Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials
We discuss the value-sharing problem, versions of the Hayman conjecture, and the uniqueness problem for p-adic meromorphic functions and their difference operators and difference polynomials.
Saved in:
| Date: | 2012 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2012
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2562 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508479589974016 |
|---|---|
| author | An, Vu Hoai Khoai, Ha Huy Ан, Ву Гоай Хоаї, Га Гуй |
| author_facet | An, Vu Hoai Khoai, Ha Huy Ан, Ву Гоай Хоаї, Га Гуй |
| author_sort | An, Vu Hoai |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:29:46Z |
| description | We discuss the value-sharing problem, versions of the Hayman conjecture, and the uniqueness problem for p-adic
meromorphic functions and their difference operators and difference polynomials. |
| first_indexed | 2026-03-24T02:25:52Z |
| format | Article |
| fulltext |
UDC 517.9
Vu Hoai An (Hai Duong College, Hai Duong, Vietnam),
Ha Huy Khoai (Ins. Math., Hanoi, Vietnam)
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS
AND THEIR DIFFERENCE OPERATORS AND DIFFERENCE POLYNOMIALS *
ЗАДАЧА ПРО СПIЛЬНI ЗНАЧЕННЯ ДЛЯ p-АДИЧНИХ МЕРОМОРФНИХ
ФУНКЦIЙ ТА ЇХ РIЗНИЦЕВИХ ОПЕРАТОРIВ I РIЗНИЦЕВИХ ПОЛIНОМIВ
We discuss the value-sharing problem, versions of the Hayman conjecture, and the uniqueness problem for p-adic
meromorphic functions and their difference operators and difference polynomials.
Дослiджено питання про спiльнi значення i єдинiсть та аналоги гiпотези Хеймена для p-адичних мероморфних
функцiй та їх рiзницевих операторiв i рiзницевих полiномiв.
1. Introduction. The problem of determining a meromorphic (or entire) function on C by its sing-
le pre-images, counting multiplicities, of finite sets is an important one and it has been studied by
many mathematicians. For instance, in 1921 G. Polya showed that an entire function on C is deter-
mined by the inverse images, counting multiplicities, of three distinct non-omitted values. In 1926,
R. Nevanlinna showed that a meromorphic function on the complex plane is uniquely determined by
the inverse images, ignoring multiplicities, of 5 distinct values.
In [16] Hayman proved the following well-known result:
Theorem 1.1. Let f be a meromorphic function on C. If f(z) 6= 0 and f (k)(z) 6= 1 for some
fixed positive integer k and for all z ∈ C, then f is constant.
Hayman also proposed the following conjecture (see [16]).
Hayman Conjecture. If an entire function f satisfies fn(z)f ′(z) 6= 1 for a positive integer n
and all z ∈ C, then f is a constant.
It has been verified for transcendental entire functions by Hayman himself for n > 1 [16], and
by Clunie for n ≥ 1 [5]. These results and some related problems caused increasingly attentions to
the value-sharing problem of meromorphic functions and their derivatives (see [2, 4, 19, 21]).
In 1997 Yang and Hua [23] studied the unicity problem for meromorphic functions and differential
monomials of the form fnf ′, when they share only one value, and obtained the following theorem.
Theorem 1.2. Let f and g be two non-constant meromorphic functions, let n ≥ 11 be an
integer, and a ∈ C be a non-zero finite value. If fnf ′ and gng′ share the value a CM, then either
f ≡ dg for some (n+1)-th root of unity d, or f = c1e
cz and g = c2e
−cz for three non-zero constants
c1, c2 and c such that (c1c2)n+1c2 = −a2.
*The work was supported by a NAFOSTED grant.
c© VU HOAI AN, HA HUY KHOAI, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2 147
148 VU HOAI AN, HA HUY KHOAI
Recently, there has been an increasing interest in studying value-sharing and uniqueness for mero-
morphic functions and their difference operators and difference polynomials. Halburd and Korhonen
[14] established a version of Nevanlinna theory based on difference operators. For an analog of Hay-
man Conjecture for difference, Laine and Yang [20] investigated the value distribution of difference
products of entire functions, and obtained the following theorem.
Theorem 1.3. Let f(z) be a transcendental entire function of finite order, and c be a non-zero
complex constant. Then n ≥ 2, f(z)nf(z + c) assumes every non-zero value a ∈ C infinitely often.
In recent years the similar problems are investigated for functions in a non-Archimedean fields
(see, for example, [3, 5]). In [22] J. Ojeda proved that for a transcendental meromorphic function
f in an algebraically closed fields of characteristic zero, complete for a non-Archimedean absolute
value K, the function f ′fn − 1 has infinitely many zeros, if n ≥ 2.
Ha Huy Khoai and Vu Hoai An [12] established a similar results for a differential monomial of
the form fn(f (k))
m
, where f is a meromorphic function in Cp.
Now let K be an algebraically closed field of characteristic zero, complete for a non-Archimedean
absolute value. We denote by A(K) the ring of entire functions in K, byM(K) the field of meromor-
phic functions, i.e., the field of fractions of A(K), and K̂ = K∪{∞}. The value-sharing problem for
meromorphic functions in K was investigated first in [1] and [8]. In recent years, many interesting
results on the value-sharing problem for meromorphic functions in K were obtained (see [17, 13]).
Let us first recall some basic definitions. For f ∈M(K) and S ⊂ K̂, we define
Ef (S) =
⋃
a∈S
{(z,m)|f(z) = a with multiplicity m}.
Let F be a nonempty subset of M(K). Two functions f, g of F are said to share S, counting
multiplicity (share S CM), if Ef (S) = Eg(S).
Now for f ∈M(K). We define difference operators of f as
∆cf = f(z + c)− f(z), ∆1
cf = ∆cf
and
∆n+1
c f = ∆nf(z + c)−∆nf(z), n = 1, 2, . . . ,
where c ∈ Cp is a non-zero constant; and difference polynomial of f as
A(z, f) =
∑
Λ∈I
aΛ(z)f(z)Λ0f(z)Λ1 . . . f(z)Λn ,
where I be a finite set of multiindex Λ = (Λ0,Λ1, . . . ,Λn) and the coefficients aΛ(z) are small with
respect to f(z) in the sence that TaΛ(r) = o(Tf (r)).
