the set S with weight k. In particular, if S = \{ a\} , then we write f, g share (a, k).
Definition 1.1 [5]. Let f, g be two meromorphic functions over \BbbC and S \subset \BbbC \cup \{ \infty \} . If
Ek
f (S) = Ek
g (S) implies f \equiv g, then S is called a unique range set for meromorphic functions with
weight k or in brief URSMk.
In particular, for k = \infty and 0 we write unique range set for meromorphic (entire) functions as
URSM (URSE) and URSM-IM (URSE-IM), respectively.
Over complex field, considering a new polynomial Frank – Reinders [10] obtained an URSM with
cardinality \geq 11 as follows:
Theorem A [10]. Let n \geq 11 be an integer and c \not = 0, 1 be a complex number. Consider the
polynomial
PFR(z) =
(n - 1)(n - 2)
2
zn - n(n - 2)zn - 1 +
n(n - 1)
2
zn - 2 - c. (1.1)
Then S = \{ z \in \BbbC | PFR(z) = 0\} is a URSM.
Let us denote
UM
CM = \{ S : S is URSM\} ,
UM
IM = \{ S : S is URSM -IM\} ,
\lambda M
CM = \mathrm{m}\mathrm{i}\mathrm{n}\{ n(S) : S \in UM
CM\} ,
and
\lambda M
IM = \mathrm{m}\mathrm{i}\mathrm{n}\{ n(S) : S \in UM
IM\} ,
where n(s) is the cardinality of S. Analogously for entire functions we can define UE
CM , UE
IM ,
\lambda E
CM , \lambda E
IM . In [17] (see Theorem 9), Li – Yang proved that \lambda M
CM \geq 5. Later in 2003, Yang – Yi
[20, p. 527] considered the following example to show that \lambda M
CM \geq 6.
Example 1.1. Let S = \{ aj , j = 1, 2, 3, 4, 5\} be an arbitrary subset of \BbbC with five distinct
elements and let f(z) = a5 +
1
ez + d
, g(z) = a5 +
1
(d - b1)(d - b2)e - z + d
provided that b1 +
+ b2 \not = b3 + b4, where bj =
1
aj - a5
, j = 1, 2, 3, 4, and d =
b3b4 - b1b2
b3 + b4 - b1 - b2
. In this case,
E\infty
f (S) = E\infty
g (S) but f \not \equiv g.
For entire functions, Li [16] obtained the following example to show that \lambda E
CM \geq 5.
Example 1.2. Let S = \{ aj ; j = 1, 2, 3, 4\} be an arbitrary subset of \BbbC with four distinct elements
and let f(z) = (d - a1)e
z + d and g(z) = (d - a2)e
- z + d provided that a1 + a2 \not = a3 + a4, where
d =
a3a4 - a1a2
a3 + a4 - a1 - a2
. It is easy to show that E\infty
f (S) = E\infty
g (S) but f \not \equiv g.
But for strictly IM sharing, finding of suitable examples is not so easy rather very complicated.
We can find only the following example to show that \lambda M
IM \geq 3.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SUBSEQUENT INVESTIGATIONS OF THE LEAST CARDINALITIES OF UNIQUE RANGE SET . . . 1589
Example 1.3. Let S = \{ 0, 1\} and f(z) =
e2z - 1
e2z + 1
, g(z) =
4e2z
(e2z + 1)2
. Notice that E0
f (S) =
= E0
g (S) but f \not \equiv g.
In the case of entire functions, if we consider f(z) = \mathrm{s}\mathrm{i}\mathrm{n} z, g(z) = \mathrm{c}\mathrm{o}\mathrm{s} z and S = \{ - 1, 0, 1\} ,
then E0
f (S) = E0
g (S), but f \not \equiv g. Thus, \lambda E
IM \geq 4.
From the above discussion we see that it is a very interesting question to find the minimum
cardinality of a URSM-IM. In this regard, in 1999, Bartels [6] considered a set whose elements are
the roots of PFR(z) to obtain the following result.
Theorem B [6]. Let n \geq 17 be an integer and c \not = 0, 1 be a complex number and consider
PFR(z) defined in Theorem A. Then S = \{ z \in \BbbC | PFR(z) = 0\} is a URSM-IM.
After that many researchers tried to reduce the lower bound of the cardinality of a URSM-IM,
but for a long time, nobody succeeded. Recently, Chakraborty [8] reduces the minimum cardinality
of the URSM-IM from 17 to 15 as follows:
Theorem C [8]. Let S = \{ z \in \BbbC | PFR(z) = 0\} , where PFR(z) is defined in (1.1). If n \geq 15,
then S is a URSM-IM.
Remark 1.1. We see that Theorem C is a huge improvement of Theorem B. But unfortunately
there is a serious error in the proof of Theorem C. In [8, p. 1805], the author uses the inequality
N(r, 1;F | \geq 2) \leq N(r, 0; f \prime | f \not = 0) (resp., N(r, 1;G | \geq 2) \leq N(r, 0; g\prime | g \not = 0)). But the
inequality is not true because, if z0 be an 1-point of F of multiplicity p(\geq 2) then z0 must be a zero
of f \prime of multiplicity p - 1. Actually the inequality would have been true had the author made the
estimations as 2N(r, 0; f \prime | f \not = 0) (resp., 2N(r, 0; g\prime | g \not = 0)) instead of N(r, 0; f \prime | f \not = 0) (resp.,
N(r, 0; g\prime | g \not = 0)). But in that case, the cardinality n of the set S becomes \geq 19. So by any means
the possible corrected version of the main theorem of [8] has no value in comparison to Theorem B.
