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MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF EDLICATION
—————————–

VANGTY NOULORVANG

SOME THEORMS ON UNIQUENESS AND
FINITENESS OF MEROMORPHIC MAPPINGS

Specialized: Geometry and Topology
Code: 9.46.10.05

SUMMARY OF THESIS DOCTOR MATHEMATICS

Hanoi, 01-2021


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The thesis is completed at: Hanoi University of Education

NScience instructor: Assoc. Prof. PHAM DUC THOAN
Assoc. Prof. PHAM HOANG HA

Rewier 1: Prof. Ha Huy Khoai
Rewier 2: Prof. Ta Thi Hoai An
Rewier 3: Prof. Tran Van Tan

The thesis was defended at the Thesis-level Thesis Judging Council meeting at .......................

Thesis can be found at: - Thesis can be found at

-Library of Hanoi University of Education


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INTRODUCTION
1. Rationale
Value distribution theory was built by the famous mathematician
R. Nevanlinna in the 20 of the last century. Since its inception, this
theory has attracted many great mathematicians around the world
to study. Many remarkable results and great applications of this
theory in different mathematical disciplines have been discovered.
The basic content of the value distribution theory is to establish the
second main theorem, nhichis about the relationship between the
zero counting function and the increase of the charateristic functions.
This theorem has many applications in studying the problem of
uniqueness, finiteness, algebraic dependence, defect relations as well
as the distribution of values of meromorphic mappings.
In order to establish a second main theorem for the meromorphic mapping from Cm into the piojective space Pn(C), they based
on the Logarithmic Derivative Lemma and the property of Wronskian’s determinant (c-Casorati and p-Casorati) and replaced the
Logarithmic Derivative Lemma by a similar lemma, which is called
the c-differences or c-differences lemma for a zero-order meromorphic map or for a meromorphic mapping of hyper-order less than 1
respectively. From there, they were able to study the uniqueness of
these such as the general Picard’s theorem. This second main theorem is called the second main theorem p-differences or c-differences
with targets. Using these approaches, in 2016, T. B. Cao and R. Korhonen have established the second main theorem p-differences for
the meromorphic mapping from Cm into the the piojective Pn(C)
intersecting the superspattice at the hyperplanes in subgeral position.
In the one-dimensional case, since R. Halburd and R. Korhonen

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have given the lemma c-differences the secmd main theorem cdifferences for the polymorphism function with super order less than
1, unique theorem Picard’s theorem similar to R. Nevanlinna’s 5
point theorem is well studied. There are many interesting results in
this direction. For example, in 2009, J. Heittokangas et al. proved
that if the f (z) meromorphic function f (z) has finite order shares
3 distinct values that count multiples with the f (z + c), then f is a
periodic function with period c, that is, f (z) = f (z + c) for every
z ∈ C. This Picard theorem is improved by these authors for the
case of sharing two multiples and one non-multiples.
In early 2016, K. S. Charak, R. J. Korhonen and G. Kumar
gave counterexamples to show that there is no unique theorem for
the case that 1 value sharing counts multiplicities and two shared
values without multiplicities. Note that, in R. Nevanlinna’s 5 point
theorem, 5 shared values don’t need Count multiplicities. A question
arises is there a Picard theorem in the case in the case the number
of shared values that are not multiplied is 4. The authors have tried
to answer the above question and they have obtained results for a
meromorphic map of hypcr-order less than 1 sharing 4 values under
one defect condition.
In 2018, W. Lin, X. Lin and A. Wu hnd a counter-example shoued
that the result is no longer correct when the multiples of shared values are interrupted. From there, they posed the problem of studying
the uniqueness of Picard’s theorem when values are truncated multiplicities. One of the goals when studying the unigcueness problem
is to reduce the number of shared values. Accordingly, we pose research problems and improve the results of W. Lin, X. Lin and A.
Wu.
The problem of the algebraic dependence of the meromorphic


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mappings from Cm to Pn(C) is studied in the paper of S. Ji and
so far have announced many results. Some of the best recent results
are of Z. Chen and Q. Yan, S. D. Quang, S. D. Quang and L. N.
Quynh. Note that, by studying the algebraic dependance of the 3
meromorphic maps, having an inverse image intersecting the 2n + 2
hyperplance in general position, it helped S. D. Quang obtained the
finiteness of that class of meromorphic mappings.
However, as noted above, reducing the number of hyperplance
in the results is one of the important goals in value distribution
theory. Therefore, we set out the purpose of studying the finiteness
of the above polymorphic maps with the number of hyperplance less
than 2n + 2 through the algebraic dependence of 3 meromorphic
mappings.
From the above reasons, we choose the thesis “Some theorems on
uniqueness and finiteness of meromorphic mappings" to study in
depth the unique problems of the meromorphic mappings and their
shifts, as well as the finite problems for the meromorphic mappings.
2. Objectives of research
The first purpose of the thesis is to give and prove some uniaueness
theorems of the meromorphic functions f (z) on the complex plane
which has hyperorder plane less than 1 share a part of the values
with its f (z + c).
Following that, the dissertation establishes some second main theorems and a some uniquenss Picard theorems for the meromorphic
mapping from Cm into projective space Pn(C) intersecting interact
with hypersurfaces.
Finally, the dissertation studies the finiteness through establishing
the theorem of algebraic dependence of 3 meromorphic mappings
from Cm into the projective space Pn(C) intersecting with 2n + 1