From now on, we assume P (z) is a non-zero polynomial on Cp of degree n. Write P (z) =
= a0(z − a1)m1(z − a2)m2 . . . (z − as)ms , a0 6= 0.
In this paper we discuss the value-sharing and versions of the Hayman Conjecture and uniqueness
for p-adic meromorphic functions and their difference operators and difference polynomials, and
prove a p-adic analog of Laine – Yang’s result. Namely, we prove the following theorems.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 149
Theorem 1.4 (A version of the Hayman Conjecture for p-adic meromorphic functions and their
difference operators). Let f be a meromorphic function on Cp and n, ki, s, q, i = 1, . . . , q, be are
integers,
s ≥ 1, q ≥ 1, ki ≥ 1, n ≥
q∑
i=1
(2ki + 1)2i + q + s+ 1− 3
q∑
i=1
ki,
and ∆qf is not identically zero. Then P (f)(∆1
cf)k1 . . . (∆q
cf)kq − a has zeros, where a ∈ Cp is a
non-zero.
Theorem 1.5 (A version of the Hayman Conjecture for p-adic meromorphic functions and their
difference polynomials). Let f be a meromorphic function on Cp and n, qi, s, k, i = 1, . . . , k, be are
integers, and
s ≥ 1, k ≥ 1, qi ≥ 1, n ≥
k∑
i=1
qi + 2k + s+ 1.
Then P (f)(f(z + c))q1 . . . (f(z + kc))qk − a has zeros, where a ∈ Cp is a non-zero.
Theorem 1.6 (A version of the Yang and Hua’s Theorem 1.2 for p-adic meromorphic functions
and their difference polynomials). Let f and g be two non-constant p-adic meromorphic functions.
(1) If Efnf(z+c)...f(z+kc)(1) = Egng(z+c)...g(z+kc)(1), with k ≥ 1 and n ≥ 5k+8 be are integers,
then f = hg with hn+k = 1 or fg = l with ln+k = 1.
(2) If Efn(f(z+c))q1 ...(f(z+kc))qk (1) = Egn(g(z+c))q1 ...(g(z+kc))qk (1), with
qi > 1, i = 1, . . . , k, k ≥ 1, n ≥
k∑
i=1
qi + 8k + 8
be are integers, then f = hg with hn+q1+...+qk = 1 or fg = l with ln+q1+...+qk = 1.
(3) If
Efn(f(z+e1c)...f(z+emc)(f(z+t1c))q1 ...(f(z+tkc))
qk (1) =
= Egn(g(z+e1c)...g(z+emc)(g(z+t1c))q1 ...(g(z+tkc))
qk (1),
with
ej ≥ 1, j = 1, . . . ,m, ti ≥ 1, qi > 1, i = 1, . . . , k, k ≥ 1,
n ≥ 5m+
k∑
i=1
qi + 8k + 8
be are integers, then f = hg with hn+m+q1+...+qk = 1 or fg = l with ln+m+q1+...+qk = 1.
The main tool of the proof is the p-adic Nevanlinna theory ([8 – 3, 17]). Therefore, in the next
section we first establish some properties of the height function (a p-adic analog of the Nevanlinna
characteristic functionrm) for p-adic meromorphic functions and their difference operator and differ-
ence polynomials for later use.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
150 VU HOAI AN, HA HUY KHOAI
2. Height of p-adic meromorphic functions. Let f be a non-constant holomorphic function on
Cp. For every a ∈ Cp, expanding f around a as f =
∑
Pi(z − a) with homogeneous polynomials
Pi of degree i, we define
vf (a) = min{i : Pi 6≡ 0}.
For a point d ∈ Cp we define the function vdf : Cp → N by
vdf (a) = vf−d(a).
Fix a real number ρ with 0 < ρ ≤ r. Define
Nf (a, r) =
1
ln p
r∫
ρ
nf (a, x)
x
dx,
where nf (a, x), as usually, is the number of the solutions of the equation f(z) = a (counting
multiplicity) in the disk Dx = {z ∈ Cp : |z| ≤ x}.
If a = 0, then set Nf (r) = Nf (0, r).
For l a positive integer, set
Nl,f (a, r) =
1
ln p
r∫
ρ
nl,f (a, x)
x
dx,
where
nl,f (a, r) =
∑
|z|≤r
min
{
vf−a(z), l
}
.
Let k be a positive integer. Define the function v≤kf from Cp into N by
v≤kf (z) =
0 if vf (z) > k,
vf (z) if vf (z) ≤ k,
and
n≤kf (r) =
∑
|z|≤r
v≤kf (z), n≤kf (a, r) = n≤kf−a(r).
Define
N≤kf (a, r) =
1
ln p
r∫
ρ
n≤kf (a, x)
x
dx.
If a = 0, then set N≤kf (r) = N≤kf (0, r).
Set
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 151
N≤kl,f (a, r) =
1
ln p
r∫
ρ
n≤kl,f (a, x)
x
dx,
where
n≤kl,f (a, r) =
∑
|z|≤r
min
{
v≤kf−a(z), l
}
.
In a like manner we define
N<k
f (a, r), N<k
l,f (a, r), N>k
f (a, r), N≥kf (a, r), N≥kl,f (a, r), N>k
l,f (a, r).
Recall that for a holomorphic function f(z) in Cp, represented by the power series
f(z) =
∞∑
0
anz
n,
for each r > 0, we define |f |r = max{|an|rn, 0 ≤ n <∞}.
Now let f =
f1
f2
be a non-constant meromorphic function on Cp, where f1, f2 be holomorphic
functions on Cp having no common zeros, we set |f |r =
|f1|r
|f2|r
. For a point d ∈ Cp ∪ {∞} we define
the function vdf : Cp → N by
vdf (a) = vf1−df2(a)
with d 6=∞, and
v∞f (a) = vf2(a).
For a point a ∈ C define
mf (∞, r) = max
{
0, log |f |r
}
, qmf (a, r) = m1/f−a(∞, r),
Nf (a, r) = Nf1−af2(r), Nf (∞, r) = Nf2(r),
Tf (r) = max
1≤i≤2
log |fi|r.
