Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes
UDC 517.5 We prove a uniqueness theorem of linearly nondegenerate holomorphic mappings from annulus to complex projective space $\mathbb{P}^n(\mathbb{C} )$ with different multiple values and a general condition on the intersections of the inverse images of these hyperplanes.
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Institute of Mathematics, NAS of Ukraine
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860506981278679040 |
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| author | Giang, Ha Huong Giang, Ha Huong Giang, Ha Huong |
| author_facet | Giang, Ha Huong Giang, Ha Huong Giang, Ha Huong |
| author_sort | Giang, Ha Huong |
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| description | UDC 517.5
We prove a uniqueness theorem of linearly nondegenerate holomorphic mappings from annulus to complex projective space $\mathbb{P}^n(\mathbb{C} )$ with different multiple values and a general condition on the intersections of the inverse images of these hyperplanes. |
| doi_str_mv | 10.37863/umzh.v73i2.107 |
| first_indexed | 2026-03-24T02:02:03Z |
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DOI: 10.37863/umzh.v73i2.107
UDC 517.5
Ha Huong Giang (Electric Power Univ., Hanoi, Vietnam)
UNIQUENESS THEOREM FOR HOLOMORPHIC MAPPINGS
ON ANNULI SHARING FEW HYPERPLANES
ТЕОРЕМА ЄДИНОСТI ДЛЯ ГОЛОМОРФНИХ ВIДОБРАЖЕНЬ
НА КIЛЬЦЯХ З КIЛЬКОМА СПIЛЬНИМИ ГIПЕРПЛОЩИНАМИ
We prove a uniqueness theorem of linearly nondegenerate holomorphic mappings from annulus to complex projective
space \BbbP n(\BbbC ) with different multiple values and a general condition on the intersections of the inverse images of these
hyperplanes.
Доведено теорему єдиностi для лiнiйно невироджених голоморфних вiдображень з кiльця до комплексного про-
ективного простору \BbbP n(\BbbC ) iз рiзними множинами значень i загальною умовою щодо перетину прообразiв гiпер-
площин.
1. Introduction. In 1975, H. Fujimoto [3] proved that if two linearly nondegenerate meromorphic
mappings of \BbbC m into \BbbP n(\BbbC ) which have the same inverse images of 3n+ 2 hyperplanes in general
position counted with multiplicities then they are identical.
In 1983, L. Smiley [9] obtained a uniqueness theorem for meromorphic mappings which share
3n + 2 hyperplanes in \BbbP n(\BbbC ) in general position without counting multiplicities (i.e., they have the
same inverse images of 3n+2 hyperplanes and are identical on these inverse images) and satisfy an
additional condition “codimension of the intersections of inverse images of two different hyperplanes
are at least two”.
Later on, the unicity problem of meromorphic mappings with truncated multiplicities has been
extended and deepened by contribution of many authors. These authors have improved the result of
L. Smiley in the case where the number of hyperplanes is replaced by a smaller one. We state here
the recent result of Z. Chen and Q. Yan [2] which is one of the best results available at present.
Take a meromorphic mapping f of \BbbC m into \BbbP n(\BbbC ) which is linearly nondegenerate over \BbbC m
such that for positive integers k, d, 1 \leq d \leq n, and q hyperplanes H1, . . . ,Hq in \BbbP n(\BbbC ) in general
position with
\mathrm{d}\mathrm{i}\mathrm{m} f - 1
\left( k+1\bigcap
j=1
Hij
\right) \leq m - 2, 1 \leq i1 < . . . < ik+1 \leq q.
Let \scrF
\bigl(
f, \{ Hi\} qi=1, k, d
\bigr)
be the set of all linearly nondegenerate over \BbbC m meromorphic maps
g : \BbbC m \rightarrow \BbbP n(\BbbC ) satisfying the conditions:
(a) \mathrm{m}\mathrm{i}\mathrm{n}(\nu (f,Hj), d) = \mathrm{m}\mathrm{i}\mathrm{n}(\nu (g,Hj), d), 1 \leq j \leq q,
(b) f(z) = g(z) on
\bigcup q
j=1 f
- 1(Hj).
Denote by \sharp S the cardinality of the set S.
Theorem A [2]. \sharp \scrF
\bigl(
f, \{ Hi\} 2n+3
i=1 , 1, 1
\bigr)
= 1.
c\bigcirc HA HUONG GIANG, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2 249
250 HA HUONG GIANG
In 2012, H. H. Giang, L. L. Quynh, and S. S. Quang [4] introduced new techniques to treat
the case k \geq 1. However, they only considered the case where the mappings f and g share all
hyperplanes with the same multiple values. Thus, our purpose of this paper is to prove a uniqueness
theorem for annulus similar to the results of Giang, Quynh, and Quang in the case where the mappings
f and g share all hyperplanes with different multiple values as following.