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hyperplanes in general position.
3. Object and scope of research
The research object of the dissertation is some uniqueness Picard
theorems and the problem of algebraic dependence as well as the
finiteness of meromorphic mappings.
The thesis is studied in the scope of Nevanlinna theory for meromorphic mapping.
4. Methodology
To solve the problems posed in the dissertation, we use the
methods of value distribution theory and complex geometry. Besides
using traditional techniques, we come up with new techniques to
achieve the goals set out in the thesis.
5. Scientific and practical significances
The dissertation contributes to deepening the results on the
uniqueness and finiteness of the meromorphic functions or the meromorphic mappings. Besides enriching these problems, the dissertation also gives new results for the algebraic dependence of 3 meromorphic mappings in the projective space with few hyperplanes.
The dissertation is a useful reference for students, graduate students and postgraduate students in this direction.

6. Structure
The structure of the dissertation consists of four main chapters.
Overview Chapter devoted to analyzing some research results related
to the content of the topic of domestic and foreign authors. The
remaining three chapters present preparation knowledge as well as
detailed evidence for the new results of the topic.
Chapter I. Overview.


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Chapter II. The uniqueness for the meromorphic function that
has hyper degree less than 1
Chapter III. The uniqueness for zero ordcr meromorphic mapping
Chapter IV.The algebraic degneracy and finiteness of meromorphic mappings.
The dissertation is based on three published articles.
7. Place of writing the dissertation
Hanoi National University of Education.


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Chapter 1: Overview
I. The uniqueness for the meromorphic function that
has hyper order less than 1
Finding the conditions for the meromorphic function f (z) on the
complex plane coincides with its shift f (z + c) has been vigorously
studied in recent years. Since the work of R. Halburd and R. Korhonen, there have been many interesting uniqueness theorems similar
to Nevanlinna’s 5 point theorem. For example, in 2009, J. Heittokangas and colleagues considered this problem for a meromorphic
function f (z) with a finite order on complex plane sharing 3 CM
values with its shift f (z + c). The results are then improved for the
case of sharing two CM values and one IM value by these authors.
In 2016, K. S. Charak, R. Korhonen and G. Kumar gave an
example to show that the case of sharing one CM value and two
IM values (and thus three IM values) is not happening in general
case.
The concept of partial sharing of values of a meromorphic function
with hyper order less than 1 was introduced by K. S. Charak,
R. Korhonen and G. Kumar. They have a uniquess theorem for
a meromorphic function with hyper order less than 1 that shares
partialhy four values the IM with its shift under the following defect

condition.
Theorem A. Let f be a nonconstant meromorphic function of
ˆ ) be
hyper-order γ(f ) < 1 and c ∈ C \ {0}. Let a1, a2, a3, a4 ∈ S(f
four distinct periodic functions with period c. If δ(a, f ) > 0 for
ˆ ) and
some a ∈ S(f
E(aj , f (z)) ⊆ E(aj , f (z + c)), j = 1, 2, 3, 4
then f (z) = f (z + c) for all z ∈ C.


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In 2018, W. Lin, X. Lin and A. Wu [11] obtained a counterexample which showed that Theorem A does not hold when the condition
"partially shared values E(aj , f (z)) ⊆ E(aj , f (z + c)), j = 1, 2"
is replaced by the condition "truncated partially shared values
E ≤k (aj , f (c)) ⊆ E ≤k (aj , f (z + c)), j = 1, 2 " with a positive integer k, even if f (z) and f (z + c) share a3, a4 CM. Then, they
introduced the following results under a reduced deficiency assump2
tion Θ(0, f ) + Θ(∞, f ) > k+1
. An example was also given to show
that this condi-tion is necessary and sharp.
Theorem B. Let f be a nonconstant meromorphic function of
hyper-order γ(f ) < 1 and c ∈ C \ {0}. Let k1, k2 be two positive
ˆ ) be four distinct
integers, and let a1, a2 ∈ S(f )\{0}, a3, a4 ∈ S(f
periodic functions with period c such that f (z) and f (z + c) share
a3, a4 CM and
E ≤kj (aj , f (z)) ⊆ E ≤kj (aj , f (z + c)), j = 1, 2.
2
If Θ(0, f ) + Θ(∞, f ) > k+1

, where k := min{k1, k2}, then
f (z) = f (z + c) for all z ∈ C.