Remark 1.2. If we closely study the proof of Theorem C, we notice that the Lemma 4.6 of [8]
plays an important role in the proof. However it can be noticed that, Lemma 2.2 of [4] exhibits better
inequality than Lemma 4.6 of [8]. So in the equation (5.5) of [8], if instead of Lemma 4.6 of [8], one
use Lemma 2.2 of [4], then proceeding in a similar manner as done in [8], we get
3
2
N(r, 1;F | \geq 2)
(resp.,
3
2
N(r, 1;G | \geq 2)) in place of N(r, 1;F | \geq 2) (resp., N(r, 1;G | \geq 2)) in [8, p. 1805].
Calculating similar procedure, we obtain n \geq 17, but this was already proved in Theorem B.
Thus we observe that, so far using the existing techniques and methods over \BbbC , it is a very
challenging question to reduce the minimum cardinality of URSM-IM from 17. Thus the following
question comes naturally.
Question 1.1. Can it be possible to find a unique range set with cardinality less than 17, if we
increase the weight of the sharing from 0 to 1?
Recently, Chakraborty – Chakraborty [9] answered the above question affirmatively and proved
that for n \geq 13, there is a URSM1 over \BbbC . The result is as follows:
Theorem D [9]. Suppose that n(\geq 1) is a positive integer. Further, suppose that S = \{ z \in
\in \BbbC | PFR(z) = 0\} , where the polynomial PFR(z) of degree n is defined by (1.1). Let f and g be
two nonconstant meromorphic functions such that f and g share (S, 1) and n \geq 13, then f \equiv g.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1590 A. BANERJEE, S. MAITY
Remark 1.3. Thus Theorem D says that S is a URSM1 with minimum cardinality 13. But the
authors did the same mistake in equation (4.4) (see [9, p. 1170]) as we have already pointed out
in Remark 1.1. The error could have been removed had the estimations of the counting functions
N(r, 1;F | \geq 2) (resp., N(r, 1;G | \geq 2)) be replace by N(r, 1;F | \geq 2) (resp., N(r, 1;G | \geq 2))
in equation (4.4) of [9], and in that case one can get n \geq 13. However, in 2016, as a consequence
of a main result of [3], the first author of this paper proved that the same set S as defined in the
last theorem is a URSM1 with cardinality \geq 12 (see Remark of [3, p. 205]). Thus we see that,
long before Chakraborty – Chakraborty’s [9] result a better result was already exhibited by Banerjee
[3]. Therefore, the result of the corrected version of Chakraborty – Chakraborty [9] is also redundant.
Actually, in equation (4.4), instead of N(r, 1;F | \geq 2) (resp., N(r, 1;G | \geq 2)) of [9] a better
estimation N(r, 1;F | \geq 3) (resp., N(r, 1;G | \geq 3)) can be used to get the cardinality \geq 12.
So, from the above discussion we see that over \BbbC the least cardinalities of URSM-IM and URSM1
are 17 and 12 respectively and there are no such fruitful method available in the literature to reduce
the same. So the next question can appear in one’s mind.
Question 1.2. Instead of \BbbC if we work on non-Archimedean field \BbbF , can it be possible to reduce
the minimum cardinality of URSM-IM and URSM1?
To seek the possible answer of Question 1.2 under the most generalized form of the range set is the
prime concern of the paper. Before approaching further, we recall some basics of non-Archimedean
field.
2. Basis of value distribution theory over non-Archimedean field. Throughout the paper we
consider \BbbF to be an algebraically closed non-Archimedean field with characteristic zero such that
it is complete for a nontrivial non-Archimedean absolute value. We denote by \mathrm{l}\mathrm{o}\mathrm{g} and \mathrm{l}\mathrm{n} as the
real logarithm of base p > 1 and e, respectively. Let us denote the collection of all meromorphic
functions on \BbbF by \scrM (\BbbF ) and \widetilde \BbbF = \BbbF \cup \{ \infty \} . The definition of CM (IM) sharing is similar as complex
field. The notion of weighted sharing over \BbbF was introduced by Meng – Liu [18] and it is similar as
over \BbbC . URSMk over \BbbF also can be defined analogously as Definition 1.1.
Definition 2.1. Let P (z) be a polynomial in \BbbF . If for any two nonconstant meromorphic func-
tions f and g, the condition P (f) \equiv cP (g) implies f \equiv g, where c is a non-zero constant, then P
is called a strong uniqueness polynomial for meromorphic functions or SUPM in brief.
To find the sufficient condition for a polynomial to be a SUPM, Fujimoto [11] introduced the
following definition and called it as “Property H” which was latter well-known as “Critical Injection
Property”.
Definition 2.2 [5]. Let P (z) be a polynomial such that P \prime (z) has l distinct zero namely
z1, z2, . . . , zl. If P (zi) \not = P (zj) for i \not = j, i, j \in \{ 1, 2, . . . , l\} , then P (z) is said to satisfy the
critical injection property.
Over the non-Archimedean field the same definitions of critical injection property can be given.
For basic terminologies of value distribution theory over non-Archimedean field, readers can
make a glance on [1, 2, 18]. Here we recall a few of them.
For a real constant \rho such that 0 < \rho \leq r, the counting function N(r, a; f) of f \in \scrM (\BbbF ) is
defined as follows:
N(r, a; f) =
1
\mathrm{l}\mathrm{n} p
r\int
\rho
n(t, a; f)
t
dt,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SUBSEQUENT INVESTIGATIONS OF THE LEAST CARDINALITIES OF UNIQUE RANGE SET . . . 1591
where n(t, a; f) is the number of solution (CM) of f(z) = a in the disk Dt = \{ z \in \BbbF : | z| \leq t\} .