In a like manner we define
Nl,f (a, r), N≤kf (a, r), N≤kl,f (a, r), N<k
f (a, r), N<k
l,f (a, r), N>k
f (a, r),
N≥kf (a, r), N≥kl,f (a, r), N>k
l,f (a, r),
with a ∈ Cp ∪ {∞}.
Then we have (see [9])
Nf (a, r) +mf (a, r) = Tf (r) +O(1)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
152 VU HOAI AN, HA HUY KHOAI
with a ∈ Cp ∪ {∞},
Tf (r) = T1/f (r) +O(1),
|f (k)|r ≤
|f |r
rk
,
mf (k)/f (∞, r) = O(1).
The following two lemmas were proved in [9].
Lemma 2.1. Let f be a non-constant holomorphic function on Cp. Then
Tf (r)− Tf (ρ) = Nf (r),
where 0 < ρ ≤ r.
Lemma 2.2. Let f be a non-constant meromorphic function on Cp and let a1, a2, . . . , aq be
distinct points of Cp. Then
(q − 1)Tf (r) ≤ N1,f (∞, r) +
q∑
i=1
N1,f (ai, r)−N0,f ′(r)− log r +O(1),
where N0,f ′(r) is the counting function of the zeros of f ′ which occur at points other than roots of
the equations f(z) = ai, i = 1, . . . , q, and 0 < ρ ≤ r.
3. Two versions of the Hayman Conjecture for p-adic meromorphic functions and their
difference operators and difference polynomials. We are going to prove Theorems 1.4 – 1.6. We
need the following lemmas.
Lemma 3.1. Let f be a non-constant p-adic meromorphic function and ∆f is not identically
zero and k, q be a positive integer. Then:
(1) mf(z+c)/f(z)(∞, r) = O(1);
(2) mf(z+kc)/f(z)(∞, r) = O(1);
(3) m∆cf/f (∞, r) = O(1);
(4) m(∆cf)q/f (∞, r) = O(1);
(5) Tf(z+c)(r) = Tf(z)(r) +O(1);
(6) Tf(z+qc)(r) = Tf(z)(r) +O(1);
(7) T∆cf/f (r) ≤ 2Tf (r) +O(1).
Proof. Set Ac =
f(z + c)
f(z)
. Then:
(1) If |c| < r. Notice that the set of r ∈ R+ such that there exist z ∈ Cp with |z| = r is
dense in R+. Therefore, without loss of generality one may assume that there exist z ∈ Cp such that
|z| = r. Then |c + z| = |z| = r. So |f(z)|r = |f(z + c)|r and |Ac| = 1. If r ≤ |c|, then |c + z| ≤
≤ max
{
|c|, |z|
}
≤ |c|. Thus |Ac| = O(1). Therefore mAc(∞, r) = max
{
0, log |Ac|r
}
= O(1).
(2) Similarly as the arguments of (1), we obtain mf(z+kc)/f(z)(∞, r) = O(1).
(3) By mf(z+c)/f(z)(∞, r) = O(1), m∆cf/f (∞, r) ≤ max
{
mf(z+c)/f(z)(∞, r), 0
}
, we have
m∆cf/f (∞, r) = O(1) .
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 153
(4) Bym∆cf/f (∞, r) = O(1), m(∆cf)q/f (∞, r) = qm∆cf/f (∞, r), we havem(∆cf)q/f (∞, r) =
= O(1).
(5) Let f =
f1
f2
be a non-constant meromorphic function on Cp, where f1, f2 be holomorphic
functions on Cp having no common zeros. Similarly as the arguments of (1), we have:
If |c| < r, then |f1(z)|r = |f1(z + c)|r and |f2(z)|r = |f2(z + c)|r. If r ≤ |c|, then |f1(z)|r ≤
≤ |f1(z)|c, |f1(z + c)|r ≤ |f1(z)|c|, and |f2(z)|r ≤ |f2(z)|c, |f2(z + c)|r ≤ |f2(z)|c. Moreover,
Tf (r) = max1≤i≤2 log |fi|r. So Tf(z+c)(r) = Tf(z)(r) +O(1).
(6) Similarly as the arguments of (5), we obtain Tf(z+qc)(r) = Tf(z)(r) +O(1).
(7) We have
T∆cf
f
(r) = m f(z+c)−f(z)
f(z)
(∞, r) +N f(z+c)−f(z)
f(z)
(∞, r) ≤
≤ m f(z+c)
f(z)
(∞, r) +N f(z+c)
f(z)
(∞, r) +O(1) ≤
≤ mf(z)(∞, r) +Nf(z)(∞, r) +mf(z+c)(∞, r) +Nf(z+c)(∞, r) +O(1) =
= Tf(z+c)(r) + Tf(z)(r) +O(1) ≤ 2Tf (r) +O(1).
Lemma 3.1 is proved.
Lemma 3.2. Let f be a non-constant p-adic meromorphic function and ∆qf is not identically
zero and k, q, m be a positive integer and P (z) is the above. Then:
(1) T∆q
cf
(r) ≤ 2qTf (r) +O(1);
(2) T∆q
cf/f
(r) ≤ 2(2q − 1)Tf (r) +O(1);
(3)
(
n+ 3
∑q
i=1
ki −
∑q
i=1
ki2
i+1
)
Tf (r) ≤ TP (f)(∆1
cf)k1 ...(∆q
cf)kq (r) +O(1);
(4)
(
n−
∑k
i=1
qi
)
Tf (r) ≤ TP (f)(f(z+c))q1 ...(f(z+kc))qk (r) +O(1).
Proof. We will prove (1) by induction on j, 1 ≤ j ≤ q − 1. With j = 1, we have T∆cf (∞, r) ≤
≤ Tf(z+c)(r) +Tf(z)(r) +O(1). By Tf(z+c)(r) = Tf(z)(r) +O(1), T∆cf (r) ≤ 2Tf (r) +O(1). Now
assume that T
∆j
cf
(r) ≤ 2jTf (r) + O(1). Moreover we have ∆j+1
c f = ∆c(∆
j
cf). From this and by
induction, T
∆j+1
c f
(r) = T
∆c(∆j
cf)
(r) ≤ T
∆j
cf(z+c)
(r) + T
∆j
cf(z)
(r) + O(1) ≤ 2.2jTf (r) + O(1) =
= 2j+1Tf (r) +O(1).