Theorem C. Let f be a nonconstant meromorphic function
of hyper-order γ(f ) < 1, Θ(∞, f ) = 1 and c ∈ C \ {0}. Let
a1, a2, a3 ∈ S(f ) be three distinct periodic functions with period
c such that f (z) and f (z + c) share a3 CM and
E ≤k (aj , f (z)) ⊆ E ≤k (aj , f (z + c)), j = 1, 2.
If k ≥ 2 then f (z) = f (z + c) for all z ∈ C.
As an application of Theorems B and C, the above authors gave
the sufficient conditions for periodicity of meromorphic functions as
follows.
Theorem D. Assume that f and g are two nonconstant meromorphic function with Θ(∞, f ) = Θ(∞, g) = 1, where f has a


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nonzero periodic c ∈ C \ {0} with hyper-order γ(f ) < 1. Let
k1, k2 be two positive integers, a1, a2, a3 ∈ S(f ) be three distinct
periodic functions with period c such that f and g share a3 CM
and
E ≤k (aj , f ) ⊆ E ≤k (aj , g), j = 1, 2.
Then g is a function with periodic T , where T ∈ {c, 2c}, that is
g(z) = g(z + T ) for all z ∈ C.
In this thesis the first aim is to generalize and improve Theorems
B and C by reducing the number of shared values. The second aim
is to give some uniqueness theorems in this direction as well as some
of their applications. Namely, we will prove the following results.
Theorem 2.2.1. Let f be a meromorphic function of hyperˆ ) be three
order γ(f ) < 1 and let c ∈ C \ {0}. Let a1, a2, a3 ∈ S(f

distinct periodic functions with period c and let k be a positive
integer. Assume that f (z) and f (z + c) share partially a1; a2 CM,
i.e.,
E(a1, f (z)) ⊆ E(a1, f (z + c)), E(a2, f (z)) ⊆ E(a2, f (z + c))
and
E ≤k (a3, f (z)) ⊆ E ≤k (a3, f (z + c)).
2
ˆ ) \ {a3, a3(a1+a2)−2a1a2 } then,
If Θ(a, f ) > k+1
for some a ∈ S(f
2a3 −(a1 +a2 )
f (z) = f (z + c) for all z ∈ C.
Corollary 2.2.2. Let f be a nonconstant meromorphic function
of hyper-order γ(f ) < 1, Θ(∞, f ) = 1 and c ∈ C \ {0}. Let
a1, a2 ∈ S(f ) be two distinct periodic functions with period c
such that f (z) and f (z + c) share partially a1 CM, i.e.,
E(a1, f (z)) ⊆ E(a1, f (z + c))
and
E ≤k (a2, f (z)) ⊆ E ≤k (a2, f (z + c)).


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If k ≥ 2, then f (z) = f (z + c) for all z ∈ C.
In the case k = ∞, we have the following theorem.
Theorem 2.2.3. Let f be a meromorphic function of hyperˆ ) be three
order γ(f ) < 1 and let c ∈ C \ {0}. Let a1, a2, a3 ∈ S(f
distinct periodic functions with period c. Assume that f (z) and
f (z + c) share partially a1, a2 CM and share partially a3 IM, i.e.,
E(a1, f (z)) ⊆ E(a1, f (z + c)) E(a2, f (z)) ⊆ E(a2, f (z + c))

and
E(a3, f (z)) ⊆ E(a3, f (z + c)).
ˆ ) \ {a3}, then f (z) = f (z + c) for
If Θ(a, f ) > 0 for some a ∈ S(f
all z ∈ C.
Omitting the deficiency assumption, we will have the following
results.
Theorem 2.2.4. Let f be a meromorphic function of hyperorder γ(f ) < 1 and let c ∈ C \ {0}. Let k, l be two positive
ˆ ) be four distinct periodic
integers and let a1, a2, a3, a4 ∈ S(f
functions with period c. Assume that f (z) and f (z + c) share
partially a1, a2 CM and
E ≤k (a3, f (z)) ⊆ E ≤k (a3, f (z+c)), E ≤l (a4, f (z)) ⊆ E ≤l (a4, f (z+c)).
Then the following statements hold:
(i) If kl > min{k, l} + 2, then f (z) = f (z + c) or

f (z)−a1
f (z)−a2

=

(z+c)−a1
− ff (z+c)−a
for all z ∈ C. Moreover, the latter occurs only when
2
a4 −a1
a3 −a1
a4 −a2 = − a3 −a2 .

(ii) If max{k, l} = ∞, then f (z) = f (z + c) for all z ∈ C.

Using the idea in the proof of Theorem D, we get a similar result
which is considered an application of Theorem 2.2.1 and Corollary
2.2.2.