For l \in \BbbZ +, define
Nl(r, a; f) =
1
\mathrm{l}\mathrm{n} p
r\int
\rho
nl(t, a; f)
t
dt,
where nl(t, a; f) =
\sum
| z| \leq r
\mathrm{m}\mathrm{i}\mathrm{n}\{ l, w(a, f ; z)\} . Thus N1(r, a; f) denotes the counting function of
a-points of f where multiplicity is counted only once, in short we call it “reduced counting function”.
Let us consider a nonconstant entire function f on \BbbF so that f has a power series expansion of the
form f =
\sum \infty
n=0
anz
n. For every r > 0, we define
| f | r = \mathrm{m}\mathrm{a}\mathrm{x}\{ | an| rn : 0 \leq n < \infty \} .
Next consider f to be a nonconstant meromorphic function over \BbbF . Thus f =
g
h
such that g, h
are entire functions in \BbbF and having no common zeros. We define | f | r =
| g| r
| h| r
. Define the proximity
function of f as follows:
m(r,\infty ; f) = \mathrm{l}\mathrm{o}\mathrm{g}+ | f | r = \mathrm{m}\mathrm{a}\mathrm{x}\{ 0, \mathrm{l}\mathrm{o}\mathrm{g} | f | r\} and m(r, a; f) = \mathrm{l}\mathrm{o}\mathrm{g}+
\bigm| \bigm| \bigm| 1
f - a
\bigm| \bigm| \bigm|
r
.
Note that w(a, f ; z) = w(0, g - ah; z), w(\infty , f ; z) = w(0, h; z), N(r, a; f) = N(r, 0; g - ah) and
N(r,\infty ; f) = N(r, 0;h). The Nevanlinna characteristic function is defined as
T (r, f) = \mathrm{m}\mathrm{a}\mathrm{x}\{ N(r,\infty ; f), N(r, 0; f)\} .
We write simply m(r, f) and N(r, f) instead of m(r,\infty ; f) and N(r,\infty ; f).
Definition 2.3. For a \in \widetilde \BbbF we denote by N(r, a; f | = 1) the counting function of simple a-
points of f. For k \in \BbbZ + we denote by N(r, a; f | \leq k)(N(r, a; f | \geq k)) the counting function of
those a-points of f whose multiplicities are not greater(less) than k where each a point is counted
according to its multiplicity. N1(r, a; f | \leq k)(N1(r, a; f | \geq k)) are defined similarly, where in
counting the a-points of f we ignore the multiplicities.
Definition 2.4. Let a \in \widetilde \BbbF , f and g be two nonconstant meromorphic functions such that f and
g share the value a IM. Let z1 be an a-point of f with multiplicity s and an a-point of g with
multiplicity t.
By NE
1 (r, a; f | = 1) we mean the reduced counting function of those a-points of f and g where
s = t = 1.
For k \in \BbbZ +, NE
1 (r, a; f | \geq k) denotes the reduced counting function of those a-points of f
and g where s = t \geq k.
By NL
1 (r, a; f) (NL
1 (r, a; g)) we mean the reduced counting function of those a-points of f and
g where s > t (t > s).
We denote by N\ast
1 (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. Note that N\ast
1 (r, a; f, g) =
= N\ast
1 (r, a; g, f) and N\ast
1 (r, a; f, g) = NL
1 (r, a; f) +NL
1 (r, a; g).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1592 A. BANERJEE, S. MAITY
3. Background and main results. In the perspective of Question 1.2, the next theorem states
that there is a URSM-IM over non-Archimedean field \BbbF of cardinality 16. Thus over \BbbF it is possible
to reduce the minimum cardinality of URSM-IM by 1 than the complex field.
Theorem E ([13], Theorem 3.47). Let S = \{ z \in \BbbF | PFR(z) = 0\} , where PFR(z) is defined in
(1.1). If n \geq 16, then S is a URSM-IM.
Now we introduce a new polynomial of degree m+ n+ 1 in the following manner:
P (z) =
n\sum
j=0
\biggl(
n
j
\biggr)
( - 1)j
m+ n+ 1 - j
zm+n+1 - jaj +
+
m\sum
i=1
n\sum
j=0
\biggl(
m
i
\biggr) \biggl(
n
j
\biggr)
( - 1)i+j
m+ n+ 1 - i - j
zm+n+1 - i - jajbi + c =
= Q(z) + c, (3.1)
where a and b be distinct such that a \in \BbbF \setminus \{ 0\} , b \in \BbbF , c \in \BbbF \setminus \{ - Q(a), - Q(b)\} . It is easy to verify
that
P \prime (z) = (z - a)n(z - b)m.
Remark 3.1. In (3.1), put a = 1, b = 0 and n = 2, the polynomial (3.1) reduces to
P1(z) =
zm+3
m+ 3
- 2
zm+2
m+ 2
+
zm+1
m+ 1
+ c,
where c \not = 0, - 2
(m+ 1)(m+ 2)(m+ 3)
. Multiplying P1(z) by
(m+ 1)(m+ 2)(m+ 3)
2
and putting
m+ 3 = t, we get
P2(z) =
(t - 1)(t - 2)
2
zt - t(t - 2)zt - 1 +
t(t - 1)
2
zt - 2 - d,
where d = - c
(m+ 1)(m+ 2)(m+ 3)
2
. As c \not = 0, - 2
(m+ 1)(m+ 2)(m+ 3)
it follows that d \not =
\not = 0, 1, and notice that P2(z) is same as PFR(z). Thus, P (z) as defined in (3.1), is a generalization
of the polynomial PFR(z).
Remark 3.2. The set of all zeros of P \prime (z) is \{ a, b\} . P (z) have only simple zeros since c \in
\in \BbbF \setminus \{ - Q(a), - Q(b)\} .