Theorem 1.1. Let f1, f2 : \BbbA (R0) \rightarrow \BbbP n(\BbbC ) be two admissible linearly nondegenerate holomor-
phic mappings, where \BbbA (R0) =
\biggl\{
z
\bigm| \bigm| \bigm| \bigm| 0 <
1
R0
< | z| < R0
\biggr\}
. Let H1, . . . ,Hq be hyperplanes in
\BbbP n(\BbbC ), located in general position and k (\leq n) be a positive integer. Let ki, 1 \leq i \leq q, be positive
integers or +\infty . Assume that:
(i) \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu 0(f1,Hi),\leq ki
\} = \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \nu 0(f2,Hi),\leq ki
\} for i = 1, . . . , q;
(ii) f - 1
1 (
\bigcap k+1
j=1 Hij ) = \varnothing , 1 \leq i1 < . . . < ik+1 \leq q;
(iii) f1 = f2 on
\bigcup q
j=1 f
- 1
1 (Hj).
Then we have f1 \equiv f2 if either q \geq 2(n+ 1)k and
q\sum
i=1
1
ki + 1
<
(q - n - 1)(q - 2k + 2kn) - 2qnk
(q - 2k + 2kn)n
or q < 2(n+ 1)k and
q\sum
i=1
1
ki + 1
<
q - n - 1 - (n+ 1)k
n
.
2. Some definitions and results from Nevanlinna theory on annuli. In this section, we will
recall some basic notions of Nevanlinna theory for meromorphic functions on annuli from [7] (see
also [1, 5, 6]).
For a divisor \nu on \BbbA (R0), which we may regard as a function on \BbbA (R0) with values in \BbbZ whose
support is discrete subset of \BbbA (R0), and for a positive integer M (maybe M = \infty ), we define the
counting function of \nu as follows:
n
[M ]
0 (t) =
\left\{
\sum
1\leq | z| \leq t
\mathrm{m}\mathrm{i}\mathrm{n}\{ M,\nu (z)\} , if 1 \leq t < R0,\sum
t\leq | z| <1
\mathrm{m}\mathrm{i}\mathrm{n}\{ M,\nu (z)\} , if
1
R0
< t < 1,
and
N
[M ]
0 (r, \nu ) =
1\int
1
r
n
[M ]
0 (t)
t
dt+
r\int
1
n
[M ]
0 (t)
t
dt, 1 < r < \infty .
For brevity we will omit the character [M ] if M = \infty .
For a divisor \nu and a positive integer k (maybe k = +\infty ), we define
\nu \leq k(z) =
\left\{ \nu (z), if \nu (z) \leq k,
0 otherwise
and \nu >k(z) =
\left\{ \nu (z), if \nu (z) > k,
0 otherwise.
For a meromorphic function \varphi , we define
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
UNIQUENESS THEOREM FOR HOLOMORPHIC MAPPINGS ON ANNULI SHARING FEW HYPERPLANES 251
\nu 0\varphi (resp., \nu \infty \varphi ) the divisor of zeros (resp., divisor of poles) of \varphi ,
\nu \varphi = \nu 0\varphi - \nu \infty \varphi ,
\nu 0\varphi ,\leq k = (\nu 0\varphi )\leq k, \nu
0
\varphi ,>k = (\nu 0\varphi )>k.
Similarly, we define \nu \infty \varphi ,\leq k, \nu
\infty
\varphi ,>k, \nu \varphi ,\leq k, \nu \varphi ,>k and their counting functions.
For a discrete subset S \subset \BbbA (R0), we consider it as a reduced divisor (denoted again by S ) whose
support is S, and denote by N0(r, S) its counting function. We also set \chi S(z) = 0 if z \not \in S and
\chi S(z) = 1 if z \in S.
Let f be a nonconstant meromorphic function on \BbbA (R0). The proximity function of f is defined
by
m0(r, f) =
1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}+
\bigm| \bigm| \bigm| \bigm| f\biggl( ei\theta
r
\biggr) \bigm| \bigm| \bigm| \bigm| d\theta + 1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}+
\bigm| \bigm| f(rei\theta )\bigm| \bigm| d\theta - 1
\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}+
\bigm| \bigm| f(ei\theta )\bigm| \bigm| d\theta
and the characteristic function of f is defined by
T0(r, f) = m0(r, f) +N0(r, \nu
\infty
f ).
Throughout this paper, we denote by Sf (r) quantities satisfying:
(i) in the case R0 = +\infty ,
Sf (r) = O
\bigl(
\mathrm{l}\mathrm{o}\mathrm{g}(rT0(r, f))
\bigr)
for r \in (1,+\infty ) except for a set \Delta r such that
\int
\Delta r
r\lambda - 1dr < +\infty for some \lambda \geq 0,
(ii) in the case R0 < +\infty ,
Sf (r) = O
\biggl(
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
T0(r, f)
R0 - r
\biggr) \biggr)
as r \rightarrow R0
for r \in (1, R0) except for a set \Delta \prime
r such that
\int
\Delta \prime
r
dr
(R0 - r)\lambda +1
< +\infty for some \lambda \geq 0.