We will prove (2) by induction on j, 1 ≤ j ≤ q−1. With j = 1, by 3.1 (7) we have T∆cf/f (r) ≤
≤ 2Tf (r) +O(1. Now assume that T
∆j
cf/f
(r) ≤ 2(2j − 1)Tf (r) +O(1). Moreover we have
T
∆
j+1
c f
f
(r) = T
∆
j+1
c f
∆
j
cf
∆
j
cf
f
(r) ≤ T
∆
j+1
c f
∆
j
cf
+ T
∆
j
cf
f
+O(1) ≤
≤ T
∆c(∆
j
cf)
∆
j
cf
+ T
∆
j
cf
f
+O(1) ≤ 2T
∆j
c
+ 2(2j − 1)Tf (r) +O(1).
By (1), T∆q
cf
(r) ≤ 2qTf (r) +O(1). Thus 2T
∆j
c
+ 2(2j − 1)Tf (r) ≤ 2(2j + 2j − 1)Tf (r) +O(1) ≤
≤ 2(2j+1 − 1)Tf (r) +O(1). Therefore
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
154 VU HOAI AN, HA HUY KHOAI
T
∆
j+1
c f
f
(r) ≤ 2(2j+1 − 1)Tf (r) +O(1).
(3) Set G = P (f)(∆1
cf)k1 . . . (∆q
cf)kq . We have
fk1 . . . fkqG = fk1+...+kqP (f)(∆1
cf)k1 . . . (∆q
cf)kq
and
1
fk1+...+kqP (f)
=
1
G
(
∆1
cf
f
)k1
. . .
(
∆q
cf
f
)kq
.
From this and (2), we obtain(
n+
q∑
i=1
ki
)
Tf (r) = T 1
fk1+...+kqP (f)
(r) +O(1) = T
1
G
(
∆1
cf
f
)k1
...
(
∆
q
cf
f
)kq (r) +O(1) ≤
≤ T1/G(r) +
q∑
i=1
T(∆i
cf/f)ki (r) +O(1) ≤ T1/G(r) +
q∑
i=1
kiT∆i
cf/f
(r) +O(1) ≤
≤ TG(r) +
q∑
i=1
ki2(2i − 1)Tf (r) +O(1).
So (
n+ 3
q∑
i=1
ki −
q∑
i=1
ki2
i+1
)
Tf (r) ≤ TP (f)(∆1
cf)k1 ...(∆q
cf)kq (r) +O(1).
(4) Set F = P (f)(f(z+ c))q1 . . . (f(z+ kc))qk . We have f q1 . . . f qkF = f q1+...+qkP (f)(f(z+
+c))q1 . . . (f(z+kc))qk and f q1+...+qkP (f) = F.
(
f(z)
f(z+c)
)q1
. . .
(
f(z)
f(z+kc)
)qk
. From this and 3.1 (5),
3.1 (6), we obtain (
n+
k∑
i=1
qi
)
Tf (r) = Tfq1+...+qkP (f)(r) +O(1) =
= T
F.
(
f(z)
f(z+c)
)q1
...
(
f(z)
f(z+kc)
)
qk
(r) +O(1) ≤ TF (r) +
k∑
i=1
T( f(z)
f(z+ic)
)qi (r) +O(1) ≤
≤ TF (r) +
k∑
i=1
qiT f(z)
f(z+ic)
(r) +O(1) ≤ TF (r) +
k∑
i=1
qi(Tf (r) + Tf(z+ic)(r)) +O(1).
Therefore (
n+
k∑
i=1
qi
)
Tf (r) ≤ TF (r) + 2
k∑
i=1
qiTf (r) +O(1).
So (
n−
k∑
i=1
qi
)
Tf (r) ≤ TP (f)(f(z+c))q1 ...(f(z+kc))qk (r) +O(1).
Lemma 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 155
Lemma 3.3. Let f and g be non-constant p-adic meromorphic functions. If Ef (1) = Eg(1),
then one of the following three cases holds:
(1) Tf (r) ≤ N1,f (∞, r) + N≥2
1,f (∞, r) + N1,f (0, r) + N≥2
1,f (0, r) + N1,g(∞, r) + N≥2
1,g (∞, r) +
+N1,g(0, r) +N≥2
1,g (0, r)− log r +O(1), the same inequality holding for Hg(r);
(2) f ≡ g;
(3) fg ≡ 1.
Proof. Set
F =
1
f − 1
, G =
1
g − 1
,
L =
f ′′
f ′
− 2
f ′
f − 1
− g”
g′
+ 2
g′
g − 1
.
(3.1)
Then
L =
F ′′
F ′
− G′′
G′
. (3.2)
Next we consider the following two cases:
Case 1: L 6≡ 0. Since Ef (1) = Eg(1), if f(a) = 1, g(a) = 1 and v1
f (a) = v1
g(a), then
L(a) = 0. We now consider the poles of L. It is clear that all poles of L are of order 1. We can easily
see from (3.1) that any simple pole of f and g is not a pole of L and the poles of L only occur at
zeros of f ′ and g′ and the multiple poles of f and g.
From (3.1) we have
mL(∞, r) = O(1),
and
N≤1
f (1, r) = N≤1
g (1, r) ≤ NL(0, r) ≤ TL(r) +O(1) ≤ NL(∞, r) +O(1). (3.3)
On the other hand, by Lemma 2.2,
Tf (r) ≤ N1,f (∞, r) +N1,f (0, r) +N1,f (1, r)−N0,f ′(r)− log r +O(1),
where N0,f ′(r) denotes the counting function of those zeros of f ′ but not that of f(f − 1). Also,
N1,0,f ′(r) is defined similarly, where in counting, each zero of f ′ is counted with multiplicity 1.
