Growth of holomorphic curves with radially distributed hypersurfaces and uniqueness of holomorphic curves in angular domain

Abstract

In this paper, we establish the results about spread relation to holomorphic curve intersecting hypersurfaces. Using the spread relation for holomorphic curve and Second Main Theorem, we prove some results for uniqueness problem of holomorphic curve by inverse image of hypersurfaces on angular domain. In my knowledge, it is the first time that uniqueness problem for holomorphic curves on angular domain into projective space intersecting with hypersurfaces is studied. Furthermore, we also study the growth of holomorphic curves with radially distributed hypersurfaces.

Keywords

Angular domain hypersurface Nevanlinna–Cartan theory subgeneral position spread relation to holomorphic curve

1. Introduction

We denote $Ω (α, β) = {z : α < \arg z < β}$ by the angle on complex plane and $\bar{Ω} (α, β)$ by its closure, where $0 < β - α ⩽ 2 π$ . Then, $Ω (α, β)$ is called an angular domain on complex plane. In 1998, by using potential method, Fedorov and Grishin studied the Second Main Theorem for holomorphic curve on half plane intersecting with hyperplanes in general position. In 2015, Zheng [19] established the value distribution of holomorphic curves on an angular domain intersecting with hyperplanes from the point of view of potential theory and established the first and second fundamental theorems corresponding to those theorems of Ahlfors–Shimizu, Nevanlinna, and Tsuji on meromorphic functions in an angular domain. We refer readers to [3,5,14,18,19] for comments on the results of the value distribution of holomorphic curves and meromorphic functions on an angular domain. In 2023, Quang [13] established some Second Main Theorems for holomorphic mappings from angular domain into projective varieties for arbitrary families of moving hypersurfaces via distributive constant. However, the applications of Second Main Theorem is not studied in [13]. The value distribution of holomorphic curve with lower order studied by some authors and has long history. We refer the readers to [2,7,9,10,12,15,17] for more details. In this paper, we will study the applications of Second Main Theorem on angular domain for holomorphic mappings intersecting with hypersurfaces in subgeneral position. In the studing of value distribution of holomorphic curve on angular domain, the spread of holomorphic curves has many applications. We refer the readers to [6] and [16] for some estimations on speard of holomorphic curve. In [8], Lin, Mori and Kohge gave some uniquenes results of meromorphic functions by using spread relation. In this work, we first extend the result of Niino [11] about spread relation to holomorphic curve intersecting hypersurfaces. Using the spread relation for holomorphic curve and a version Second Main Theorem with good level of truncated function, we establish some results for uniqueness problem of holomorphic curve by inverse image of hypersurfaces on angular domain. In my knowledge, it is the first time that uniqueness problem for holomorphic curves on angular domain into projective space intersecting with hypersurfaces is studied. Furthermore, we also study the growth of holomorphic curves with radially distributed hypersurfaces.

2. Characteristics of holomorphic curve on angular domain

In this paper, we consider the holomorphic curve $f = (f_{0} : \dots : f_{n}) : C \to P^{n} (C)$ with reduced representation $\tilde{f} = (f_{0}, \dots, f_{n})$ , where $f_{0}, \dots, f_{n}$ are holomorphic functions and without common zeros in $C$ . The characteristic function $T_{f} (r)$ of f is given by

\begin{aligned} T_{f} (r) = \frac{1}{2 π} \int_{0}^{2 π} \log ‖ \tilde{f} (r e^{i θ}) ‖ d θ, \end{aligned}

where

‖ \tilde{f} (z) ‖ = max {| f_{0} (z) |, \dots, | f_{n} (z) |}

. The above definition is independent, up to an additive constant, of the choice of the reduced representation of f. The order and lower order of f is defined by

\begin{aligned} ρ (f) = \underset{r \to \infty}{lim sup} \frac{\log T_{f} (r)}{\log r} and τ (f) = \underset{r \to \infty}{lim inf} \frac{\log T_{f} (r)}{\log r}, \end{aligned}

respectively. The holomorphic curve f is called transcendental if

\begin{aligned} lim_{r \to \infty} \frac{T_{f} (r)}{\log r} = \infty . \end{aligned}

Let r be a real number satisfying $1 ⩽ r < \infty$ . Set $ω = \frac{π}{β - α}$ and

\begin{aligned} Ω (α, β; r) = Ω (α, β) \cap {1 < | z | < r} . \end{aligned}

Let

f = (f_{0} : \dots : f_{n}) : \bar{Ω} (α, β) \to P^{n} (C)

be a holomorphic curve with reduced representation

\tilde{f} = (f_{0}, \dots, f_{n})

, where

f_{0}, \dots, f_{n}

are holomorphic functions and without common zeros in

\bar{Ω} (α, β)