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Theorem 2.3.1. Assume that f and g are two nonconstant
meromorphic func-tions with Θ(∞, f ) = Θ(∞, g) = 1, where
f has a nonzero periodic c ∈ C \ {0} with hyper-order γ(f ) < 1.
Let k be a positive integer and let a1, a2 ∈ S(f ) be two distinct
periodic functions with period c such that f and g share partially
a1 CM and
E ≤k (a2, f ) ⊆ E ≤k (a2, g).
If k ≥ 2, then g g is a function with periodic c, that is g(z) =
g(z + c) for all z ∈ C.
By the same argument as in the proof of Theorem 2.3.1, we also
get a result in this form from applying Theorem 2.2.4.
Theorem 2.3.2 Assume that f and g are two nonconstant
meromorphic func-tions, where f has a nonzero periodic c ∈
C\{0} with hyper-order γ(f ) < 1. Let k, l be two positive integers
and let a1, a2 ∈ S(f ) \ {0} be two distinct periodic functions with
period c such that
E(0, f ) ⊆ E(0, g), E(∞, f ) ⊆ E(∞, g)
and
E ≤k (a1, f ) ⊆ E ≤k (a1, g), E ≤l (a2, f ) ⊆ E ≤l (a2, g).
Then the following statements hold:
(i) If kl > min{k, l} + 2, then g is a function with periodic T ,
where T ∈ {c, 2c}, that is g(z) = g(z + T ) for all z ∈ C.
(ii) If max{k, l} = ∞, then g is a function with periodic c, that

is g(z) = g(z + c) for all z ∈ C.
II. The uniqueness for zero order meromorphic mapping
In recent years, the Second Main Theorem for meromorphic
mappings intersecting hypersurfaces has been investigated by many


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authors such as T. V. Tan and V. V. Truong, M. Ru, S. D. Quang
and other authors. For example, in 2004, M. Ru proved a second
main theorem for the nondegenerate mappings mappings to Pn(C)
intersecting with the hyperface family in general position, this is a
breakthrough result. In 2017, S. D. Quang obtaineda second main
theorem for the general case of meromorphic mappings intersecting
by hypersurfaces ingenrdl position using the Chow weights.
For the purpose of studying the uniqueness or the Picard theorem
of the meromorphic maps from Cm into Pn(C) having order 0
intersecting hypersurfaces, we have studied and gave some results
for the distribution the q-differences value of the complex-variable
meromorphic mapping intersecting with hypersurfaces located in
subgcneral position based on the ideas of M. Ru and S. D. Quang.
Theorem 3.2.1. Let q = (q1, . . . , qm) ∈ Cm with qj = 0 for all
1 ≤ j ≤ m and let f : Cm → Pn(C) be a meromorphic mapping
with zero-order. Assume that f is algebraically nondegenerate
over the field φ0q . Denote by f˜ = (f0 : · · · : fn) a reduced
(local) representation of f . Let Qj be hypersurfaces of degree
dj (1 ≤ j ≤ p) located in N -subgeneral position in Pn(C). Let d
be the least common multiple of all dj . Then there exists a large
positive u which is divisible by d, such that
p

(q − (N − n + 1)(n + 1)) Tf (r) ≤
i=1

−

1
N ˜ (r)
di Qi(f )

N −n+1
un+1
(n+1)!

+

O(un)

NCq (f I1 ,...,f IM )(r)

+ o (Tf (r))
on a set of logarithmic density 1, where Ij = (ij0, . . . , ijn),
u+d
|Ij | = ij0 + · · · + ijn = u and M =
.
u


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We have the following result similar to the Nochka-Cartan theorem
with truncated multiplicities.
Theorem 3.2.3. Let q = (q1, . . . , qm) ∈ Cm with qj = 0 for all
1 ≤ j ≤ m and let f : Cm → Pn(C) be a meromorphic mapping
with zero-order. Assume that f is algebraically nondegenerate
over the field φ0q . Denote by f˜ = (f0 : · · · : fn) a reduced(local)
representation of f. Let Qj be hypersurfaces of degree dj (1 ≤
j ≤ p), located in N -subgeneral position in Pn(C). Let d be the
least common multiple of all dj . Then, for every > 0, we have
p

(p − (N − n + 1)(n + 1) − ) Tf (r) ≤
j=1

1 ¯ [M0,q]
N
(r) + o (Tf (r))
dj Qj (f˜)

on a set of logarithmic density 1, where
M0 = 4(ed(N − n + 1)(n + 1)2I( −1))n − 1.
Here, by the notation I(x) we denote the smallest integer number
which is not smaller than the real number x.
Theorem 3.3.1. Let q = (q1, . . . , qm) ∈ Cm with qj = 0, 1
for all j ∈ {1, . . . , m} and let f : Cm → Pn(C) be a zeroorder meromorphic mapping. Let Q1, . . . , Qp be hypersurfaces in
Pn(C), located in N -subgeneral position with common degree d.
n+d
Put M =
− 1. Assume that f is forward invariant over
n

