On Fundamental Theorems for Holomorphic Curves on the Annuli
We prove some fundamental theorems for holomorphic curves on the annuli crossing a finite set of fixed hyperplanes in the general position in $ℙ_n (ℂ)$ with ramification.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507953539317760 |
|---|---|
| author | Phuong, Ha Tran Thin, N. V. Пхуонг, Ха Тран Тхін, Н. В. |
| author_facet | Phuong, Ha Tran Thin, N. V. Пхуонг, Ха Тран Тхін, Н. В. |
| author_sort | Phuong, Ha Tran |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:49:13Z |
| description | We prove some fundamental theorems for holomorphic curves on the annuli crossing a finite set of fixed hyperplanes in the general position in $ℙ_n (ℂ)$ with ramification. |
| first_indexed | 2026-03-24T02:17:30Z |
| format | Article |
| fulltext |
UDC 517.9
H. T. Phuong, N. V. Thin (Thai Nguyen Univ. Education, Vietnam)
ON FUNDAMENTAL THEOREMS FOR HOLOMORPHIC CURVES ON ANNULI*
ФУНДАМЕНТАЛЬНI ТЕОРЕМИ ДЛЯ ГОЛОМОРФНИХ
КРИВИХ НА КIЛЬЦЯХ
We prove some fundamental theorems for holomorphic curves on annuli intersecting a finite set of fixed hyperplanes in
general position in Pn(C) with ramification.
Доведено деякi фундаментальнi теореми для голоморфних кривих на кiльцях, що перетинають скiнченну множину
фiксованих гiперплощин загального положення в Pn(C) з розгалудженням.
1. Introduction and main results. In 1933, H. Cartan (see [3]) proved the second main theorem for
holomorphic curves with targets being hyperplanes in general position in Pn(C). Since that times, the
problem which studies the characteristics of holomorphic maps has been attracted by many authors.
For example, in 1983, E. I. Nochka (see [6]) proved the second main theorem in the case of the
hyperplanes is in N -subgeneral position in P(C) with ramification. In 2003, M. Ru and T. Y. Wang
(see [7]) proved an inequality of the second main theorem type, with ramification for a holomorphic
curve intersecting a finite set of moving or fixed hyperplanes. In 2004, M. Ru (see [9]) showed the
second main theorem for holomorphic curves with targets being hypersurfaces in general position
in P(C) without ramification. In 2009, T. T. H. An and H. T. Phuong (see [1]) gave the result on
the second main theorem for holomorphic curves from C to Pn(C) intersecting hypersurfaces with
ramification.
Recently, there exist the results about of the characteristics of meromorphic functions on annuli in
complex plane C. In 2005, A. Y. Khrystiyanyn and A. A. Kondratyuk (see [4, 5]) showed some results
about of the fundamental theorems and defect relation, which were considered again by T. B. Cao and
Z. S. Deng in [2] and by Y. Tan and Q. Zhang in [11]. Our idea is to prove some fundamental theorems
for holomorphic mappings from annuli ∆ ⊂ C to Pn(C) intersecting a finite set of hyperplanes. To
state our results, we first introduce some notations.
Let R0 > 1 be a fixed positive real number or +∞, set
∆ =
{
z ∈ C :
1
R0
< |z| < R0
}
,
be a annuli in C, and for any real number r such that 1 < r < R0 we denote
∆r =
{
z ∈ C :
1
r
< |z| < r
}
, ∆1,r =
{
z ∈ C :
1
r
< |z| ≤ 1
}
,
∆2,r =
{
z ∈ C : 1 < |z| < r
}
.
Let f = (f0 : . . . : fn) : ∆ → Pn(C) be a holomorphic map where f0, . . . , fn are holormorphic
functions and without common zeros in ∆. For 1 < r < R0, characteristic function Tf (r) of f is
* This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED)
under grant number 101. 04-2014.41.
c© H. T. PHUONG, N. V. THIN, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7 981
982 H. T. PHUONG, N. V. THIN
defined by
Tf (r) =
1
2π
2π∫
0
log ‖f(reiθ)‖dθ +
1
2π
2π∫
0
log ‖f(r−1eiθ)‖dθ,
where ‖f(z)‖ = max{|f0(z)|, . . . , |fn(z)|}. The above definition is independent, up to an additive
constant, of the choice of the reduced representation of f.