From (3.1), (3.2) and (3.3) we deduce that
N≤1
f (1, r) ≤ N≥2
1,f (∞, r)+
+N≥2
1,g (∞, r) +N1,0,f ′(r) +N1,0,g′(r) +N≥2
1,f (0, r) +N≥2
1,g (0, r) +O(1). (3.4)
Since Ef (1) = Eg(1),
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
156 VU HOAI AN, HA HUY KHOAI
N1,f (1, r) = N≤1
f (1, r) +N≥2
1,g (1, r).
Then
Tf (r) ≤ N1,f (∞, r) +N1,f (0, r) +N≤1
f (1, r) +N≥2
1,g (1, r)−N0,f ′(r)− log r +O(1). (3.5)
Now we consider N≥2
1,g (1, r).
By Lemma 2.1,
Ng′(0, r)−Ng(0, r) +N1,g(0, r) = N g′
g
(0, r) ≤ T g′
g
(r) +O(1) =
= N g′
g
(∞, r) +m g′
g
(∞, r) +O(1) =
= N1,g(∞, r) +N1,g(0, r) +O(1).
Therefore
Ng′(0, r) ≤ N1,g(∞, r) +Ng(0, r) +O(1).
Moreover
N0,g′(r) +N≥2
1,g (1, r) +N≥2
g (0, r)−N≥2
1,g (0, r) ≤ Ng′(0, r).
The above two inequalities yield
N0,g′(r) +N≥2
1,g (1, r) ≤ N1,g(∞, r) +N1,g(0, r) +O(1).
Combining this inequality and (3.4) and (3.5), we obtain (1).
Case 2: L ≡ 0. Then
f ′′
f ′
− 2
f ′
f − 1
≡ g′′
g′
− 2
g′
g − 1
. (3.6)
By (3.6) we have
F ′′
F ′
≡ G′′
G′
.
Thus
f ≡ ag + b
cg + d
,
where a, b, c, d ∈ Cp satisfying ad− bc 6= 0. Then Tf (r) = Tg(r) +O(1).
Next we consider the following subcases:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 157
Subcase 1: ac 6= 0. Then
f − a
c
≡
b− ad
c
cg + d
.
By Lemma 2.3
Tf (r) ≤ N1,f (∞, r) +N1,f−a
c
(0, r) +N1,f (0, r) +O(1) =
= N1,f (∞, r) +N1,g(∞, r) +N1,f (0, r) +O(1).
We get (1).
Subcase 2: a 6= 0, c = 0. Then f ≡ ag + b
d
. If b 6= 0, by Lemma 2.2,
Tf (r) ≤ N1,f (∞, r) +N1,f− b
d
(0, r) +N1,f (0, r) +O(1) =
= N1,f (∞, r) +N1,g(0, r) +N1,f (0, r) +O(1).
We get (1). If b = 0, then f ≡ ag
d
. If
a
d
= 1, then f ≡ g. We obtain (2). If
a
d
6= 1, then by
Ef (1) = Eg(1) and Lemma 2.3
f 6= 1, f 6= a
d
,
Tf (r) ≤ N1,f (∞, r) +N1,f (
a
d
, r) +N1,f (1, r) +O(1) = N1,f (∞, r) +O(1).
We get (1).
Subcase 3: a = 0, c 6= 0. Then f ≡ b
cg + d
. If d 6= 0 , by Lemma 2.2,
Tf (r) ≤ N1,f (∞, r) +N1,f− b
d
(0, r) +N1,f (0, r) +O(1) =
= N1,f (∞, r) +N1,g(0, r) +N1,f (0, r) +O(1).
We obtain (1).
If d = 0, then f ≡ b
cg
. If
b
c
= 1, then fg ≡ 1. We obtain (3).
If
b
c
6= 1, then by Ef (1) = Eg(1) and Lemma 2.2,
f 6= 1, f 6= b
c
,
Tf (r) ≤ N1,f (∞, r) +N1,f
(
b
c
, r
)
+N1,f (1, r) +O(1) = N1,f (∞, r) +O(1).
We get (1).
Lemma 3.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
158 VU HOAI AN, HA HUY KHOAI
Now we use the above lemmas to prove the main result of the paper.
Proof of Theorem 1.4. From P (z) = a0(z − a1)m1(z − a2)m2 . . . (z − as)ms , a0 6= 0, P (f) =
= a0(f −a1)m1(f −a2)m2 . . . (f −as)ms . Set G = P (f)(∆1
cf)k1 . . . (∆q
cf)kq . We see that any pole
of G can occur only at poles of f, f(z + c), f(z + 2c), . . . , f(z + qc), and any zero of G can occur
only at zeros of f − a1, f − a2, . . . , f − as, ∆1
cf, . . . ,∆
q
cf . From this and by Lemmas 2.1, 2.2,
3.1 (5), 3.1 (6), 3.2 (1), 3.2 (3) we have(
n+ 3
q∑
i=1
ki −
q∑
i=1
ki2
i+1
)
Tf (r) ≤ TG(r) +O(1) ≤
≤ N1,G(∞, r) +N1,G(0, r) +N1,G(a, r)− log r +O(1) ≤
≤ N1,f (∞, r) +
q∑
i=1
N1,f(z+ic)(∞, r) +
s∑
i=1
N1,f (ai, r)+
+
q∑
i=1
N1,∆i
cf
+N1,G(a, r)− log r +O(1) ≤
≤ Tf (r) + qTf (r) + sTf (r) +
q∑
i=1
2iTf (r) +N1,G(a, r)− log r +O(1) =
=
(
q∑
i=1
2i + q + s+ 1
)
Tf (r) +N1,G(a, r)− log r +O(1).
Therefore(
n+ 3
q∑
i=1
ki −
q∑
i=1
(2ki + 1)2i − q − s− 1
)
Tf (r) + log r ≤ N1,G(a, r) +O(1).
Since and
n ≥
q∑
i=1
(2ki + 1)2i + q + s+ 1− 3
q∑
i=1
ki,
we obtain
P (f)(∆1
cf)k1 . . . (∆q
cf)kq − a
has zeros.
Theorem 1.4 is proved.