Remark 3.3. Notice that P (z) - P (b) = (z - b)m+1W1(z), where W1(b) \not = 0 and W1(z) has
no multiple zero. Similarly, P (z) - P (a) = (z - a)n+1W2(z), where W2(a) \not = 0 and W2(z) has no
multiple zero. If possible, let P (a) = P (b), then this implies (z - a)n+1W2(z) = (z - b)m+1W1(z).
As a \not = b so W2(z) has a factor (z - b)m+1, hence the degree of P (z) is at least m+ n+ 2, which
is a contradiction. Thus, P (a) \not = P (b). Therefore, P (z) is a critically injective polynomial.
From Remark 3.1 we see that as P (z) is a generalization of PFR(z), it will be natural to
investigate analogous results of Theorem E under P (z). In this respect we have the following result.
Theorem 3.1. Let f, g be two nonconstant meromorphic functions on \BbbF and m,n be two
positive integers such that n \geq 2, m \geq n+ 2 and m+ n \geq 15. Consider the polynomial (3.1), then
the set \widetilde S = \{ z \in \BbbF | P (z) = 0\} is URSM-IM.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SUBSEQUENT INVESTIGATIONS OF THE LEAST CARDINALITIES OF UNIQUE RANGE SET . . . 1593
From Theorem 3.1 we can deduce the following corollaries.
Corollary 3.1. (i) Let m \geq 13. Consider the polynomial
P1(z) =
zm+3
m+ 3
- 2
zm+2
m+ 2
+
zm+1
m+ 1
+ c,
where c \not = 0, - 2
(m+ 1)(m+ 2)(m+ 3)
. Then the set \widetilde S1 = \{ z \in \BbbF | P1(z) = 0\} is a URSM-IM.
(ii) Let t \geq 16. Consider the polynomial
P2(z) =
(t - 1)(t - 2)
2
zt - t(t - 2)zt - 1 +
t(t - 1)
2
zt - 2 - d,
where d \not = 0, 1. Then the set \widetilde S2 = \{ z \in \BbbF | P2(z) = 0\} is a URSM-IM.
Remark 3.4. Theorem 3.1 exhibits a URSM-IM of cardinality \geq 16.
Remark 3.5. Corollary 3.1(ii) is actually Theorem E. So Theorem 3.1extends Theorem E at a
large extent.
In the next theorem instead of IM sharing we increase the weight only by 1 to investigate the
least cardinality of the range set, under the periphery of the smallest positive integer weight.
Theorem 3.2. Let f, g be two nonconstant meromorphic functions on \BbbF and m,n be two
positive integers such that n \geq 2, m \geq n+ 2 and m+ n \geq 10. Consider the polynomial (3.1), then
the set \widetilde S = \{ z \in \BbbF | P (z) = 0\} is URSM1.
From Theorem 3.2, it is seen that the cardinality of URSM1 is \geq 11. Thus, for non-Archimedean
field cardinality of URSM1 reduces by 1 than complex field.
Corollary 3.2. (i) Let m \geq 8. Consider the polynomial P1(z) as in Corollary 3.1. Then the set\widetilde S1 = \{ z \in \BbbF | P1(z) = 0\} is a URSM1.
(ii) Let t \geq 11. Consider the polynomial P2(z) as in Corollary 3.1. Then the set \widetilde S2 = \{ z \in
\in \BbbF | P2(z) = 0\} is a URSM1.
4. Lemmas.
Lemma 4.1 [13]. Let f(z) be a nonconstant meromorphic function on \BbbF and a1, a2, . . . , an \in \widetilde \BbbF
are distinct points. Then
(n - 2)T (r, f) \leq
n\sum
i=1
N1(r, ai; f) - N0
\bigl(
r, 0; f \prime \bigr) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1),
where N0(r, 0; f \prime ) denotes the counting function of zeros of f \prime which are not ai, i = 1, 2, . . . , n,
points of f.
Lemma 4.2 [7]. Let f(z) be a nonconstant meromorphic function on \BbbF and Q(z) be a polyno-
mial of degree n over \BbbF . Then T (r,Q(f)) = nT (r, f) +O(1).
The next lemma follows from the equivalence of (i) and (iv) of Theorem 1 of Wang [19].
Lemma 4.3 [19]. Let f, g be two nonconstant meromorphic functions on \BbbF and P (z) be a
critically injective polynomial such that derivative of P (z) is of the form (z - \alpha )m(z - \beta )n and let
\mathrm{m}\mathrm{i}\mathrm{n}\{ m,n\} \geq 2. If P (f) \equiv P (g) then f \equiv g.
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1594 A. BANERJEE, S. MAITY
Lemma 4.4 [14]. Let f, g be two nonconstant meromorphic functions on \BbbF and P (z) be a
polynomial with no multiple zero and derivative of P (z) is of the form (z - \alpha )m(z - \beta )n, also let
\mathrm{m}\mathrm{i}\mathrm{n}\{ m,n\} \geq 2. Assume that there exist constants c1 \not = 0 and c2 such that
1
P (f)
=
c1
P (g)
+ c2.
Then c2 = 0.
Lemma 4.5. Let N(r, 0; f \prime | f \not = 0) denotes the counting function of zeros of f \prime which are not
the zeros of f. Then
N(r, 0; f \prime | f \not = 0) \leq N1(r,\infty ; f) +N1(r, 0; f) +O(1).
Proof. Using the lemma of logarithmic derivative, we get
N(r, 0; f \prime | f \not = 0) \leq N
\biggl(
r, 0;
f \prime
f
\biggr)
\leq
\leq T
\biggl(
r,
f \prime
f
\biggr)
\leq
\leq N
\biggl(
r,
f \prime
f
\biggr)
+m
\biggl(
r,
f \prime
f
\biggr)
+O(1) \leq
\leq N1(r,\infty ; f) +N1(r, 0; f) +O(1).