Let H be a hyperplane in Pn(C) and
L(z0, . . . , zn) =
n∑
j=0
ajzj
be linear form defined H, where aj ∈ C, j = 0, . . . , n, be constants. Denote by a = (a0, . . . , an) the
non-zero associated vector with H. And denote
(H, f) = (a, f) =
n∑
j=0
ajfj .
Under the assumption that (a, f) 6≡ 0, for 1 < r < R0, the proximity function of f with respect to
H is defined as
mf (r,H) =
1
2π
2π∫
0
log
‖f(reiθ)‖
|(a, f)(reiθ)|
dθ +
1
2π
2π∫
0
log
‖f(r−1eiθ)‖
|(a, f)(r−1eiθ)|
dθ,
where the above definition is independent, up to an additive constant, of the choice of the reduced
representation of f.
Next, we denote by n1,f (r,H) the number of zeros of (a, f) in ∆1,r, counting multiplicity and
by n2,f (r,H) the number of zeros of (a, f) in ∆2,r, counting multiplicity too. Set
N1,f (r,H) = N1,f (r, L) =
1∫
r−1
n1,f (t,H)
t
dt,
N2,f (r,H) = N2,f (r, L) =
r∫
1
n2,f (t,H)
t
dt.
The counting function function of f is defined by
Nf (r,H) = N1,f (r,H) +N2,f (r,H).
Now let δ be a positive integer, we denote by nδ1,f (r,H) and nδ2,f (r,H) be the numbers of zeros of
(a, f) in ∆1,r and ∆2,r respectively, where any zero of multiplicity greater than δ is “truncated" and
counted as if it only had multiplicity δ. Set
N δ
1,f (r,H) = N δ
1,f (r, L) =
1∫
r−1
nδ1,f (t,H)
t
dt,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON FUNDAMENTAL THEOREMS FOR HOLOMORPHIC CURVES ON ANNULI 983
N δ
2,f (r,H) = N δ
2,f (r, L) =
r∫
1
nδ2,f (t,H)
t
dt.
The truncated counting function of f is defined by
N δ
f (r,H) = N δ
1,f (r,H) +N δ
2,f (r,H).
Recall that hyperplanes H1, . . . ,Hq, q > n, in Pn(C) are said to be in general position if for any
distinct i1, . . . , in+1 ∈ {1, . . . , q},
n+1⋂
k=1
supp(Hik) = ∅,
this is equivalence to the Hi1 , . . . ,Hin+1 being linearly independent.
In this paper, a notation “‖” in the inequality is mean that for R0 = +∞, the inequality holds
for r ∈ (1,+∞) outside a set ∆′r satisfying
∫
∆′r
rλ−1dr < +∞, and for R0 < +∞, the inequality
holds for r ∈ (1, R0) outside a set ∆′r satisfying
∫
∆′r
1
(R0 − r)λ+1
dr < +∞, where λ ≥ 0.
Our main results are:
Theorem 1.1. Let H be a hyperplane in Pn(C) and f = (f0 : . . . : fn) : ∆ → Pn(C) be a
holomorphic curve whose image is not contained H. Then we have for any 1 < r < R0,
Tf (r) = mf (r,H) +Nf (r,H) +O(1).
Theorem 1.2. Let f = (f0 : . . . : fn) : ∆ → Pn(C) be a linearly nondegenerate holomorphic
curve and H1, . . . ,Hq be hyperplanes in Pn(C) in general position. Then we have∥∥∥∥∥∥ (q − n− 1)Tf (r) ≤
q∑
j=1
Nn
f (r,Hj) +Of (r) ,
where
Of (r) =
O(log r + log Tf (r)) if R0 = +∞,
O
(
log
1
R0 − r
+ log Tf (r)
)
if R0 < +∞.