Proof of Theorem 1.5. From P (z) = a0(z − a1)m1(z − a2)m2 . . . (z − as)ms , a0 6= 0, P (f) =
= a0(f − a1)m1(f − a2)m2 . . . (f − as)ms . Set F = P (f)(f(z+ c))q1 . . . (f(z+ kc))qk . We see that
any pole of F can occur only at poles of f, f(z + c), f(z + 2c), . . . , f(z + kc), and any zero of G
can occur only at zeros of f − a1, f − a2, . . . , f − as, f(z + c), f(z + 2c), . . . , f(z + kc). From
this and by Lemmas 2.1, 2.2, 3.1 (5), 3.1 (6), 3.2 (4) we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 159(
n−
k∑
i=1
qi
)
Tf (r) ≤ TF (r) +O(1) ≤
≤ N1,F (∞, r) +N1,F (0, r) +N1,F (a, r)− log r +O(1) ≤
≤ N1,f (∞, r) +
k∑
i=1
N1,f(z+ic)(∞, r) +
s∑
i=1
N1,f (ai, r) +
k∑
i=1
N1,f(z+ic)(0, r)+
+N1,F (a, r)− log r +O(1) ≤ Tf (r) + kTf (r) + sTf (r) + kTf (r) +N1,F (a, r)−
− log r +O(1) = (2k + s+ 1)Tf (r) +N1,F (a, r)− log r +O(1).
Therefore (
n−
k∑
i=1
qi − 2k − s− 1
)
Tf (r) + log r ≤ N1,F (a, r) +O(1).
Since and
n ≥
k∑
i=1
qi + 2k + s+ 1
we obtain
P (f)(f(z + c))q1 . . . (f(z + kc))qk − a
has zeros.
Theorem 1.5 is proved.
Proof of Theorem 1.6. (1) Set A = fnf(z + c) . . . f(z + kc), B = gng(z + c) . . . g(z + kc).
It suffices to consider the following cases:
Case 1:
TA(r) +O(1) ≤ N1,A(∞, r) +N≥2
1,A(∞, r) +N1,A(0, r) +N≥2
1,A(0, r)+
+N1,B(∞, r) +N≥2
1,B(∞, r) +N1,B(0, r) +N≥2
1,B(0, r)− log r +O(1).
By Lemmas 3.2 (4), 3.3,
(n− k)Tf (r) ≤ TA(r) +O(1) ≤ N1,A(∞, r) +N≥2
1,A(∞, r) +N1,A(0, r) +N≥2
1,A(0, r)+
+N1,B(∞, r) +N≥2
1,B(∞, r) +N1,B(0, r) +N≥2
1,B(0, r)− log r +O(1),
(n− k)Tg(r) ≤ TB(r) +O(1) ≤ N1,A(∞, r) +N≥2
1,A(∞, r) +N1,A(0, r)+
+N≥2
1,A(0, r) +N1,B(∞, r) +N≥2
1,B(∞, r) +N1,B(0, r) +N≥2
1,B(0, r)− log r +O(1).
(3.7)
We see that any pole of A can occur only at poles of
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
160 VU HOAI AN, HA HUY KHOAI
f, f(z + c), f(z + 2c), . . . , f(z + kc).
From this and by Lemmas 2.1, 3.1 (5), 3.1 (6) we have
N1,A(∞, r) +N≥2
1,A(∞, r) ≤
≤ 2N1,f (∞, r) +
k∑
i=1
(N1,f(z+ic)(∞, r) +N≥2
1,f(z+ic)(∞, r)) +O(1) ≤
≤ 2Nf (∞, r) +
k∑
i=1
Nf(z+ic)(∞, r) +O(1) ≤
≤ 2Tf (r) +
k∑
i=1
Tf(z+ic)(r) +O(1) ≤ (k + 2)Tf (r) +O(1).
So
N1,A(∞, r) +N≥2
1,A(∞, r) ≤ (k + 2)Tf (r) +O(1). (3.8)
Similarly, and note that any zero of A can occur only at zeros of
f, f(z + c), f(z + 2c), . . . , f(z + kc),
we obtain
N1,A(0, r) +N≥2
1,A(0, r) ≤ (k + 2)Tf (r) +O(1). (3.9)
Similarly we obtain
N1,B(∞, r) +N≥2
1,B(∞, r) ≤ (k + 2)Tg(r) +O(1),
N1,B(0, r) +N≥2
1,B(0, r) ≤ (k + 2)Tg(r) +O(1).
(3.10)
From (3.7) – (3.10) we have
(n− k)Tf (r) ≤ (2k + 4)(Tf (r) + Tg(r))− log r +O(1).
Similarly
(n− k)Tg(r) ≤ (2k + 4)(Tf (r) + Tg(r))− log r +O(1).
So
(n− k)(Tf (r) + Tg(r)) ≤ (4k + 8)(Tf (r) + Tg(r))− 2 log r +O(1),
(n− 5k − 8)(Tf (r) + Tg(r)) + 2 log r ≤ O(1).
By n ≥ 5k + 8 we obtain a contradiction.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 161
Case 2: A = fnf(z + c) . . . f(z + kc) ≡ B = gng(z + c) . . . g(z + kc). Set h =
f
g
. Assume
that h is not a constant. Then we get
hn =
1
h(z + c) . . . h(z + kc)
.
Thus, by Lemma 3.1 (5), we get
nTh(r) = Thn = T 1
h(z+c)...h(z+kc)
≤
k∑
i=1
Th(z+ic)(r) +O(1) ≤ kTh(r),
which is a contradiction with n ≥ 5k+ 8. Hence h must be a constant, which implies that hn+k = 1,
thus f = hg with hn+k = 1.
Case 3: fnf(z+ c) . . . f(z+kc).gng(z+ c) . . . g(z+kc) ≡ 1. From this we have (fg)n(f(z+
+ c)g(z + c)) . . . (f(z + kc)g(z + kc)) = 1. Set l = fg. Assume that l is not a constant. Then we
get
ln =
1
l(z + c) . . . l(z + kc)
.
Similar as above, l must be a constant. Thus fg = l with ln+k = 1.
(2) Set C = fn(f(z + c))q1 . . . (f(z + kc))qk , D = gn(g(z + c))q1 . . . (g(z + kc))qk .