Qj with respect to the rescaling τq (z) = qz. If p ≥ M + 2N − n + 1
then the image of f is contained in a linear subspace over the
field φ0q of dimension ≤ M − n − 1 +

p

.
[
]+1
In the case of hyperplanes in subgeneral position in Pn(C), we
have M = n. Moreover, when |qi| = 1 for all i ∈ {1, . . . , m},
then f (z) = f (qz) implies that f must be a constant mapping.
Immediately, we have the following corollary.
p−N −1
M −n+1


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Corollary 3.3.5. Let f be a zero-order meromorphic mapping of
Cm into Pn(C), and let τq (z) = qz, where q = (q1, . . . , qm) ∈ Cm
with qj = 0 for all j ∈ {1, . . . , m}. Assume that τq ((f, Hj )−1) ⊂
(f, Hj )−1 (counting multiplicity) hold for distinct hyperplanes
{Hj }pj=1 in N -subgeneral position in Pn(C). If p > 2N then
f (qz) = f (z). In particular, if |qi| = 1 for all i ∈ {1, . . . , m},
then f is constant.
III. The finiteness of meromorphic mappings
The problem of the algebraic dependence of the complex sereralvariable meromorphic mapping on the complex project space for
fixed targets was first studied by S. Ji and W. Stoll. After that,
their results were developed by many authors such as H. Fujimoto,

Z. Chen and Q. Yan, S. D. Quang, S. D. Quang and L. N. Quynh.
More specifically, H. Fujimoto introduced the degeneracy theorem
for n + 2 meromorphic mappings sharing 2n + 2 hypnplanes with
+n. Recently, S. D. Quang has
truncated multiplicities to level n(n+1)
2
obtained the algebraic degenerate theorem for three meromorphic
maps and used it to give results on the finiteness of meromorphic
mapping sharing 2n + 2 hyperplane located in genwral position
without multiplicities.
In 2019, S. D. Quang proved the following theorem, in which he
did not need to count all zeros with multiples greater than a certain
value.
Theorem E Let H1, . . . , H2n+2 be hyperplanes in general
position in Pn(C). Assume that
2n+2

j=1

n+1
1
<
.
kj + 1 n(3n + 1)

Then for three mappings f 1, f 2, f 3 ∈ F(f, {Hj , kj }2n+2
j=1 , 1), we


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have f 1 ∧ f 2 ∧ f 3 ≡ 0 on Cm.
In 2015, S. D. Quang and L. N. Quynh found a sucient condition
for the algebraic dependence of three meromorphic maps sharing
less than 2n + 2 hyperplanes in general position as follows.
Theorem F Let f 1, f 2, f 3 ∈ F(f, {Hj }qj=1, n) and let {Hi}qi=1 be
a family √
of q hyperplanes of Pn(C) in general position. If q >
2n + 5 + 28n2 + 20n + 1
then one of the following assertions
4
holds:
(i) There exist

q
3

+ 1 hyperplanes Hi1 , . . . , Hi

[ 3q ]+1

such that

(f u, Hi q )
(f , Hi1 ) (f , Hi2 )
[ 3 ]+1
=
=
·
·

·
=
,
(f v , Hi1 ) (f v , Hi2 )
(f v , Hi q )
[ 3 ]+1
u

u

(ii) f 1 ∧ f 2 ∧ f 3 ≡ 0 on Cm.
Clearly, to obtain the assertion (ii) in Theorem B, they need
to assume that (i) does not occur. The question is whether we
can ignore this condition for q < 2n + 2? The first aim of this
section is to give a positive answer for the above question. For our
purpose, we will rearrange hyperplanes into suitable groups and use
the technique “rearranging counting functions” due to D. D. Thai
and S. D. Quang as well as consider a new auxiliary function. These
help us obtain a complete theorem for the algebraic dependence of
three meromorphic sharing 2n + 1 hyperplanes in general position
as follows.
Theorem 4.2.1. Let H1, . . . , H2n+1 be hyperplanes in general
position in Pn(C) (n ≥ 5). Let f 1, f 2, f 3 : Cm → Pn(C) be