Theorem 1.1 is first main theorem, and Theorem 1.2 is second main theorem for holomorphic
curves from annuli ∆ to Pn(C) intersecting a collection of fixed hyperplanes in general position with
truncated counting functions. When one applies inequalities of second main theorem type, it is often
crucial to the application to have the inequality with truncated counting functions. For example, all
existing constructions of unique range sets depend on a second main theorem with truncated counting
functions.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
984 H. T. PHUONG, N. V. THIN
2. Some preliminaries in Nevanlinna theory for meromorphic functions. In order to prove
theorems, we need the following lemmas. Let f be a meromorphic function on ∆, we recall that
m
(
r,
1
f − a
)
=
1
2π
2π∫
0
log+ 1
|f(reiθ)− a|
dθ,
m(r, f) = m(r,∞) =
1
2π
2π∫
0
log+ |f(reiθ)|dθ,
where log+ x = max{0, log x}, a ∈ C and r ∈ (R−1
0 ;R0). For r ∈ (1, R0), we denote
m0
(
r,
1
f − a
)
= m
(
r,
1
f − a
)
+m
(
1
r
,
1
f − a
)
,
m0(r, f) = m(r, f) +m(r−1, f).
Denote by n1
(
t,
1
f − a
)
the number of zeros of f − a in {z ∈ C : t < |z| ≤ 1}, n2
(
t,
1
f − a
)
the number of zeros of f − a in {z ∈ C : 1 < |z| < t}, n1(t,∞) the number of poles in {z ∈ C :
t < |z| ≤ 1} and n2(t,∞) the number of poles in {z ∈ C : 1 < |z| < t}. For any r : 1 < r < R0,
put
N1
(
r,
1
f − a
)
=
1∫
1/r
n1
(
t,
1
f − a
)
t
dt, N2
(
r,
1
f − a
)
=
r∫
1
n2
(
t,
1
f − a
)
t
dt,
and
N1(r, f) = N1(r,∞) =
1∫
1/r
n1(t,∞)
t
dt, N2(r, f) = N2(r,∞) =
r∫
1
n2(t,∞)
t
dt.
Let
N0
(
r,
1
f − a
)
= N1
(
r,
1
f − a
)
+N2
(
r,
1
f − a
)
,
N0(r, f) = N1(r, f) +N2(r, f).
Denote the Nevanlinna characteristic of f by
T0(r, f) = m0(r, f)− 2m(1, f) +N0(r, f).
Lemma 2.1 [4]. Let f be a nonconstant meromorphic function on ∆. Then for any r ∈ (1, R0),
we have
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON FUNDAMENTAL THEOREMS FOR HOLOMORPHIC CURVES ON ANNULI 985
N0
(
r,
1
f
)
−N0(r, f) =
1
2π
2π∫
0
log |f(reiθ)|dθ +
1
2π
2π∫
0
log |f(r−1eiθ)|dθ−
− 1
π
2π∫
0
log |f(eiθ)|dθ.
Lemma 2.2 [5]. Let f be a nonconstant meromorphic function on ∆ and λ ≥ 0. Then for any
r ∈ (1, R0)
(i) if R0 = +∞, ∥∥∥∥ m0
(
r,
f ′
f
)
= O(log r + log T0(r, f)) ;
(ii) if R0 < +∞, ∥∥∥∥ m0
(
r,
f ′
f
)
= O
(
log
1
R0 − r
+ log T0(r, f)
)
.
Lemma 2.3 [4]. Let f be a nonconstant meromorphic function on ∆. Then we have for any
r ∈ (1, R0)
T0(r, f1 + f2) ≤ T0(r, f1) + T0(r, f2) +O(1),
T0
(
r,
f1
f2
)
≤ T0(r, f1) + T0(r, f2) +O(1).
3. Proofs of Theorems 1.1 and 1.2. Proof of Theorem 1.1. Let a = (a0, . . . , an) is
the associated vector with H. First we note that N0(r, (H, f)) = 0. By the definitions of Tf (r),
Nf (r,H), mf (r,H) and apply to Lemma 2.1 for (H, f), we have
Nf (r,H) = N0
(
r,
1
(H, f)
)
=
=
1
2π
2π∫
0
log |(a, f)(reiθ)|dθ +
1
2π
2π∫
0
log |(a, f)(r−1eiθ)|dθ +O(1).