The function f is said to be admissible if it satisfies
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r\rightarrow +\infty
T0(r, f)
\mathrm{l}\mathrm{o}\mathrm{g} r
= +\infty in the case R0 = +\infty
or
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r\rightarrow R -
0
T0(r, f)
- \mathrm{l}\mathrm{o}\mathrm{g}(R0 - r)
= +\infty in the case 1 < R0 < +\infty .
Thus for an admissible meromorphic function f on the annulus \BbbA (R0), we have Sf (r) = o(T0(r, f))
as r \rightarrow R0 for all 1 \leq r < R0 except for the set \Delta r or the set \Delta \prime
r mentioned above, respectively
(cf. [1]).
A meromorphic function a on \BbbA (R0) is said to be small with respect to f if
T0(r, a) = Sf (r).
Through this paper, by notation “\| P ”, we mean that the asseartion P holds for all 1 \leq r < R0
except for the set \Delta r or the set \Delta \prime
r mentioned above, respectively.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
252 HA HUONG GIANG
Lemma 2.1 (Lemma on logarithmic derivatives [1, 5 – 7]). Let f be a nonzero meromorphic
function on \BbbA (R0). Then for each k \in \BbbN we have
m0
\Biggl(
r,
f (k)
f
\Biggr)
= Sf (r), 1 \leq r < R0.
Theorem 2.1 (First main theorem for meromorphic functions and values [1, 5 – 7]). Let f be a
meromorphic function on \BbbA (R0). Then for each a \in \BbbC we have
T0(r, f) = T0
\biggl(
r,
1
f - a
\biggr)
+ Sf (r), 1 \leq r < R0.
Then for every small (with respect to f ) function a (a \not \equiv \infty ) on \BbbA (R0), we obtain
T0(r, f) \leq T0(r, f - a) + T0(r, a) = T0
\biggl(
r,
1
f - a
\biggr)
+ S1/(f - a)(r) + Sf (r).
Similarly, we get
T0
\biggl(
r,
1
f - a
\biggr)
= T0(r, f - a) + S1/(f - a)(r) \leq
\leq T0(r, f) + T0(r, - a) + S1/(f - a)(r) + Sf (r).
Therefore, we have the first main theorem for meromorphic functions and small function as follows.
Theorem 2.2 (First main theorem for meromorphic functions and small functions). Let f be a
meromorphic function on \BbbA (R0) and let a be a small function with respect to f. Then we have
T0(r, f) = T0
\biggl(
r,
1
f - a
\biggr)
+ Sf (r), 1 \leq r < R0.
3. Nevanlinna theory for holomorphic mappings from an annulus into a projective space.
Let f be a holomorphic mapping from an annulus \BbbA (R0) into \BbbP n(\BbbC ) with a reduced representation
f = (f0 : . . . : fn). The characteristic function of f is defined by
T0(r, f) =
1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| f(rei\theta )\bigm\| \bigm\| d\theta + 1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| \bigm\| \bigm\| f\biggl( 1
r
ei\theta
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| d\theta - 1
\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| f(ei\theta )\bigm\| \bigm\| d\theta ,
where \| f\| =
\bigl(
| f0| 2 + . . .+ | fn| 2
\bigr) 1
2 .
Let H be a hyperplane in \BbbP n(\BbbC ) given by H =
\bigl\{
(\omega 0 : . . . : \omega n) | a0\omega 0 + . . .+ aN\omega n = 0
\bigr\}
. We
set (f,H) = a0f0 + . . .+ anfn. The proximity function of f with respect to H is defined by
m0(r, f,H) =
1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| f(rei\theta )\bigm\| \bigm\| \| H\|
| (f,H)(rei\theta )|
d\theta +
+
1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| \bigm\| \bigm\| f\biggl( 1
r
ei\theta
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| \| H\| \bigm| \bigm| \bigm| \bigm| (f,H)
\biggl(
1
r
ei\theta
\biggr) \bigm| \bigm| \bigm| \bigm| d\theta -
1
\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| f(ei\theta )\bigm\| \bigm\| \| H\| \bigm| \bigm| (f,H)(ei\theta )
\bigm| \bigm| d\theta ,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
UNIQUENESS THEOREM FOR HOLOMORPHIC MAPPINGS ON ANNULI SHARING FEW HYPERPLANES 253
where \| H\| =
\bigl(
| a0| 2 + . . .+ | an| 2
\bigr) 1
2 .
By Jensen formula, we obtain
T0(r, f) = m0(r, f,H) +N0(r, f
\ast H),
where f\ast H denotes the pull back divisor of H by f.
Lemma 3.1. Let f be as above. Let H1 and H2 be two distinct hyperplanes of \BbbP n(\BbbC ), then we
have
T0
\biggl(
r,
(f,H1)
(f,H2)
\biggr)
\leq T0(r, f) +O(1).