15

meromorphic mappings which belong to F(f, {Hj , kj }2n+1
j=1 , n). If
2n+1

i=1

1
n−4
<
,
ki + 1 2n(2n + 1)

then f 1 ∧ f 2 ∧ f 3 ≡ 0 on Cm.
We would like to emphasize that Theorem A plays an essential role
in S. D. Quang ’s proof for finiteness of meromorphic mappings. Here
is his result.
Theorem G Let f be a linearly nondegenerate meromorphic
mappings of Cm into Pn(C). Let H1, . . . , H2n+2 be 2n + 2 hyperplanes of Pn(C) in general position and k1, . . . , kn+2 be positive
integers or +∞. Assume that
2n+2

i=1

n + 1 5n − 9
n2 − 1
1
< min
,
,
.
ki + 1
3n2 + n 24n + 12 10n2 + 8n

Then F(f, {Hi, ki}2n+2
i=1 , 1) ≤ 2.
The following question arises naturally at this moment: By using
Quang’s techniques in [7] and by Theorem 1.0.11, do we obtain
a result on finiteness for meromorphic mappings sharing 2n + 1
hyperplanes in general position with truncated multiplicities to level
n? The second aim of this section is also to give a positive answer
for this question. Namely, we have the following
Theorem 4.3.1. Let f be a linearly nondegenerate meromorphic mappings of Cm into Pn(C). Let H1, . . . , H2n+1 be 2n + 1
hyperplanes of Pn(C) in general position and let k1, . . . , kn+1 be
positive integers or +∞ such that
2n+1

i=1

1
n−4
<
.
ki + 1 2n(2n + 1)


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If n ≥ 8 then F(f, {Hi, ki}2n+1
i=1 , n) ≤ 2.

Chapter 2: The uniqueness of the zero rder
meromorphic mapping
Chapter 2 is written based on the article [1] in the published works

related to the thesis.
2.1 Some preparation knowledge
Lemma 2.1.3. Let f be a nonconstant meromorphic function
on C. Let a1, a2, . . . , aq (q ≥ 3) be q distinct small meromorphic
functions of f on C. Then the following holds
q

(q − 2)T (r, f ) ≤

N r,
i=1

1
+ S(r, f ).
f − ai

Lemma 2.1.4. Let f be a nonconstant meromorphic function
and c ∈ C. If f is of finite order, then
m r,

log r
f (z + c)
=O
T (r, f )
f (z)
r

for all r outside of a subset E zero logarithmic density. If the
hyper-order γ(f ) of f is less than one, then for each > 0, we
have

T (r, f )
f (z + c)
m r,
= o 1−γ(f )−
f (z)
r
for all r outside of a subset finite logarithmic measure.
Lemma 2.1.5. Let T : R+ → R+ be a non-decreasing continuous
function, and let s ∈ (0, +∞) such that hyper-order of T is
strictly less than one, i.e.,
log+ log+ T (r)
γ = lim sup
< 1,
log r
r→∞


17

Then

T (r)
,
r1−γ−
where > 0 and r → ∞ outside a subset of finite logarithmic
measure.
T (r + s) = T (r) + o

For each meromorphic function f , we denote fc(z) = f (z + c) and
∆cf := fc − f.

Lemma 2.1.6. Let c ∈ C and let f be a meromorphic function
of hyper-order γ(f ) < 1 such that ∆cf ≡ 0. Let q ≥ 2 and
a1(z), . . . , aq (z) be distinct meromorphic periodic small functions
of f with period c. Then
q

m(r, f ) +

m r,
k=1

1
≤ 2T (r, f ) − Npair (r, f ) + S1(r, f ),
f − ak

where
Npair (r, f ) = 2N (r, f ) − N (r, ∆cf ) + N r,

1
.
∆cf

2.2 The uniqueness for the meromorphic function with
hyper order less than 1
In this section, we give and prove the Theorems 1.0.1, 1.0.3, 1.0.4.
2.3 Periodic property of the meromorphic functions with
hyper order less than 1
In this section, we give some necessary conditions for a meromorphic function less than 1 to be periodic. This is an application of
the Theorem 1.0.1 and corollary 1.0.2. Specifically, we prove the
Theorem 1.0.5, 1.0.6.

Chapter 3: The uniqueness problem for the
meromorphic mapping with zero order


18

Chapter 3 is written based on the article [2] in the published works
related to the thesis.
3.1. Some preparation knowledge
Lemma 3.1.1. Let q ∈ Cm \ {0}. For a positive integer M ,
set Iα = {(i0, . . . , in) ∈ Nn+1
: i0 + · · · + in = α}, Ij ∈ Iα
0
for all j ∈ {1, . . . , M }. Then the meromorphic mapping f =
(f0 : · · · : fn) : Cm → Pn(C) with zero-order satisfies Cq (f ) =
Cq f I1 , . . . , f IM ≡ 0 if and only if the functions f0, . . . , fn are
alge-braically dependent over the field φ0q .
Lemma 3.1.2. Let Q1, . . . , Qk+1 be hypersurfaces in Pn(C) of
the same degree d such that
k+1

Qi = ∅.
i=1

Then there exist n hypersurfaces P2, . . . , Pn+1 of the forms
k−n+t

ctj Qj , ctj ∈ C, t = 2, . . . , n + 1,

Pt =
j=2

such that

n+1
i=1 Pi

= ∅.