Hence, we get
Nf (r,H) +mf (r,H) =
=
1
2π
2π∫
0
log
‖f(reiθ)‖
|(a, f)(reiθ)|
dθ +
1
2π
2π∫
0
log
‖f(r−1eiθ)‖
|(a, f)(r−1eiθ)|
dθ+
+
1
2π
2π∫
0
log |(a, f)(reiθ)|dθ +
1
2π
2π∫
0
log |(a, f)(r−1eiθ)|dθ +O(1) =
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
986 H. T. PHUONG, N. V. THIN
=
1
2π
2π∫
0
log ‖f(reiθ)‖dθ +
1
2π
2π∫
0
log ‖f(r−1eiθ)‖dθ +O(1) =
= Tf (r) +O(1).
Theorem 1.1 is proved.
To prove Theorem 1.2, we need some lemmas. First we recall the property of Wronskian. Let
f = (f0 : . . . : fn) : ∆ → Pn(C) be holomorphic curves, the determining of Wronskian of f is
defined by
W = W (f) = W (f0, . . . , fn) =
∣∣∣∣∣∣∣∣∣∣∣
f0(z) f1(z) . . . fn(z)
f ′0(z) f ′1(z) . . . f ′n(z)
. . . . . . . . . . . .
f
(n)
0 (z) f
(n)
1 (z) . . . f
(n)
n (z)
∣∣∣∣∣∣∣∣∣∣∣
.
We denote by NW (r, 0) the counting function of zeros of W (f0, . . . , fn) in ∆r, namely
NW (r, 0) = N0
(
r,
1
W
)
+O(1).
Let L0, . . . , Ln are linearly independent forms of z0, . . . , zn. For j = 0, . . . , n, set
Fj(z) := Lj(f(z)).
By the property of Wronskian there exists a constant C 6= 0 such that
W (F0, . . . , Fn) = CW (f0, . . . , fn).
Lemma 3.1. Let f = (f0 : . . . : fn) : ∆ → Pn(C) be a linearly nondegenerate holomorphic
curve and H1, . . . ,Hq be hyperplanes in Pn(C) in general position. Let aj is the associated vector
with Hj for j = 1, . . . , q. Then we have∥∥∥∥∥∥
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
2π∫
0
max
K
∑
j∈K
log
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
≤
≤ (n+ 1)Tf (r)−NW (r, 0) +Of (r),
where
Of (r) =
O(log r + log Tf (r)) if R0 = +∞,
O
(
log
1
R0 − r
+ log Tf (r)
)
if R0 < +∞.
Here the maximum is taken over all subsets K of {1, . . . , q} such that aj , j ∈ K, are linearly
independent.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON FUNDAMENTAL THEOREMS FOR HOLOMORPHIC CURVES ON ANNULI 987
Proof. We prove the case R0 = +∞, case R0 < +∞ can be proved similarly. First, we prove∥∥∥∥∥∥
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
1
2π
2π∫
0
log |W (f)(reiθ)|dθ ≤
≤ (n+ 1)
1
2π
2π∫
0
log ‖f(reiθ)‖dθ +O(log r + log Tf (r)), (3.1)
holds for any r ∈ (1, R0). Let K ⊂ {1, . . . , q} such that aj , j ∈ K, are linearly independent. Without
loss of generality, we may assume that q ≥ n+ 1 and #K = n+ 1. Let T is the set of all injective
maps µ : {0, 1, . . . , n} → {1, . . . , q}. Noting that #T < +∞, then we have
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
=
=
2π∫
0
max
µ∈T
n∑
j=0
log
‖f(reiθ)‖
|(aµ(j), f)(reiθ)|
dθ
2π
=
=
2π∫
0
max
µ∈T
log
n∏
j=0
‖f(reiθ)‖
|(aµ(j), f)(reiθ)|
dθ
2π
=
=
2π∫
0
max
µ∈T
log
‖f(reiθ)‖n+1∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
=
=
2π∫
0
log
max
µ∈T
‖f(reiθ)‖n+1∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
+O(1) ≤
≤
2π∫
0
log
∑
µ∈T
‖f(reiθ)‖n+1∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
+O(1) =
=
2π∫
0
log
∑
µ∈T
|W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)|∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
+
+
2π∫
0
log
∑
µ∈T
‖f(reiθ)‖n+1
|W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)|
dθ
2π
+O(1).