Let \{ Hi\} qi=1, q \geq n + 2, be a set of q hyperplanes in \BbbP n(\BbbC ). We say that the family \{ Hi\} qi=1 is
in general position if
\bigcap n+1
j=1 Hij = \varnothing for any 1 \leq i1 < . . . < in+1 \leq q. Using the same argument
as in the proof of the Second Main Theorem for holomorphic curves from \BbbC into \BbbP n(\BbbC ) (see [8],
Theorem 3.1), we have the following Second Main Theorem for holomorphic curves from an annulus
into \BbbP n(\BbbC ).
Theorem 3.1. Let f : \BbbA (R0) \rightarrow \BbbP n(\BbbC ) be a linearly nondegenerate holomorphic mapping. Let
\{ Hi\} qi=1, q \geq n+ 2, be a set of q hyperplanes in \BbbP n(\BbbC ) in general position. Then
(q - n - 1)T0(r, f) \leq
q\sum
i=1
N
[n]
0 (r, f\ast Hi) + Sf (r), 1 \leq r < R0,
where f\ast Hi denotes the pull back divisor of Hi by f.
4. Proof of Theorem 1.1. In order to prove Theorem 1.1, we need the following.
Lemma 4.1. Let f be nonconstant holomorphic mappings of \BbbA (R0) into \BbbP n(\BbbC ). Let H be a
hyperplane in \BbbP n(\BbbC ) in general position and k (\geq n) be a positive integer. Then
N
[n]
0
\bigl(
r, \nu 0(f,H)
\bigr)
\leq n
\biggl(
1 - n
k + 1
\biggr)
N
[1]
0
\bigl(
r, \nu 0(f,H),\leq k
\bigr)
+
n
k + 1
N0
\bigl(
r, \nu 0(f,H)
\bigr)
and
N
[n]
0
\bigl(
r, \nu 0(f,H)
\bigr)
\leq n
\biggl(
1 - n
k + 1
\biggr)
N
[1]
0
\bigl(
r, \nu 0(f,H),\leq k
\bigr)
+
n
k + 1
T0(r, f) + Sf (r).
Proof. From
N
[n]
0
\bigl(
r, \nu 0(f,H)
\bigr)
= N
[n]
0
\bigl(
r, \nu 0(f,H),\leq k
\bigr)
+N
[n]
0 (r, \nu 0(f,H),>k)
and
N
[n]
0 (r, \nu 0(f,H),>k) \leq
n
k + 1
N0(r, \nu
0
(f,H),>k) \leq
n
k + 1
\Bigl(
N0
\bigl(
r, \nu 0(f,H)
\bigr)
- N
[n]
0
\bigl(
r, \nu 0(f,H),\leq k
\bigr) \Bigr)
we deduce that
N
[n]
0
\bigl(
r, \nu 0(f,H)
\bigr)
\leq
\biggl(
1 - n
k + 1
\biggr)
N
[n]
0
\bigl(
r, \nu 0(f,H),\leq k
\bigr)
+
n
k + 1
N0
\bigl(
r, \nu 0(f,H)
\bigr)
\leq
\leq n
\biggl(
1 - n
k + 1
\biggr)
N
[1]
0
\bigl(
r, \nu 0(f,H),\leq k
\bigr)
+
n
k + 1
N0
\bigl(
r, \nu 0(f,H)
\bigr)
.
This completes the proof of the first inequality of the lemma. The second inequality of the lemma
follows immediately because of N0
\bigl(
r, \nu 0(f,H)
\bigr)
\leq T0(r, f) + Sf (r).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
254 HA HUONG GIANG
Lemma 4.2. Let f1, f2 be nonconstant admissible holomorphic mappings of \BbbA (R0) into \BbbP n(\BbbC ).
Let \{ Hi\} qi=1, q \geq n+2, be hyperplanes in \BbbP n(\BbbC ) in general position. Let ki, 1 \leq i \leq q, be positive
integers or +\infty . Assume that
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,Hi),\leq ki
, 1
\Bigr\}
= \mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f2,Hi),\leq ki
, 1
\Bigr\}
for all 1 \leq i \leq q.
If
\sum q
i=1
1
ki + 1
<
q - n - 1
n
, then \| T0(r, f2) = O(T0(r, f1)) and \| T0(r, f1) = O(T0(r, f2)).
Proof. By the Second Main Theorem, we have
\bigm\| \bigm\| (q - n - 1)T0(r, f2) \leq
q\sum
i=1
N
[n]
0
\bigl(
r, \nu 0(f2,Hi)
\bigr)
+ Sf2(r) \leq
\leq
q\sum
i=1
\biggl(
n
\biggl(
1 - n
ki + 1
\biggr)
N
[1]
0
\bigl(
r, \nu 0(f2,H),\leq ki
\bigr)
+
n
ki + 1
T0(r, f2)
\biggr)
+Sf2(r) \leq
\leq
q\sum
i=1
\biggl(
nN
[1]
0
\bigl(
r, \nu 0(f1,Hi),\leq ki
\bigr)
+
n
ki + 1
T0(r, f2)
\biggr)
+ Sf2(r) \leq
\leq qn T0(r, f1) + n
q\sum
i=1
1
ki + 1
T0(r, f2) + Sf2(r).