Lemma 3.1.3. Let {Qi}i∈R be a family of hypersurfaces in Pn(C)
of the common degree d, and let f be a meromorphic mapping
of Cm into Pn(C). Assume that i∈R Qi = ∅. Then there exist
positive constants α and β such that
α||f˜||d ≤ max |Qi(f˜)| ≤ β||f˜||d.
i∈R

The following lemma is called the q-difference logarithmic derivative lemma, which is analogous to the logarithmic derivative lemma.
Lemma 3.1.4. Let f be a non-constant zero-order meromorphic
mapping of Cm into C and q = (q1, . . . , qm) ∈ Cm with qj = 0 for


19

all j, then
f (qz)
= o(T (r, f (z)))
f (z)
on a set of logarithmic density 1.
m r,

3.2 The Second main theorem for q-differences
In this section, we will give and prove Theorem 1.0.7, 1.0.8.
3.3 Picard’s theorem
In this section, we will give and prove Theorems 1.0.9 and Corollary 1.0.10.
To prove Theorem 1.0.9, we need the following main lemma.
Lemma 3.3.2. Let Q1, . . . , Qp be hypersurfaces of the common
degree d in Pn(C), located in N -subgeneral position. Put M =
n+d
p−N −1
− 1 and let p˜ = M
−n+1 + N + 1. If p ≥ M + 2N − n + 1
n
then exists a subset U ⊂ {1, . . . , p} with |U | ≥ p˜ satisfying the
condition: (∗) for every subset R ⊂ U with |R| ≤ p˜ − N − 1, we
have span{Q∗j }j∈R ∩ span{Q∗i }i∈R∗ = {0}, where R∗ = U \ R and
Q∗j is a homogeneous polynomial defining Qj .
Lemma 3.3.3. Let f : Cm → Pn(C) be a meromorphic mapping
with a reduced representation f˜ = (f0 : · · · : fn) and let
q = (q1, . . . , qm) ∈ Cm with qj = 0 for all j. Assume that σ(f ) = 0
and all zeros of f0, . . . , fnare forward invariant with respect to the
n+d
rescaling τq (z) = qz. Let d ∈ N∗ and put M =
− 1. If
n
f Ii
for each d, I ∈ φ0q for all i, j ∈ {0, . . . , M } such that Ii = Ij ,
f j
then f0, . . . , fn are algebraically nondegenerate over the field φ0q .
Lemma 3.3.4. Let f = (f0 : · · · : fn) be a meromorphic mapping

of Cm to Pn(C) such that σ(f ) = 0 and all q = (q1, . . . , qm) ∈ Cm


20

with qj = 0, 1 for all j. Assume that all zeros of f0, . . . , fn are
forward invariant with respect to the rescaling τq (z) = qz. Let
n+d
d ∈ N∗, put M =
− 1. Let S1 ∪ · · · ∪ Sl be a portion of
n
{0, . . . , M } formed in such a way that i and j are in the same
f Ii
class Sk if and only if I ∈ φ0q . If f I0 + · · · + f IM = 0 then
f j
f Ij = 0 for all k ∈ {1, . . . , l}.
j∈Sk

Chapter 4: The algebraic degeneracy and
finiteness of meromorphic mappings
Chapter 4 is written based on the article [3] in the published works
related to the dissertation.
In this chapter, we prove a theorem of algebraic degeneracy of
three meromorphic mappings from Cm to Pn(C) intersecting 2n + 1
hyperplane in general position with truncated to leveln. That said,
the intersections with a multiple greater than actually a certain
value will not need to be counted. As an application we have a
theorem on the finiteness of these meromorphic mappings.
4.1 Some basic properties and additional results in
Nevanlinna theory

Theorem 4.1.1. [The first main theorem]. Let f : Cm → Pn(C)
be a linearly nondegenerate meromorphic mapping and H be a
hyperplane in Pn(C). Then
N(f,H)(r) + mf,H (r) = Tf (r), r > 1.

Theorem 4.1.2. [The second main theorem]. Let f : Cm →
Pn(C) be a linearly nondegenerate meromorphic mapping and


21

H1, . . . , Hq be hyperplanes in general position in Pn(C). Then
q
[n]

||(q − n − 1)Tf (r) ≤

N(f,Hi)(r) + o(Tf (r)).
i=1

Lemma 4.1.3. If Φα (F, G, H) = 0 and Φα F1 , G1 , H1 = 0 for
all α with |α| ≤ 1, then one of the following assertions holds:
(i) F = G, G = H or H = F .
(ii)

F G
G, H

and

H
F

are all constants.