By the property of Wronskian, we see that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
988 H. T. PHUONG, N. V. THIN
|W ((aµ(0), f), . . . , (aµ(n), f))| = |C| |W (f0, . . . , fn)|,
where C 6= 0 is constant. So we obtain
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
≤
≤
2π∫
0
log
∑
µ∈T
|W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)|∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
+
+
2π∫
0
log
‖f(reiθ)‖n+1
|W (f0, . . . , fn)(reiθ)|
dθ
2π
+O(1). (3.2)
We have
W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)∏n
j=0
(aµ(j), f)(reiθ)
=
=
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1 1 . . . 1
(aµ(0), f)′
(aµ(0), f)
(aµ(1), f)′
(aµ(1), f)
. . .
(aµ(n), f)′
(aµ(n), f)
. . . . . . . . . . . .
(aµ(0), f)(n)
(aµ(0), f)
(aµ(1), f)(n)
(aµ(1), f)
. . .
(aµ(n), f)(n)
(aµ(n), f)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
(reiθ). (3.3)
We see that ∥∥∥∥∥ m
(
r,
(aµ(j), f)(k)
(aµ(j), f)
)
≤ m0
(
r,
(aµ(j), f)(k)
(aµ(j), f)
)
=
= m0
(
r,
(aµ(j), f)(k)
(aµ(j), f)(k−1)
(aµ(j), f)(k−1)
(aµ(j), f)(k−2)
. . .
(aµ(j), f)′
(aµ(j), f)
)
≤
≤
k∑
l=1
m0
(
r,
(aµ(j), f)(l)
(aµ(j), f)(l−1)
)
. (3.4)
By Lemma 2.2, we get
m0
(
r,
(aµ(j), f)′
(aµ(j), f)
)
= O(log r + log T0(r, (aµ(j), f)). (3.5)
From the definition of T0(r, (aµ(j), f)′), N((aµ(j), f)′) = 0 and (3.5), we obtain
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON FUNDAMENTAL THEOREMS FOR HOLOMORPHIC CURVES ON ANNULI 989
T0(r, (aµ(j), f)′) = m0(r, (aµ(j), f)′) =
= m0
(
r,
(aµ(j), f)′
(aµ(j), f)
(aµ(j), f)
)
≤
≤ m0(r, (aµ(j), f)) +O(log r + log T0(r, (aµ(j), f)) =
= T0(r, (aµ(j), f)) +O(log r + log T0(r, (aµ(j), f)). (3.6)
Similarly, again using Lemma 2.2 and (3.6), we have
T0(r, (aµ(j), f)′′) = m0(r, (aµ(j), f)′′) =
= m0
(
r,
(aµ(j), f)′′
(aµ(j), f)′
(aµ(j), f)′
)
≤
≤ m0(r, (aµ(j), f)′) +O(log r + log T0(r, (aµ(j), f)′) =
= T0(r, (aµ(j), f)) +O(log r + log T0(r, (aµ(j), f)). (3.7)
By argument as (3.7) and using inductive method, we obtain that the inequality
T0(r, (aµ(j), f)(l)) ≤ T0(r, (aµ(j), f)) +O(log r + log T0(r, (aµ(j), f)) (3.8)
holds for all l ∈ N∗. Furthemore, by Lemma 2.2, we also have the equality
m0
(
r,
(aµ(j), f)(l+1)
(aµ(j), f)(l)
)
= O(log r + log T0(r, (aµ(j), f)(l))), (3.9)
holds for all l ∈ N. Combining (3.4), (3.8) and (3.9), we get for any k ∈ {1, . . . , n} and j ∈
∈ {0, . . . , n}, ∥∥∥∥∥ m
(
r,
(aµ(j), f)(k)
(aµ(j), f)
)
≤ O(log r + log T0(r, (aµ(j), f)) . (3.10)
By the definition of T0(r, ft), Tf (r), we have for any t ∈ {0, . . . , n},
T0(r, ft) +O(1) = m0(r, ft) = m(r, ft) +m
(
1
r
, ft
)
≤
≤ O
1
2π
2π∫
0
log ‖f(reiθ)‖dθ +
1
2π
2π∫
0
log ‖f(r−1eiθ)‖dθ
= O(Tf (r))
and
T0(r, (aµ(j), f)) ≤
n∑
t=0
T0(r, ft) +O(1).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
990 H. T. PHUONG, N. V. THIN
Then we have from (3.10),∥∥∥∥∥ m
(
r,
(aµ(j), f)(k)
(aµ(j), f)
)
≤ O(log r + log Tf (r)) .