It suffices to consider the following cases:
Case 1:
TC(r) +O(1) ≤ N1,C(∞, r) +N≥2
1,C(∞, r) +N1,C(0, r) +N≥2
1,C(0, r)+
+N1,D(∞, r) +N≥2
1,D(∞, r) +N1,D(0, r) +N≥2
1,D(0, r)− log r +O(1).
By Lemmas 3.2 (4), (
n−
k∑
i=1
qi
)
Tf (r) ≤ TC(r) +O(1),
(
n−
k∑
i=1
qi
)
Tg(r) ≤ TD(r) +O(1).
(3.11)
By qi ≥ 2, i = 1, . . . , k,
N1,(f(z+ic))qi (∞, r) +N≥2
1,(f(z+ic))qi (∞, r)) ≤ 2Nf(z+ic)(∞, r).
From this and similar as (3.8) we obtain
N1,C(∞, r) +N≥2
1,C(∞, r) ≤ (2k + 2)Tf (r) +O(1),
N1,C(0, r) +N≥2
1,C(0, r) ≤ (2k + 2)Tf (r) +O(1),
N1,D(∞, r) +N≥2
1,D(∞, r) ≤ (2k + 2)Tg(r) +O(1),
N1,D(0, r) +N≥2
1,D(0, r) ≤ (2k + 2)Tg(r) +O(1).
(3.12)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
162 VU HOAI AN, HA HUY KHOAI
Since (3.11), (3.12), and similar as in (1) we obtain
TC(r) ≤ (4k + 4)(Tf (r) + Tg(r))− log r +O(1),
TD(r) ≤ (4k + 4)(Tf (r) + Tg(r))− log r +O(1),(
n−
k∑
i=1
qi − 8k − 8
)
(Tf (r) + Tg(r)) + 2 log r ≤ O(1).
By
n ≥
k∑
i=1
qi + 8k + 8
we obtain a contradiction.
Case 2: C = fn(f(z + c))q1 . . . (f(z + kc))qk ≡ D = gn(g(z + c))q1 . . . (g(z + kc))qk .
Similar as Case 2 of (1) we get f = hg with hn+q1+...+qk = 1.
Case 3: fn(f(z + c))q1 . . . (f(z + kc))qk .gn(g(z + c))q1 . . . (g(z + kc))qk ≡ 1.
Similar as Case 3 of (1) we get fg = l with ln+q1+...+qk = 1.
(3) Set E = fnf(z + e1c) . . . f(z + emc)(f(z + t1c))
q1 . . . (f(z + tkc))
qk , H = gng(z +
+ e1c) . . . g(z + emc)(g(z + t1c))
q1 . . . (g(z + tkc))
qk .
It suffices to consider the following cases:
Case 1:
TE(r) +O(1) ≤ N1,E(∞, r) +N≥2
1,E(∞, r) +N1,E(0, r) +N≥2
1,E(0, r)+
+N1,H(∞, r) +N≥2
1,H(∞, r) +N1,H(0, r) +N≥2
1,H(0, r)− log r +O(1).
By Lemma 3.2 (4)
(
n−m−
k∑
i=1
qi
)
Tf (r) ≤ TE(r) +O(1),
(
n−m−
k∑
i=1
qi
)
Tg(r) ≤ TD(r) +O(1).
(3.13)
Similar as Case 1 of (1) and (2) we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VALUE-SHARING PROBLEM FOR p-ADIC MEROMORPHIC FUNCTIONS . . . 163
N1,E(∞, r) +N≥2
1,E(∞, r) ≤
≤ 2Nf (∞, r) +
m∑
i=1
Nf(z+eic)(∞, r) + 2
k∑
i=1
Nf(z+tic)(∞, r)O(1) ≤
≤ (m+ 2k + 2)Tf (r) +O(1),
N1,E(0, r) +N≥2
1,E(0, r) ≤ (m+ 2k + 2)Tf (r) +O(1),
N1,H(∞, r) +N≥2
1,H(∞, r) ≤ (m+ 2k + 2)Tg(r) +O(1),
N1,H(0, r) +N≥2
1,H(0, r) ≤ (m+ 2k + 2)Tg(r) +O(1).
(3.14)
Since (3.13), (3.14), and similar as in (1), (2), we obtain(
n−m−
k∑
i=1
qi
)
Tf (r) ≤ 2(m+ 2k + 2)(Tf (r) + Tg(r))− log r +O(1),
(
n−m−
k∑
i=1
qi
)
Tg(r) ≤ 2(m+ 2k + 2)(Tf (r) + Tg(r))− log r +O(1),
(
n−m−
k∑
i=1
qi
)
(Tf (r) + Tg(r)) ≤ 4(m+ 2k + 2)(Tf (r) + Tg(r))− 2 log r +O(1),
(
n− 5m−
k∑
i=1
qi − 8k − 8
)
(Tf (r) + Tg(r)) + 2 log r ≤ +O(1).
Which is a contradiction with
n ≥ 5m+
k∑
i=1
qi + 8k + 8.
Case 2: Prove is similarly as in Case 2 of (1) and (2) we get f = hg with hn+m+q1+...+qk = 1.
Case 3: Prove is similarly as in Case 2 of (1) and (2) we get fg = l with ln+m+q1+...+qk = 1.
Theorem 1.6 is proved.
1. Adam W. W., Straus E. G. Non-Archimedean analytic functions taking the same values at the same points // Ill. J.
Math. – 1971. – 15. – P. 418 – 424.
2. Alotaibi A. On the Zeros of af(f (k))
n
for n ≥ 2 // Comput. Methods and Funct. Theory. – 2004. – 4, № 1. –
P. 227 – 235.
3. Boutabaa A., Escassut A. Uniqueness problems and applications of the ultrametric Nevanlinna theory // Contemp.
Math. – 2003. – 319. – P. 53 – 74.
4. Chen H. H., Fang M. L. On the value distribution of fnf ′ // Sci. China Ser. A. – 1995. – 38. – P. 789 – 798.
5. Clunie J. On a result of Hayman // J. London Math. Soc. – 1967. – 42. – P. 389 – 392.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
164 VU HOAI AN, HA HUY KHOAI
6. Escassut A., Ojeda J., Yang C. C. Functional equations in a p-adic context // J. Math. Anal. and Appl. – 2009. – 351,
№ 1. – P. 350 – 359.