Now let us consider two nonconstant meromorphic functions \scrF and \scrG on \BbbF such that \scrF = P (f)
and \scrG = P (g), where P (z) is defined as in (3.1). Besides this we also consider a function \scrH as
follows:
\scrH =
\biggl(
\scrF \prime \prime
\scrF \prime -
2\scrF \prime
\scrF
\biggr)
-
\biggl(
\scrG \prime \prime
\scrG \prime -
2\scrG \prime
\scrG
\biggr)
. (4.1)
Lemma 4.6. Let \scrH \not \equiv 0 and \scrF , \scrG share (0, 0), then
NE(r, 0;\scrF | = 1) = NE(r, 0;\scrG | = 1) \leq N(r,\infty ;\scrH ) +O(1).
Proof. As \scrF and \scrG share (0, 0), so each simple zero of \scrF is also simple zero of \scrG and vice
versa. Now each simple zero of \scrF (i.e., simple zero of \scrG ) is a zero of \scrH . Note that m(r,\scrH ) = O(1).
Hence,
NE(r, 0;\scrF | = 1) = NE(r, 0;\scrG | = 1) \leq N(r, 0;\scrH ) \leq T (r,\scrH ) \leq N(r,\infty ;\scrH ) +O(1).
Lemma 4.7. Let \widetilde S = \{ z \in \BbbF | P (z) = 0\} , where P (z) is defined as in (3.1). Let \scrH \not \equiv 0 and f,
g be any two nonconstant meromorphic functions on \BbbF such that E0
f (
\widetilde S) = E0
g (
\widetilde S). Then
N1(r,\infty ;\scrH ) \leq N1(r,\infty ; f) +N1(r,\infty ; g) +N1(r, a; f) +N1(r, a; g) +N1(r, b; f) +
+ N1(r, b; g) +N\ast
1 (r, 0;\scrF ,\scrG ) +N0
1
\bigl(
r, 0; f \prime \bigr) +N0
1
\bigl(
r, 0; g\prime
\bigr)
,
where N0
1 (r, 0; f
\prime ) denotes reduced counting function of those zeros of f \prime which are not zeros of
\scrF (f - a)(f - b) and N0
1 (r, 0; g
\prime ) denotes similar counting function.
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Proof. Note that \scrF \prime = P \prime (f) = (f - a)n(f - b)mf \prime . The lemma directly follows by calculating
all the possible poles of \scrH and observe that all poles of \scrH are simple.
Lemma 4.8. Let m, n be two positive integers such that n \geq 2 and m \geq n + 2. Consider the
polynomial P (z) as defined in (3.1), then P (z) is a SUPM.
Proof. Consider P (z) as defined in (3.1). In view of Remarks 3.2 and 3.3 we see that P (z) is a
critically injective polynomial whose all zeros are simple. Let us assume
P (f) \equiv AP (g), (4.2)
for some constant A \not = 0. By Lemma 4.2 and (4.2) we get
T (r, f) = T (r, g) +O(1). (4.3)
Case 1. Let P (b) \not = 1. Now suppose A \not = 1.
Subcase 1.1. First assume A = P (a). From (4.2)
P (f) - P (a) \equiv P (a)(P (g) - 1). (4.4)
Consider the polynomial P (z) - 1. Note that P (a) - 1 \not = 0 as A = P (a) \not = 1 and P (b) - 1 \not = 0. So
all zeros of P (z) - 1 are simple. Let us denote those simple zeros by \alpha j , j = 1, 2, . . . ,m+n+1. On
the other hand, P (z) - P (a) = (z - a)n+1W2(z), where W2(z) has no multiple zero and W2(a) \not = 0.
So P (z) - P (a) has one zero at a with multiplicity n + 1, and suppose the other simple zeros are
\beta j , j = 1, 2, . . . ,m. So from (4.4), we have
N1(r, a; f) +
m\sum
j=1
N1(r, \beta j ; f) =
m+n+1\sum
j=1
N1(r, \alpha j ; g). (4.5)
Using Lemma 4.1 and the equation (4.5), we obtain
(m+ n - 1)T (r, g) \leq
m+n+1\sum
j=1
N1(r, \alpha j ; g) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) =
= N1(r, a; f) +
m\sum
j=1
N1(r, \beta j ; f) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq (m+ 1)T (r, f) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1). (4.6)
Now, by (4.3) and (4.6), we get
(n - 2)T (r, g) + \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1),
this is a contradiction as n \geq 2.
Subcase 1.2. Next assume A \not = P (a). From (4.2)
P (f) - AP (b) \equiv A(P (g) - P (b)). (4.7)
Now we consider the following two subcases.
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1596 A. BANERJEE, S. MAITY
Subcase 1.2.1. First assume P (a) \not = AP (b). Consider the polynomial P (z) - AP (b). Now as
c \not = - Q(b), i.e., P (b) \not = 0 it follows that P (b) - AP (b) = P (b)(1 - A) \not = 0. So all zeros of
P (z) - AP (b) are simple and denote them by \zeta j , j = 1, 2, . . . ,m + n + 1. On the other hand,
from the discussion of Remark 3.3 we have P (z) - P (b) has one zero at b with multiplicity m+ 1
and other simple zeros are \eta j , j = 1, 2, . . . , n. Now proceeding similarly as (4.5) and (4.6), we get
(m - 2)T (r, f) + \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1), but this is a contradiction as n \geq 2 and m \geq n+ 2.
Subcase 1.2.2. Next we assume P (a) = AP (b). Consider the polynomial P (z) - AP (b). Note
that a is a zero of P (z) - AP (b) with multiplicity n + 1 and other zeros are simple say \lambda j ,
j = 1, 2, . . . ,m. Next for the polynomial P (z) - P (b), b is a zero of multiplicity m + 1 and all
other zeros are simple namely \eta j , j = 1, 2, . . . , n. Again proceeding in a similar manner as in (4.5)
and (4.6), we deduce (m - n - 2)T (r, g) + \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1), is a contradiction as m \geq n+ 2.