Hence for any µ ∈ T , we have from (3.3)∥∥∥∥∥∥
2π∫
0
log+ |W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)|∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
≤ O(log r + log Tf (r)) .
This implies that ∥∥∥∥∥∥
2π∫
0
log
∑
µ∈T
|W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)|∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
≤
≤
2π∫
0
log+
∑
µ∈T
|W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)|∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
≤
≤
∑
µ∈T
2π∫
0
log+ |W ((aµ(0), f), . . . , (aµ(n), f))(reiθ)|∏n
j=0
|(aµ(j), f)(reiθ)|
dθ
2π
+O(1) ≤
≤ O(log r + log Tf (r)). (3.11)
We may obtain the inequality (3.1) from (3.4) and (3.11). Similarly, we get∥∥∥∥∥∥
2π∫
0
max
K
∑
j∈K
log
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
+
1
2π
2π∫
0
log |W (f)(r−1eiθ)|dθ ≤
≤ (n+ 1)
1
2π
2π∫
0
log ‖f(r−1eiθ)‖dθ +O(log r + log Tf (r)) (3.12)
holds for any r ∈ (1, R0). Combining (3.1) and (3.12) we obtain∥∥∥∥∥∥
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
2π∫
0
max
K
∑
j∈K
log
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
≤
≤ (n+ 1)
1
2π
2π∫
0
log ‖f(reiθ)‖dθ +
1
2π
2π∫
0
log ‖f(r−1eiθ)‖dθ
−
− 1
2π
2π∫
0
log |W (f)(reiθ)|dθ +
2π∫
0
log |W (f)(r−1eiθ)|dθ
+
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON FUNDAMENTAL THEOREMS FOR HOLOMORPHIC CURVES ON ANNULI 991
+O(log r + log Tf (r)).
Since
NW (r, 0) =
1
2π
2π∫
0
log |W (f)(reiθ)|dθ +
1
2π
2π∫
0
log |W (f)(r−1eiθ)|dθ +O(1).
Lemma 3.1 is proved.
Lemma 3.2. Let f = (f0 : . . . : fn) : ∆ −→ Pn(C) be a linearly nondegenerate holomorphic
curve and H1, . . . ,Hq be hyperplanes in Pn(C) in general position and let aj is the associated vector
with Hj for j = 1, . . . , q. Then we have
q∑
j=1
mf (r,Hj) ≤
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
+
2π∫
0
max
K
∑
j∈K
log
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
+O(1).
Proof. Let aj = (aj,0, . . . , aj,n) is the associated vector of Hj , 1 ≤ j ≤ q, and let T is the set
of all injective maps µ : {0, 1, . . . , n} −→ {1, . . . , q}. By hypothesis that H1, . . . ,Hq are in general
position that for any µ ∈ T , the vectors aµ(0), . . . , aµ(n) are linearly independent.
Let µ ∈ T , we have
(f, aµ(t)) = aµ(t),0f0 + . . .+ aµ(t),nfn, t = 0, 1, . . . , n. (3.13)
Solve the system of linear equations (3.13), we get
ft = bµ(t),0(aµ(0), f) + . . .+ bµ(t),n(aµ(n), f), t = 0, 1, . . . , n,
where
(
bµ(t),j
)n
t,j=0
is the inverse matrix of
(
aµ(t),j
)n
t,j=0
. So there is a constant Cµ satisfying
‖f(z)‖ ≤ Cµ max
0≤t≤n
|(aµ(t), f)(z)|.