Thus \bigm\| \bigm\| \bigm\| \bigm\| \biggl( q - n - 1 - n
q\sum
i=1
1
ki + 1
\biggr)
T0(r, f2) \leq qnT0(r, f1) + Sf2(r).
Case 1. If R0 = +\infty , then
\bigm\| \bigm\| Sf2(r) = O
\bigl(
\mathrm{l}\mathrm{o}\mathrm{g}(rT0(r, f2))
\bigr)
. Therefore there exists a positive
constant K such that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow +\infty
Sf2(r)
T0(r, f2)
= \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow +\infty
K \mathrm{l}\mathrm{o}\mathrm{g}(rT0(r, f2))
T0(r, f2)
= 0.
Case 2. If 1 < R0 < +\infty then
\bigm\| \bigm\| Sf2(r) = O
\biggl(
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
T0(r, f2)
R0 - r
\biggr) \biggr)
as r \rightarrow R0. Therefore there
exists a positive constant K such that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow R0, r /\in \Delta \prime
r
Sf2(r)
T0(r, f2)
= \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow R0, r /\in \Delta \prime
r
K \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
T0(r, f2)
R0 - r
\biggr)
T0(r, f2)
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r\rightarrow R0
K \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
R0 - r
\biggr)
T0(r, f2)
= 0.
Hence \| T0(r, f2) = O(T0(r, f1)). Similarly, we get \| T0(r, f1) = O(T0(r, f2)).
Lemma 4.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
UNIQUENESS THEOREM FOR HOLOMORPHIC MAPPINGS ON ANNULI SHARING FEW HYPERPLANES 255
Proof of Theorem 1.1. Assuming that f1 \not \equiv f2. By changing indices if necessary, we may
assume that
(f1, H1)
(f2, H1)
\equiv (f1, H2)
(f2, H2)
\equiv . . . \equiv (f1, Hk1)
(f2, Hk1)\underbrace{} \underbrace{}
group 1
\not \equiv (f1, Hk1+1)
(f2, Hk1+1)
\equiv . . . \equiv (f1, Hk2)
(f2, Hk2)\underbrace{} \underbrace{}
group 2
\not \equiv
\not \equiv (f1, Hk2+1)
(f2, Hk2+1)
\equiv . . . \equiv (f1, Hk3)
(f2, Hk3)\underbrace{} \underbrace{}
group 3
\not \equiv . . . \not \equiv
(f1, Hks - 1+1)
(f2, Hks - 1+1)
\equiv . . . \equiv (f1, Hks)
(f2, Hks)\underbrace{} \underbrace{}
group s
,
where ks = q.
For each 1 \leq i \leq q, we set
\sigma (i) =
\left\{ i+ n, if i+ n \leq q,
i+ n - q, if i+ n > q,
and
Pi = (f1, Hi)(f2, H\sigma (i)) - (f2, Hi)(f1, H\sigma (i)).
Since f1 \not \equiv f2, the number of elements of each group is at most n. Then
(f1, Hi)
(f2, Hi)
and
(f1, H\sigma (i))
(f2, H\sigma (i))
belong to distinct groups. Therefore Pi \not \equiv 0, 1 \leq i \leq q. We set
P =
q\prod
i=1
Pi \not \equiv 0
and
S =
\bigcup
1\leq i1<...<ik+1\leq q
f - 1
1
\left( k+1\bigcap
j=1
Hij
\right) .
Then S is an analytic set of codimension at most 2. By Jensen formula and by the definition of the
characteristic function, we have
NP (r) \leq
1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| P (rei\theta )
\bigm\| \bigm\| d\theta + 1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| \bigm\| \bigm\| P\biggl( 1
r
ei\theta
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| d\theta - 1
\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm\| \bigm\| P (ei\theta )
\bigm\| \bigm\| d\theta \leq
\leq 1
2\pi
q\sum
i=1
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\Bigl(
| (f1, Hi)| 2 + | (f1, H\sigma (i)| 2
\Bigr) 1
2
d\theta +
+
1
2\pi
q\sum
i=1
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\Bigl(
| (f2, Hi)| 2 + | (f2, H\sigma (i)| 2
\Bigr) 1
2
d\theta \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
256 HA HUONG GIANG
\leq 1
2\pi
q\sum
i=1
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\Bigl(
\| f1\|
\bigl(
\| Hi\| 2 + \| H\sigma (i)\| 2
\bigr) 1
2
\Bigr)
d\theta +
+
1
2\pi
q\sum
i=1
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\Bigl(
\| f2\|
\bigl(
\| Hi\| 2 + \| H\sigma (i)\| 2
\bigr) 1
2
\Bigr)
d\theta =
= q
\left( 1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}(\| f1\| )d\theta +
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}(\| f2\| )d\theta
\right) +O(1) =
= q
\bigl(
T0(r, f1) + T0(r, f2)
\bigr)
+O(1). (4.1)
On the other hand, we let \xi :=
1
z
, then f1(\xi ), f2(\xi ) are holomorphic mappings on \BbbA (R0). By
applying the Second Main Theorem we get, for i = 1, 2,
(q - n - 1)T0(r, fi) \leq
q\sum
j=1
N
[n]
0
\bigl(
r, \nu 0(fi,Hj)
\bigr)
+ Sfi(r). (4.2)
We put Sf (r) = Sf1(r) + Sf2(r), 1 \leq r < R0.