Thus, from Case 1, we conclude that A = 1. Hence, from (4.2), we obtain P (f) \equiv P (g). Now
applying Lemma 4.3, we get f \equiv g.
Case 2. Let P (b) = 1. Now suppose A \not = 1.
Subcase 2.1. First assume A = P (a). From (4.2)
P (f) - 1 \equiv P (a)
\biggl(
P (g) - 1
P (a)
\biggr)
. (4.8)
Consider the polynomial P (z) - 1
P (a)
. Note that P (a) \not = 1.
Subcase 2.1.1. Let P (a) \not = - 1. Hence P (a) - 1
P (a)
\not = 0 as P (a) \not = 1, - 1. Also P (b) -
- 1
P (a)
= 1 - 1
P (a)
\not = 0. So all the zeros of P (z) - 1
P (a)
are simple. Let us denote them by \gamma j ,
j = 1, 2, . . . ,m + n + 1. On the other hand, as P (b) = 1, P (z) - 1 has only one multiple zero
at b with multiplicity m + 1 and remaining all zeros are simple namely \delta j , j = 1, 2, . . . , n. Now
using the similar steps as in (4.5) and (4.6), we obtain (m - 2)T (r, g) + \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1), which is a
contradiction as n \geq 2 and m \geq n+ 2.
Subcase 2.1.2. Let P (a) = - 1. As P (a) - 1
P (a)
= 0, so a is a zero of P (z) - 1
P (a)
with
multiplicity n + 1, and other zeros are simple say \theta j , j = 1, 2, . . . ,m. On the other hand, as
P (b) = 1, P (z) - 1 has only one multiple zero at b with multiplicity m + 1 and remaining all
zeros are simple say \delta j , j = 1, 2, . . . , n. Now proceeding similarly as in (4.5) and (4.6), we obtain
(m - n - 2)T (r, g) + \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1), which is also contradiction as m \geq n+ 2.
Subcase 2.2. Let A \not = P (a). From (4.2)
P (f) - A \equiv A(P (g) - 1). (4.9)
Note that, by the assumption of Case 2, A \not = P (a) and we also have A \not = 1, so P (a) - A \not = 0 and
P (b) - A = 1 - A \not = 0. Thus, all zeros of P (z) - A are simple namely \mu j , j = 1, 2, . . . ,m+n+1.
On the other hand, P (z) - 1 = P (z) - P (b) has a multiple zero at b with multiplicity m + 1 and
rest all zeros are simple namely \eta j , j = 1, 2, . . . , n. Again by the same argument as used in (4.5)
and (4.6), we get (m - 2)T (r, f) + \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1), this is a contradiction as n \geq 2 and m \geq n+ 2.
Thus, from Case 2, it is clear that A = 1. Therefore, from (4.2), we get P (f) \equiv P (g). Now
applying Lemma 4.3, we conclude f \equiv g.
Therefore, from both Cases 1 and 2, we get that P (z) is a SUPM.
Lemma 4.8 is proved.
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SUBSEQUENT INVESTIGATIONS OF THE LEAST CARDINALITIES OF UNIQUE RANGE SET . . . 1597
5. Proofs of the theorems.
Proof of Theorem 3.1. Let \widetilde S = \{ z \in \BbbF | P (z) = 0\} . Consider two functions \scrF := P (f) and
\scrG := P (g).
Case 1. First assume \scrH \not \equiv 0. As E0
f (
\widetilde S) = E0
g (
\widetilde S), \scrF and \scrG share (0, 0). By using Lemmas 4.1,
4.2, 4.6, 4.7 and Definition 2.4, we deduce
(m+ n+ 2)T (r, f) \leq
\leq N1(r,\infty ; f) +N1(r, 0;\scrF ) +N1(r, a; f) +N1(r, b; f) - N0
\bigl(
r, 0; f \prime \bigr) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq N1(r,\infty ; f) +NE
1 (r, 0;\scrF | = 1) +NL
1 (r, 0;\scrF ) +NL
1 (r, 0;\scrG ) +NE
1 (r, 0;\scrF | \geq 2) +
+ N1(r, a; f) +N1(r, b; f) - N0
\bigl(
r, 0; f \prime \bigr) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq N1(r,\infty ; f) +N1(r,\infty ;\scrH ) +NL
1 (r, 0;\scrF ) +NL
1 (r, 0;\scrG ) +NE
1 (r, 0;\scrF | \geq 2) +
+ N1(r, a; f) +N1(r, b; f) - N0
\bigl(