7. Gross F., Yang C. C. On pre-images and range sets of meromorphic functions // Proc. Jap. Acad. – 1982. – 58. –
P. 17 – 20.
8. Ha Huy Khoai. On p-adic meromorphic functions // Duke Math. J. – 1983. – 50. – P. 695 – 711.
9. Ha Huy Khoai. Height of p-adic holomorphic functions and applications // Int. Symp. "Holomorphic Mappings,
Diophantine Geometry and Related Topics” (Kyoto, 1992). Surikaisekikenkyusho Kokyuroku. – 1993. – № 819. –
P. 96 – 105.
10. Ha Huy Khoai, Mai Van Tu. p-Adic Nevanlinna – Cartan theorem // Int. J. Math. – 1995. – 6. – P. 719 – 731.
11. Ha Huy Khoai, Vu Hoai An. Value distribution for p-adic hypersurfaces // Taiwan. J. Math. – 2003. – 7, № 1. –
P. 51 – 67.
12. Ha Huy Khoai, Vu Hoai An. Value distribution problem for p-adic meromorphic functions and their derivatives //
Ann Toulouse (to appear).
13. Ha Huy Khoai, My Vinh Quang. On p-adic Nevanlinna theory // Lect. Notes Math. – 1988. – 1351. – P. 146 – 158.
14. Halburd R. G., Korhonen R. J. Nevanlinna theory for the difference operator // Ann. Acad. Sci. Fenn. Math. – 2006.
– 31. – P. 463 – 478.
15. Han Q., Hu P.-C. Unicity of meromorphic functions related to their derivatives // Bull. Belg. Math. Soc. Simon
Stevin. – 2007. – 14. – P. 905 – 918.
16. Hayman W. K. Research problems in function theory. – London: Athlone Press Univ. London, 1967.
17. Hu P.-C., Yang C.-C. Meromorphic functions over non-Archimedean fields. – Dordrech: Kluwer Acad., 2000.
18. Liu K., Yang L. Z. Value distribution of the difference operator // Arch. Math. – 2009. – 92, № 3. – P. 270 – 278.
19. Lahiri I., Dewan S. Value distribution of the product of a meromorphic function and its derivative // Kodai Math. J.
– 2003. – 26. – P. 95 – 100.
20. Lahiri I.. Yang C.-C. Value distribution of difference polynomials // Proc. Jap. Acad. Ser. A. – 2007. – 83, № 8. –
P. 148 – 151.
21. Nevo Sh., Pang X. C., Zalcman L. Picard – Hayman behavior of derivatives of meromorphic functions with multiple
zeros // Electron. Res. Announcements Amer. Math. Soc. – 2006. – 12. – P. 37 – 43.
22. Ojeda J. Hayman’s conjecture in a p-adic field // Taiwan. J. Math. – 2008. – 12, № 9. – P. 2295 – 2313.
23. Yang C.-C., Hua X. H. Uniqueness and value-sharing of meromorphic functions // Ann. Acad. Sci. Fenn. Math. –
1997. – 22. – P. 395 – 406.
Received 21.08.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
|
| id | umjimathkievua-article-2562 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:52Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/07/b8f7b9806b97bcaa4cce5efa7d223307.pdf |
| spelling | umjimathkievua-article-25622020-03-18T19:29:46Z Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials Задача про спiльнi значення для p-адичних мероморфних функцiй та їх рiзницевих операторiв i рiзницевих полiномiв An, Vu Hoai Khoai, Ha Huy Ан, Ву Гоай Хоаї, Га Гуй We discuss the value-sharing problem, versions of the Hayman conjecture, and the uniqueness problem for p-adic meromorphic functions and their difference operators and difference polynomials. Дослiджено питання про спiльнi значення i єдинiсть та аналоги гiпотези Хеймена для p-адичних мероморфних функцiй та їх рiзницевих операторiв i рiзницевих полiномiв. Institute of Mathematics, NAS of Ukraine 2012-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2562 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 2 (2012); 147-164 Український математичний журнал; Том 64 № 2 (2012); 147-164 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2562/1883 https://umj.imath.kiev.ua/index.php/umj/article/view/2562/1884 Copyright (c) 2012 An Vu Hoai; Khoai Ha Huy |
| spellingShingle | An, Vu Hoai Khoai, Ha Huy Ан, Ву Гоай Хоаї, Га Гуй Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials |
| title | Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials |
| title_alt | Задача про спiльнi значення для p-адичних мероморфних функцiй та їх рiзницевих операторiв i рiзницевих полiномiв |
| title_full | Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials |
| title_fullStr | Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials |
| title_full_unstemmed | Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials |
| title_short | Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials |
| title_sort | value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2562 |
| work_keys_str_mv | AT anvuhoai valuesharingproblemforpadicmeromorphicfunctionsandtheirdifferenceoperatorsanddifferencepolynomials AT khoaihahuy valuesharingproblemforpadicmeromorphicfunctionsandtheirdifferenceoperatorsanddifferencepolynomials AT anvugoaj valuesharingproblemforpadicmeromorphicfunctionsandtheirdifferenceoperatorsanddifferencepolynomials AT hoaígaguj valuesharingproblemforpadicmeromorphicfunctionsandtheirdifferenceoperatorsanddifferencepolynomials AT anvuhoai zadačaprospilʹniznačennâdlâpadičnihmeromorfnihfunkcijtaíhriznicevihoperatoriviriznicevihpolinomiv AT khoaihahuy zadačaprospilʹniznačennâdlâpadičnihmeromorfnihfunkcijtaíhriznicevihoperatoriviriznicevihpolinomiv AT anvugoaj zadačaprospilʹniznačennâdlâpadičnihmeromorfnihfunkcijtaíhriznicevihoperatoriviriznicevihpolinomiv AT hoaígaguj zadačaprospilʹniznačennâdlâpadičnihmeromorfnihfunkcijtaíhriznicevihoperatoriviriznicevihpolinomiv |