Fix a point z \not \in I(f1)\cup I(f2)\cup S. We assume that z is a zero of functions (f1, Hi1), . . . , (f1, Hit)
with multiplicities m1, . . . ,mt, respectively, where 1 \leq i1 < . . . < it \leq q, t \leq k, and z is not zero
of any (f1, Hi) for i \not \in \{ i1, . . . , it\} . For an index i \in \{ 1, . . . , q\} , we distinguish the following four
cases:
Case 1: i, \sigma (i) \not \in \{ i1, . . . , it\} . Then z is a zero point of Pi with multiplicity at least 1, since
f1(z) = f2(z). We denote v(z) the number of indices i in this case. It is easy to see that v(z) \geq
\geq q - 2t.
Case 2: i \in \{ i1, . . . , it\} and \sigma (i) \not \in \{ i1, . . . , it\} . Then z is a zero point of Pi with multiplicity
at least \mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,Hi),\leq ki
, \nu 0(f2,Hi),\leq ki
\Bigr\}
.
Case 3: \sigma (i) \in \{ i1, . . . , it\} and i \not \in \{ i1, . . . , it\} . Then z is a zero point of Pi with multiplicity
at least \mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,H\sigma (i)),\leq ki
, \nu 0(f2,H\sigma (i)),\leq ki
\Bigr\}
.
Case 4: i, \sigma (i) \in \{ i1, . . . , it\} . Then z is a zero point of Pi with multiplicity at least
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,Hi),\leq ki
, \nu 0(f2,Hi),\leq ki
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,H\sigma (i)),\leq ki
, \nu 0(f2,H\sigma (i)),\leq ki
\Bigr\}
.
Therefore, from the above four cases, it follows that
\nu P (z) \geq 2
t\sum
j=1
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,Hij
),\leq kij
(z), \nu 0(f2,Hij
),\leq kij
(z)
\Bigr\}
+ v(z) \geq
\geq 2
t\sum
j=1
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,Hij
),\leq kij
(z), \nu 0(f2,Hij
),\leq kij
(z)
\Bigr\}
+ q - 2t.
We consider the following two cases.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
UNIQUENESS THEOREM FOR HOLOMORPHIC MAPPINGS ON ANNULI SHARING FEW HYPERPLANES 257
Case 1. If q \geq 2(n + 1)k, then by using the fact that, for any two positive integer a and b,
\mathrm{m}\mathrm{i}\mathrm{n}\{ a, b\} \geq \mathrm{m}\mathrm{i}\mathrm{n}\{ a, n\} +\mathrm{m}\mathrm{i}\mathrm{n}\{ b, n\} - n, we get
\nu 0P (z) \geq 2
t\sum
j=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hij
),\leq kij
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hij
),\leq kij
\Bigr\}
- n
\Bigr)
+ q - 2t =
= 2
t\sum
j=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hij
),\leq kij
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hij
),\leq kij
\Bigr\} \Bigr)
- 2nt+ q - 2t \geq
\geq 2
t\sum
j=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hij
),\leq kij
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hij
),\leq kij
\Bigr\} \Bigr)
+ q - 2(n+ 1)k \geq
\geq
\biggl(
2 +
q - 2(n+ 1)k
2nk
\biggr) q\sum
i=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hi),\leq ki
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hi),\leq ki
\Bigr\} \Bigr)
=
=
q - 2k + 2kn
2nk
q\sum
i=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hi),\leq ki
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hi),\leq ki
\Bigr\} \Bigr)
for all z outside the analytic set I(f1) \cup I(f2) \cup S.
Integrating both sides of the above inequality, we get
NP (r) \geq
q - 2k + 2kn
2nk
q\sum
i=1
\Bigl(
N
[n]
0
\bigl(
r, \nu 0(f1,Hi),\leq ki
\bigr)
+N
[n]
0
\bigl(
r, \nu 0(f2,Hi),\leq ki
\bigr) \Bigr)
=
=
q - 2k + 2kn
2nk
q\sum
i=1
\Bigl(
N
[n]
0
\bigl(
r, \nu 0(f1,Hi)
\bigr)
- N
[n]
0
\bigl(
r, \nu 0(f1,Hi),>ki
\bigr)
+
+N
[n]
0
\bigl(
r, \nu 0(f2,Hi)
\bigr)
- N
[n]
0
\bigl(
r, \nu 0(f2,Hi),>ki
\bigr) \Bigr)
\geq
\geq q - 2k + 2kn
2nk
q\sum
i=1
\biggl(
N
[n]
0
\bigl(
r, \nu 0(f1,Hi)
\bigr)
+N
[n]
0
\bigl(
r, \nu 0(f2,Hi)
\bigr)
- n
ki + 1
\bigl(
T0(r, f1) + T0(r, f2)
\bigr) \biggr)
.