Theorem 4.1.4. Let f 1, f 2, f 3 be three mappings in F(f, {Hi, ki}qi=1, 1).
Suppose that there exist s, t, l ∈ {1, . . . , q} such that
(f 1, Hs) (f 1, Ht) (f 1, Hl )
P := (f 2, Hs) (f 2, Ht) (f 2, Hl ) ≡ 0.
(f 3, Hs) (f 3, Ht) (f 3, Hl )
Then we have
[1]

T (r) ≥ NP (r) ≥

(N (r, min {ν(f u,Hi),≤ki }) − N(f 1,H ),≤k (r))
i

1≤u≤3

i=s,t,l
q

i

[1]

+2

N(f 1,H ),≤k (r),

i

i

i=1
3
u=1 Tf u (r).
1
4.1.5. If 2n+1
i=1 ki +1

where T (r) =

Lemma
<
is linearly nondegenerate and

n−1
n ,

then every g ∈ F(f, {Hi, ki}qi=1, n)

||Tg (r) = O(Tf (r)) và ||Tf (r) = O(Tg (r)).
4.2 The algerbraic degeneracy of thrre meromorphic
mappings sharing 2n + 1 hypreplanes
In this section, we prove Theorem 1.0.11. However, we need the
following Lemma.


22

Lemma 4.2.2. Let q, N be two integers satisfying q ≥ 2N + 2,
N ≥ 2 and q be even. Let {a1, a2, . . . , aq } be a family of vectors in
a 3-dimensional vector space such that rank{aj }j∈R = 2 for any
subset R ⊂ Q = {1, . . . , q} with cardinality R = N + 1. Then
q/2
there exists a partition j=1 Ij of {1, . . . , q} satisfying Ij = 2
and rank{ai}i∈Ij = 2 for all j = 1, . . . , q/2.
4.3 The finitenees of meromorphic mappings sharing
2n + 1 hypreplanes
In this section, we prove Theorem 1.0.12. However, we need the
following Lemma.
Lemma 4.3.2. With the assumption of Theorem 1.0.12, let h
and g be two elements of the family F(f, {Hi, ki}2n+1
i=1 , n). If there
exists a constant λ and two indices i, j such that
(g, Hi)
(h, Hi)
=λ
(h, Hj )
(g, Hj )
then λ = 1.
2n+1
Lemma 4.3.3. Let f 1, f 2, f 3 be three elements of F(f, {Hi, ki}i=1
, n),
where ki (1 ≤ i ≤ 2n + 1) are positive integers or +∞. Suppose
that f 1 ∧ f 2 ∧ f 3 ≡ 0 and Vi ∼ Vj for some distinct indices i and
j. Then f 1, f 2, f 3 are not distinct.

Lemma 4.3.4. With the assumption of Theorem 1.0.12, let

f 1, f 2, f 3 be three maps in F(f, {Hi, ki}2n+1
i=1 , n). Suppose that
f 1, f 2, f 3 are distinct and there are two indices i, j ∈ {1, 2, . . . , 2n+
1} (i = j) such that Vi ∼
= Vj and
Φαij := Φα (F1ij , F2ij , F3ij ) ≡ 0
for every α = (α1, . . . , αm) ∈ Z+
m with |α| = 1. Then for every
t ∈ {1, . . . , 2n + 1} \ {i}, the following assertions hold:
(i) Φαit ≡ 0 for all |α ≤ 1|,


23

(ii) if Vi ∼
= Vt, then F1ti, F2ti, F3ti are distinct and
[1]

[1]

N(f,Hi),≤ki (r) ≥

[1]

N(f,Hs),≤ks (r) − N(f,Ht),≤kt (r)
s=i,t
3
[1]

−2

N(f u,Hs),>ks (r) + o(T (r)).
u=1 s=i,t

Lemma 4.3.5. With the assumption of Theorem 1.0.12, let
f 1, f 2, f 3 be three maps in F(f, {Hi, ki}2n+1
i=1 , n). Suppose that
f 1, f 2, f 3 are distinct and there are two indices i, j ∈ {1, 2, . . . , 2n+
α
1} (i = j) and α ∈ Z+
m with |α| = 1 such that Φij ≡ 0. Then we
have
3

3
[n]
N(f u,Hi),≤ki (r)

T (r) ≥
u=1

[n]

+

[1]

N(f u,Hj ),≤kj (r) + 2
u=1

N(f,Ht),≤kt (r)
t=1,t=i,j

1
1
− (2n + 1)N(f,H
(r) − (n + 1)N(f,H
(r) + N (r, νj )
i ),≤ki
j ),≤kj
3

−

1+
u=1

2n − 2 [1]
n − 1 [1]
N(f u,Hj ),>kj + 1 +
N(f u,Hi),>ki
3
3

+ o(T (r)),
where νj := {z : ki ≥ ν(f u,Hi)(z) = ν(f t,Hi)(z)} for each permutation (u, v, t) of (1, 2, 3).

Conclusions and recommendations
Conclusions The thesis researches some unigueness problems such
that, algebraic dependence and finiteness of the meromorphic mappings. The dissertation has achieved some of the following results:

• Prove some uniqueness theorem for the meromorphic function with
hyper order less than 1 in the complex plane.
• Prove the second main theorem for the meromorphic mapping from