Set C = maxµ∈T Cµ. Then for any µ ∈ T , we have
‖f(z)‖ ≤ C max
0≤t≤n
|(aµ(t), f)(z)|.
For any z ∈ ∆r, there exists the mapping µ ∈ T such that
0 < |(aµ(0), f)(z)| ≤ |(aµ(1), f)(z)| ≤ . . . . ≤ |(aµ(n), f)(z)| ≤ |(aj , f)(z)|,
for j /∈ {µ(0), . . . , µ(n)}. Hence
q∏
j=1
‖f(z)‖
|(aj , f)(z)|
≤ Cq−n−1 max
µ∈T
n∏
t=0
‖f(z)‖
|(aµ(t), f)(z)|
.
We have
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
992 H. T. PHUONG, N. V. THIN
q∑
j=1
mf (r,Hj) =
=
q∑
j=1
1
2π
2π∫
0
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
q∑
j=1
1
2π
2π∫
0
log
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
=
=
1
2π
2π∫
0
log
q∏
j=1
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
1
2π
2π∫
0
log
q∏
j=1
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
≤
≤ 1
2π
2π∫
0
log max
µ∈T
n∏
t=0
‖f(reiθ)‖
|(aµ(t), f)(reiθ)|
dθ
2π
+
+
1
2π
2π∫
0
log max
µ∈T
n∏
t=0
‖f(r−1eiθ)‖
|(aµ(t), f)(r−1eiθ)|
dθ
2π
+O(1) =
=
1
2π
2π∫
0
max
µ∈T
log
n∏
t=0
‖f(reiθ)‖
|(aµ(t), f)(reiθ)|
dθ
2π
+
+
1
2π
2π∫
0
max
µ∈T
log
n∏
t=0
‖f(r−1eiθ)‖
|(aµ(t), f)(r−1eiθ)|
dθ
2π
+O(1) =
=
1
2π
2π∫
0
max
µ∈T
n∑
t=0
log
‖f(reiθ)‖
|(aµ(t), f)(reiθ)|
dθ
2π
+
+
1
2π
2π∫
0
max
µ∈T
n∑
t=0
log
‖f(r−1eiθ)‖
|(aµ(t), f)(r−1eiθ)|
dθ
2π
+O(1).
So we obtain
q∑
j=1
mf (r,Hj) ≤
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
+
2π∫
0
max
K
∑
j∈K
log
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
+O(1).
Lemma 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON FUNDAMENTAL THEOREMS FOR HOLOMORPHIC CURVES ON ANNULI 993
Proof of Theorem 1.2. We prove for R0 = +∞, the case R0 < +∞ can be proved similarly.
By Lemmas 3.1 and 3.2, we obtain∥∥∥∥∥∥
q∑
j=1
mf (r,Hj) ≤
2π∫
0
max
K
∑
j∈K
log
‖f(reiθ)‖
|(aj , f)(reiθ)|
dθ
2π
+
+
2π∫
0
max
K
∑
j∈K
log
‖f(r−1eiθ)‖
|(aj , f)(r−1eiθ)|
dθ
2π
+O(1) ≤
≤ (n+ 1)Tf (r)−NW (r, 0) +O(log r + log Tf (r)). (3.14)
By Theorem 1.1, we get that
Tf (r) = Nf (r,Hj) +mf (r,Hj) +O(1)
for any j ∈ {1, . . . , q}. So from (3.14), we have∥∥∥∥∥∥ (q − n− 1)Tf (r) ≤
q∑
j=1
Nf (r,Hj)−NW (r, 0) +O(log r + log Tf (r)) . (3.15)
For z0 ∈ ∆r, we may assume that (aj , f) vanishes at z0 for 1 ≤ j ≤ q1, (aj , f) does not vanish
at z0 for j > q1. Hence, there exists a nonnegative integer kj and nowhere vanishing holomorphic
function gj in neighborhood U of z such that
(aj , f)(z) = (z − z0)kjgj(z), for j = 1, . . . , q,
here kj = 0 for q1 < j ≤ q. We may assume that kj ≥ n for 1 ≤ j ≤ q0, and 1 ≤ kj < n for
q0 < j ≤ q1. By property of the Wronskian, we have
W (f) = CW ((a0, f), . . . , (an, f)) =
q0∏
j=1
(z − z0)kj−nh(z),
where h(z) is holomorphic function on U. Then W (f) is vanishes at z0 with order at least
q0∑
j=1
(kj − n) =
q0∑
j=1
kj − q0n.