(4.3)
Combining (4.2) and (4.3), it shows that
NP (r) \geq
q - 2k + 2kn
2nk
\Biggl( \biggl(
q - n - 1 -
q\sum
i=1
n
ki + 1
\biggr) \bigl(
T0(r, f1) + T0(r, f2)
\bigr)
- Sf (r)
\Biggr)
. (4.4)
Thus, by (4.1) and (4.4) we have
q
\bigl(
T0(r, f1) + T0(r, f2)
\bigr)
\geq
\geq q - 2k + 2kn
2nk
\Biggl( \biggl(
q - n - 1 -
q\sum
i=1
n
ki + 1
\biggr) \bigl(
T0(r, f1) + T0(r, f2)
\bigr)
- Sf (r)
\Biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
258 HA HUONG GIANG
This implies that\Biggl(
q - n - 1 -
q\sum
i=1
n
ki + 1
- 2qnk
q - 2k + 2kn
\Biggr) \bigl(
T0(r, f1) + T0(r, f2)
\bigr)
\leq Sf (r). (4.5)
Case 2. If q < 2(n+ 1)k, then we get
\nu 0P (z) \geq
\biggl(
2 - q
(n+ 1)k
\biggr) t\sum
j=1
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\nu 0(f1,Hij
),\leq kij
, \nu 0(f2,Hij
),\leq kij
\Bigr\}
+
+
q
(n+ 1)k
t\sum
j=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hij
),\leq kij
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hij
),\leq kij
\Bigr\}
- n
\Bigr)
+ q - 2t \geq
\geq
\biggl(
2 - q
(n+ 1)k
\biggr)
t - qnt
(n+ 1)k
+ q - 2t+
+
q
(n+ 1)k
t\sum
j=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hij
),\leq kij
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hij
),\leq kij
\Bigr\} \Bigr)
\geq
\geq q
(n+ 1)k
q\sum
i=1
\Bigl(
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f1,Hi),\leq ki
\Bigr\}
+\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
n, \nu 0(f2,Hij
),\leq ki
\Bigr\} \Bigr)
for all z outside the analytic set I(f1) \cup I(f2) \cup S.
Integrating both sides of the above inequality, we obtain
NP (r) \geq
q
(n+ 1)k
q\sum
i=1
\Bigl(
N
[n]
0
\bigl(
r, \nu 0(f1,Hi),\leq ki
\bigr)
+N
[n]
0
\bigl(
r, \nu 0(f2,Hi),\leq ki
\bigr) \Bigr)
=
=
q
(n+ 1)k
q\sum
i=1
(N
[n]
0
\bigl(
r, \nu 0(f1,Hi)
\bigr)
- N
[n]
0
\bigl(
r, \nu 0(f1,Hi),>ki
\bigr)
+
+N
[n]
0
\bigl(
r, \nu 0(f2,Hi)
\bigr)
- N
[n]
0
\bigl(
r, \nu 0(f2,Hi),>ki
\bigr) \bigr)
\geq
\geq q
(n+ 1)k
q\sum
i=1
\biggl(
N
[n]
0
\bigl(
r, \nu 0(f1,Hi)
\bigr)
+N
[n]
0
\bigl(
r, \nu 0(f2,Hi)
\bigr)
- n
ki + 1
\bigl(
T0(r, f1) + T0(r, f2)
\bigr) \biggr)
. (4.6)
Combining (4.2) and (4.6), it shows that
NP (r) \geq
q
(n+ 1)k
\Biggl( \Biggl(
q - n - 1 -
q\sum
i=1
n
ki + 1
\Biggr) \bigl(
T0(r, f1) + T0(r, f2)
\bigr)
- Sf (r)
\Biggr)
. (4.7)
Thus, by (4.1) and (4.7) we have
q
\bigl(
T0(r, f1) + T0(r, f2)
\bigr)
\geq
\geq q
(n+ 1)k
\Biggl( \Biggl(
q - n - 1 -
q\sum
i=1
n
ki + 1
\Biggr) \bigl(
T0(r, f1) + T0(r, f2)
\bigr)
- Sf (r)
\Biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
UNIQUENESS THEOREM FOR HOLOMORPHIC MAPPINGS ON ANNULI SHARING FEW HYPERPLANES 259
This implies that\Biggl(
q - n - 1 -
q\sum
i=1
n
ki + 1
- (n+ 1)k
\Biggr) \bigl(
T0(r, f1) + T0(r, f2)
\bigr)
\leq Sf (r). (4.8)
We consider the following two cases.