r, 0; f \prime \bigr) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq 2N1(r,\infty ; f) +N1(r,\infty ; g) + 2N1(r, a; f) + 2N1(r, b; f) +N1(r, a; g) +N1(r, b; g) +
+ N0
1
\bigl(
r, 0; g\prime
\bigr)
+NE
1 (r, 0;\scrF | \geq 2) +N\ast
1 (r, 0;\scrF ,\scrG ) +NL
1 (r, 0;\scrF ) +NL
1 (r, 0;\scrG ) -
- \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq 2N1(r,\infty ; f) +N1(r,\infty ; g) + 2N1(r, a; f) + 2N1(r, b; f) +N1(r, a; g) +N1(r, b; g) +
+ N0
1
\bigl(
r, 0; g\prime
\bigr)
+NE
1 (r, 0;\scrF | \geq 2) + 2NL
1 (r, 0;\scrF ) + 2NL
1 (r, 0;\scrG ) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1). (5.1)
Now we deduce
N0
1
\bigl(
r, 0; g\prime
\bigr)
+NE
1 (r, 0;\scrF | \geq 2) + 2NL
1 (r, 0;\scrF ) + 2NL
1 (r, 0;\scrG ) \leq
\leq N0
1
\bigl(
r, 0; g\prime
\bigr)
+NE
1 (r, 0;\scrG | \geq 2) + 2NL
1 (r, 0;\scrG ) + 2NL
1 (r, 0;\scrF ) \leq
\leq N0
1
\bigl(
r, 0; g\prime
\bigr)
+N1(r, 0;\scrG | \geq 2) +NL
1 (r, 0;\scrG ) + 2NL
1 (r, 0;\scrF ) \leq
\leq N(r, 0; g\prime | g \not = 0) +NL
1 (r, 0;\scrG ) + 2NL
1 (r, 0;\scrF ). (5.2)
By using Lemma 4.5, we get
NL
1 (r, 0;\scrG ) \leq N1(r, 0;\scrG | \geq 2) \leq
\leq N(r, 0; g\prime | g \not = 0) \leq
\leq N1(r,\infty ; g) +N1(r, 0; g) +O(1)
and similarly NL
1 (r, 0;\scrF ) \leq N1(r,\infty ; f) +N1(r, 0; f) +O(1) holds. Thus, from (5.2),
N0
1
\bigl(
r, 0; g\prime
\bigr)
+NE
1 (r, 0;\scrF | \geq 2) + 2NL
1 (r, 0;\scrF ) + 2NL
1 (r, 0;\scrG ) \leq
\leq 2\{ N1(r,\infty ; g) +N1(r, 0; g) +N1(r,\infty ; f) +N1(r, 0; f)\} +O(1). (5.3)
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1598 A. BANERJEE, S. MAITY
Combining (5.1) and (5.3), we obtain
(m+ n+ 2)T (r, f) \leq
\leq 4N1(r,\infty ; f) + 3N1(r,\infty ; g) + 2N1(r, a; f) + 2N1(r, b; f) +N1(r, a; g) +N1(r, b; g) +
+ 2N1(r, 0; f) + 2N1(r, 0; g) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq 10T (r, f) + 7T (r, g) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1). (5.4)
Similarly, we can have
(m+ n+ 2)T (r, g) \leq 10T (r, g) + 7T (r, f) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1). (5.5)
Thus, adding (5.4) and (5.5), we get
(m+ n - 15)(T (r, f) + T (r, g)) + 2 \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1),
which is a contradiction as m+ n \geq 15.
Case 2. Now consider the case \scrH \equiv 0. Integrating (4.1), we obtain
1
\scrF
\equiv c1
\scrG
+ c2,
where c1 \not = 0, c2 are two constants. From Lemma 4.4 we get c2 = 0. This implies P (f) \equiv 1
c1
P (g).
Now by Lemma 4.8 we have that P (z) is a SUPM, therefore f \equiv g.
Theorem 3.1 is proved.
Proof of Corollary 3.1. (i) From Remark 3.1 we know that putting a = 1, b = 0 and n = 2 the
polynomial (3.1) reduces to
P1(z) =
zm+3
m+ 3
- 2
zm+2
m+ 2
+
zm+1
m+ 1
+ c,
where c \not = 0, - 2
(m+ 1)(m+ 2)(m+ 3)
. By Theorem 3.1 we get, when m \geq 13, then the set \widetilde S1
becomes a URSM-IM. Therefore, for m \geq 13 and c \not = 0, - 2
(m+ 1)(m+ 2)(m+ 3)
, the set \widetilde S1 is
URSM-IM.
(ii) In the proof of (i), assuming m+ 3 = t, we obtain
P1(z) =
2
t(t - 1)(t - 2)
\biggl[
(t - 1)(t - 2)
2
zt - t(t - 2)zt - 1 +
t(t - 1)
2
zt - 2 - d
\biggr]
=
=
2
t(t - 1)(t - 2)
P2(z),
where d = - c
t(t - 1)(t - 2)
2
. Noticing the fact that t \geq 16, from (i) we see that if c \not = 0,
- 2
t(t - 1)(t - 2)
, P1(z) is a SUPM and this implies, whenever d \not = 0, 1; P2(z) is also a SUPM.
Therefore, for t \geq 16 and d \not = 0, 1, the set \widetilde S2 is URSM-IM.
Corollary 3.1 is proved.
Proof of Theorem 3.2. Let \widetilde S = \{ z \in \BbbF | P (z) = 0\} . Consider two functions \scrF and \scrG as
Theorem 3.1.