By the definition of Nf (r,H), NW (r, 0) and Nn
f (r,H), we get
q∑
j=1
Nf (r,Hj)−NW (r, 0) =
=
q∑
j=1
N1,f (r,Hj)−N1
(
r,
1
W
)+
q∑
j=1
N2,f (r,Hj)−N2
(
r,
1
W
)+O(1) ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
994 H. T. PHUONG, N. V. THIN
≤
q∑
j=1
Nn
1,f (r,Hj) +
q∑
j=1
Nn
2,f (r,Hj) +O(1) =
q∑
j=1
Nn
f (r,Hj) +O(1).
So from (3.15), we obtain∥∥∥∥∥∥ (q − n− 1)Tf (r) ≤
q∑
j=1
Nn
f (r,Hj) +O(log r + log Tf (r)) .
Theorem 1.2 is proved.
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P. 774 – 786.
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Proc. Indian Acad. Sci. (Math. Sci.). – 2012. – 122, № 2. – P. 203 – 220.
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1933. – 7. – P. 80 – 103.
4. Khrystiyanyn A. Y., Kondratyuk A. A. On the Nevanlinna theory for meromorphic functions on annuli I // Mat. Stud. –
2005. – 23. – P. 19 – 30.
5. Khrystiyanyn A. Y., Kondratyuk A. A. On the Nevanlinna theory for meromorphic functions on annuli II // Mat.
Stud. – 2005. – 24. – P. 57 – 68.
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Received 29.03.12,
after revision — 14.05.15
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
|
| id | umjimathkievua-article-2037 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:17:30Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6d/15cfc62bb89bb1fe7effa92bb0dab06d.pdf |
| spelling | umjimathkievua-article-20372019-12-05T09:49:13Z On Fundamental Theorems for Holomorphic Curves on the Annuli Фундаментальні теореми для голоморфних кривих на кільцях Phuong, Ha Tran Thin, N. V. Пхуонг, Ха Тран Тхін, Н. В. We prove some fundamental theorems for holomorphic curves on the annuli crossing a finite set of fixed hyperplanes in the general position in $ℙ_n (ℂ)$ with ramification. Доведено дєякі фундаментальні теореми для голоморфних кривих на кільцях, що перетинають скінченну множину фіксованих гіперплощин загального положення в $ℙ_n (ℂ)$ з розгалудженням. Institute of Mathematics, NAS of Ukraine 2015-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2037 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 7 (2015); 981-994 Український математичний журнал; Том 67 № 7 (2015); 981-994 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2037/1092 https://umj.imath.kiev.ua/index.php/umj/article/view/2037/1093 Copyright (c) 2015 Phuong Ha Tran; Thin N. V. |
| spellingShingle | Phuong, Ha Tran Thin, N. V. Пхуонг, Ха Тран Тхін, Н. В. On Fundamental Theorems for Holomorphic Curves on the Annuli |
| title | On Fundamental Theorems for Holomorphic Curves on the Annuli |
| title_alt | Фундаментальні теореми для голоморфних кривих на кільцях |
| title_full | On Fundamental Theorems for Holomorphic Curves on the Annuli |
| title_fullStr | On Fundamental Theorems for Holomorphic Curves on the Annuli |
| title_full_unstemmed | On Fundamental Theorems for Holomorphic Curves on the Annuli |
| title_short | On Fundamental Theorems for Holomorphic Curves on the Annuli |
| title_sort | on fundamental theorems for holomorphic curves on the annuli |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2037 |
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