Case 1. If R0 = +\infty , then \| Sf (r) = O(\mathrm{l}\mathrm{o}\mathrm{g}(r
\bigl(
T0(r, f1) + T0(r, f2)
\bigr)
). Therefore there exists a
positive constant K such that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow +\infty
Sf (r)
T0(r, f1) + T0(r, f2)
= \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow +\infty
K \mathrm{l}\mathrm{o}\mathrm{g}(r
\bigl(
T0(r, f1) + T0(r, f2)
\bigr)
)
T0(r, f1)) + T0(r, f2)
= 0.
Letting r \rightarrow \infty , we get two subcases.
Subcase 1.1. If q \geq 2(n+ 1)k, then by (4.5) we have
q - n - 1 -
q\sum
i=1
n
ki + 1
- 2qnk
q - 2k + 2kn
\leq 0,
i.e.,
q\sum
i=1
1
ki + 1
\geq (q - n - 1)(q - 2k + 2kn) - 2qnk
(q - 2k + 2kn)n
.
This is a contradiction.
Subcase 1.2. If q < 2(n+ 1)k, then by (4.8) we get
q - n - 1 -
q\sum
i=1
n
ki + 1
- (n+ 1)k \leq 0,
i.e.,
q\sum
i=1
1
ki + 1
\geq q - n - 1 - (n+ 1)k
n
.
This is a contradiction.
Case 2. If 1 < R0 < +\infty , then\bigm\| \bigm\| Sf (r) = O
\biggl(
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
T0(r, f1) + T0(r, f2)
R0 - r
\biggr) \biggr)
as r \rightarrow R0.
Therefore, there exists a positive constant K such that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow R0, r /\in \Delta \prime
r
Sf (r)
T0(r, f1) + T0(r, f2)
= \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow R0, r /\in \Delta \prime
r
K \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
T0(r, f1) + T0(r, f2)
R0 - r
\biggr)
T0(r, f1) + T0(r, f2)
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r\rightarrow R0
K \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
R0 - r
\biggr)
T0(r, f1) + T0(r, f2))
= 0.
Letting r \rightarrow R0, by repeating the same arguments of the Case 1, we get a contradiction.
Hence, from the above two cases, it follows that f1 \equiv f2.
Theorem 1.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
260 HA HUONG GIANG
References
1. T. B. Cao, Z. S. Deng, On the uniqueness of meromorphic functions that share three or two finite sets on annuli,
Proc. Indian Acad. Sci., 122, No 2, 203 – 220 (2012).
2. Z. Chen, Q. Yan, Uniqueness theorem of meromorphic mappings into \BbbP N (\BbbC ) sharing 2N+3 hyperplanes regardless
of multiplicities, Internat. J. Math., 20, 717 – 726 (2009).
3. H. Fujimoto, The uniqueness problem of meromorphic maps into the complex projective space, Nagoya Math. J., 58,
1 – 23 (1975).
4. H. H. Giang, L. N. Quynh, S. D. Quang, Uniqueness theorems for meromorphic mappings sharing few hyperplanes,
J. Math. Anal. and Appl., 393, 445 – 456 (2012).
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23, № 01, 19 – 30 (2005).
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Received 21.05.18,
after revision — 08.03.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
|
| id | umjimathkievua-article-107 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:03Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6e/5063fd1aa825b72503552d8422fd816e.pdf |
| spelling | umjimathkievua-article-1072025-03-31T08:48:28Z Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes Giang, Ha Huong Giang, Ha Huong Giang, Ha Huong UDC 517.5 We prove a uniqueness theorem of linearly nondegenerate holomorphic mappings from annulus to complex projective space $\mathbb{P}^n(\mathbb{C} )$ with different multiple values and a general condition on the intersections of the inverse images of these hyperplanes. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; УДК 517.5 Теорема єдиності для голоморфних відображень на кільцях з кількома спільними гіперплощинами Доведено теорему єдиності для лінійно невироджених голоморфних відображень з кільця до комплексного про\-ективного простору $\mathbb{P}^n(\mathbb{C} )$ із різними множинами значень і загальною умовою щодо перетину прообразів гіперплощин. Institute of Mathematics, NAS of Ukraine 2021-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/107 10.37863/umzh.v73i2.107 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 2 (2021); 249 - 260 Український математичний журнал; Том 73 № 2 (2021); 249 - 260 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/107/8943 Copyright (c) 2021 Giang Huong Ha |
| spellingShingle | Giang, Ha Huong Giang, Ha Huong Giang, Ha Huong Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| title | Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| title_alt | Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| title_full | Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| title_fullStr | Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| title_full_unstemmed | Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| title_short | Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| title_sort | uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/107 |
| work_keys_str_mv | AT gianghahuong uniquenesstheoremforholomorphicmappingsonannulisharingfewhyperplanes AT gianghahuong uniquenesstheoremforholomorphicmappingsonannulisharingfewhyperplanes AT gianghahuong uniquenesstheoremforholomorphicmappingsonannulisharingfewhyperplanes |