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SUBSEQUENT INVESTIGATIONS OF THE LEAST CARDINALITIES OF UNIQUE RANGE SET . . . 1599
Case 1. First assume \scrH \not \equiv 0. Since E1
f (
\widetilde S) = E1
g (
\widetilde S), then \scrF and \scrG share (0, 1), this implies
NE(r, 0;\scrF | = 1) = N(r, 0;\scrF | = 1). By using Lemmas 4.1, 4.2, 4.6, 4.7 and Definition 2.4, we
get
(m+ n+ 2)T (r, f) \leq
\leq N1(r,\infty ; f) +N1(r, 0;\scrF ) +N1(r, a; f) +N1(r, b; f) - N0
\bigl(
r, 0; f \prime \bigr) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq N1(r,\infty ; f) +N1(r, 0;\scrF | = 1) +N1(r, 0;\scrF | \geq 2) +N1(r, a; f) +N1(r, b; f) -
- N0
\bigl(
r, 0; f \prime \bigr) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq N1(r,\infty ; f) +N1(r,\infty ;\scrH ) +N1(r, 0;\scrG | \geq 2) +N1(r, a; f) +N1(r, b; f) -
- N0
\bigl(
r, 0; f \prime \bigr) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq 2N1(r,\infty ; f) +N1(r,\infty ; g) + 2N1(r, a; f) + 2N1(r, b; f) +N1(r, a; g) +N1(r, b; g) +
+ N0
1
\bigl(
r, 0; g\prime
\bigr)
+N1(r, 0;\scrG | \geq 2) +N\ast
1 (r, 0;\scrF ,\scrG ) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1). (5.6)
Now using Lemma 4.5, we deduce
N0
1
\bigl(
r, 0; g\prime
\bigr)
+N1(r, 0;\scrG | \geq 2) +N\ast
1 (r, 0;\scrF ,\scrG ) \leq
\leq N0
1
\bigl(
r, 0; g\prime
\bigr)
+N1(r, 0;\scrG | \geq 2) +N1(r, 0;\scrG | \geq 3) +N1(r, 0;\scrF | \geq 3) \leq
\leq N(r, 0; g\prime | g \not = 0) +
1
2
N(r, 0; f \prime | f \not = 0) \leq
\leq N1(r,\infty ; g) +N1(r, 0; g) +
1
2
\{ N1(r,\infty ; f) +N1(r, 0; f)\} +O(1). (5.7)
Combining (5.6) and (5.7), we have
(m+ n+ 2)T (r, f) \leq
\leq 5
2
N1(r,\infty ; f) + 2N1(r,\infty ; g) + 2N1(r, a; f) + 2N1(r, b; f) +N1(r, a; g) +N1(r, b; g) +
+
1
2
N1(r, 0; f) +N1(r, 0; g) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1) \leq
\leq 7T (r, f) + 5T (r, g) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1). (5.8)
Similarly, we can obtain
(m+ n+ 2)T (r, g) \leq 7T (r, g) + 5T (r, f) - \mathrm{l}\mathrm{o}\mathrm{g} r +O(1). (5.9)
Adding (5.8) and (5.9), we get
(m+ n - 10)(T (r, f) + T (r, g)) + 2 \mathrm{l}\mathrm{o}\mathrm{g} r \leq O(1),
this is a contradiction as m+ n \geq 10.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1600 A. BANERJEE, S. MAITY
Case 2. Similar as Case 2 of Theorem 3.1 we get f \equiv g.
Theorem 3.2 is proved.
Proof of Corollary 3.2. We omit the proof as the same can be carried out in the line of proof of
Corollary 3.1.
6. Acknowledgements. Abhijit Banerjee is thankful to DST-PURSE II programme for financial
assistance. Sayantan Maity is thankful to Council of Scientific and Industrial Research (India) for
their financial support under File No. 09/106(0191)/2019-EMR-I.
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Received 02.05.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
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| id | umjimathkievua-article-6717 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:29:55Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/5a/a76b892af9ddc923399b85c3f685a85a.pdf |
| spelling | umjimathkievua-article-67172023-01-23T14:02:45Z Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field Banerjee, A. Maity, S. Banerjee, A. Maity, S. Non-Archimedean meromorphic function unique range set UDC 517.53 First of all, we indicate a severe error in the analysis of the main results of both&nbsp; Chakraborty [Ukr. Math. J., 72, No. 11, 1794–1806 (2021)] and Chakraborty–Chakraborty [Ukr. Math. J., 72, No. 7, 1164–1174 (2020)], to show that both these&nbsp; papers cease to be true.&nbsp;&nbsp;Further, pertinent to the results of these two papers, we&nbsp; deal with the unique range set of a meromorphic function over a non-Archimedean field with the smallest possible weights 0 and 1 under the aegis of its most&nbsp; generalized form to improve the existing result. УДК 517.53 Подальші дослідження найменших потужностей множини унікальності за двома мінімальними &nbsp;вагами над неархімедовим полем&nbsp; Насамперед вказано на грубу помилку в аналізі основних результатів, що наведені в статтях Chakraborty [Ukr. Math. J., 72, № 711, 1794–1806 (2021)] та Chakraborty–Chakraborty [Ukr. Math. J., 72, № 77, 1164–1174 (2020)], щоб показати, що обидві статті втрачають силу.&nbsp;&nbsp;Далі, що стосується результатів&nbsp; цих двох статей, розглянуто&nbsp; множину унікальності мероморфної функції над неархімедовим полем з найменшими можливими вагами 0 і 1 під егідою його найбільш загальної форми для того, щоб покращити існуючий результат. Institute of Mathematics, NAS of Ukraine 2023-01-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6717 10.37863/umzh.v74i12.6717 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 12 (2022); 1587 - 1600 Український математичний журнал; Том 74 № 12 (2022); 1587 - 1600 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6717/9338 Copyright (c) 2023 Abhijit Banerjee, SAYANTAN MAITY |
| spellingShingle | Banerjee, A. Maity, S. Banerjee, A. Maity, S. Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field |
| title | Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field |
| title_alt | Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field |
| title_full | Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field |
| title_fullStr | Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field |
| title_full_unstemmed | Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field |
| title_short | Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field |
| title_sort | subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-archimedean field |
| topic_facet | Non-Archimedean meromorphic function unique range set |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6717 |
| work_keys_str_mv | AT banerjeea subsequentinvestigationsoftheleastcardinalitiesofuniquerangesetfortwominimumweightsoveranonarchimedeanfield AT maitys subsequentinvestigationsoftheleastcardinalitiesofuniquerangesetfortwominimumweightsoveranonarchimedeanfield AT banerjeea subsequentinvestigationsoftheleastcardinalitiesofuniquerangesetfortwominimumweightsoveranonarchimedeanfield AT maitys subsequentinvestigationsoftheleastcardinalitiesofuniquerangesetfortwominimumweightsoveranonarchimedeanfield |