. The counting function

C_{α β, f} (r, D)

of f with respect to D is defined as

\begin{aligned} C_{α β, f} (r, D) = 2 \sum_{1 ⩽ ρ_{n} ⩽ r, α ⩽ ψ_{n} ⩽ β} (\frac{1}{{ρ_{n}}^{ω}} - \frac{{ρ_{n}}^{ω}}{r^{2 ω}}) \sin ω (ψ_{n} - α), \end{aligned}

where the

{ρ_{n} e^{i ψ_{n}}}

are the zeros of

Q (\tilde{f})

\bar{Ω} (α, β)

counting with multiplicity. For each zero

ρ_{n} e^{i ψ_{n}}

Q (\tilde{f})

\bar{Ω} (α, β)

with multiple m, then term

2 (\frac{1}{{ρ_{n}}^{ω}} - \frac{{ρ_{n}}^{ω}}{r^{2 ω}}) \sin ω (ψ_{n} - α)

is counted m times in

C_{α β, f} (r, D)

Now let δ be a positive integer, the truncated counting function of f is defined by

\begin{aligned} C_{α β, f}^{δ} (r, D) = 2 \sum_{1 ⩽ ρ_{n} ⩽ r, α ⩽ ψ_{n} ⩽ β, min {{ord}_{Q (\tilde{f})} (ρ_{n} e^{i ψ_{n}}), δ}} (\frac{1}{{ρ_{n}}^{ω}} - \frac{{ρ_{n}}^{ω}}{r^{2 ω}}) \sin ω (ψ_{n} - α), \end{aligned}

where any zero of multiplicity greater than δ of

Q (\tilde{f})

\bar{Ω} (α, β)

is “truncated” and counted as if it only had multiplicity δ. This means that for each zero

ρ_{n} e^{i ψ_{n}}

Q (\tilde{f})

\bar{Ω} (α, β)

with multiple m, the terms

2 (\frac{1}{{ρ_{n}}^{ω}} - \frac{{ρ_{n}}^{ω}}{r^{2 ω}}) \sin ω (ψ_{n} - α)

is counted

min {m, δ}

times in

C_{α β, f} (r, D)

The angular proximity Nevanlinna of f with respect to D is defined as following:

\begin{aligned} A_{α β, f} (r, D) = \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) \log \frac{‖ \tilde{f} (t e^{i α}) ‖^{d} ‖ \tilde{f} (t e^{i β}) ‖^{d} ‖ Q ‖^{2}}{| Q (\tilde{f}) (t e^{i α}) Q (\tilde{f}) (r e^{i β}) |} \frac{d t}{t} \end{aligned}

and

\begin{aligned} B_{α β, f} (r, D) = \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log \frac{‖ \tilde{f} (r e^{i φ}) ‖^{d} ‖ Q ‖}{| Q (\tilde{f}) (r e^{i φ}) |} \sin (ω (φ - α)) d φ, \end{aligned}

where

‖ \tilde{f} (z) ‖ = max {| f_{0} (z) |, \dots, | f_{n} (z) |}

The angular Nevanlinna characteristic $S_{α β, f} (r)$ of f is defined by

\begin{aligned} S_{α β, f} (r) & = \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) \log ‖ \tilde{f} (t e^{i α}) ‖ ‖ \tilde{f} (t e^{i β}) ‖ \frac{d t}{t} \\ + \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log ‖ \tilde{f} (r e^{i φ}) ‖ . \sin (ω (φ - α)) d φ . \end{aligned}

Definition 1.
Assume that $N ⩾ n$ and $q ⩾ N + 1$ . Let $D_{1}, \dots, D_{q}$ be hypersurfaces in $P^{n} (C)$ . The hypersurfaces $D_{1}, \dots, D_{q}$ are said to be in N-sugeneral position in $P^{n} (C)$ if for any subset $I \subset {1, \dots, q}$ with $# I = N + 1$ , we have $⋂_{i \in I} Supp D_{i} = \emptyset$ . When $N = n$ , we said that the hypersurfaces $D_{1}, \dots, D_{q}$ are said to be in general position in $P^{n} (C)$ .
Definition 2.
A map $f : \bar{Ω} (α, β) \to P^{n} (C)$ is said to be algebraically nondegenerate if the image of f is not contained in any proper subvarieties of $P^{n} (C)$ .

In this paper, a notation “‖” in the inequality is mean that the inequality holds for $r \in (1, \infty)$ outside a set with measure finite.

Our results are given as follows:
Theorem 3.
Let D be a hypersurface in $P^{n} (C)$ and $f = (f_{0} : \dots : f_{n}) : \bar{Ω} (α, β) \to P^{n} (C)$ be a holomorphic curve whose image is not contained D. Then we have for any $1 < r < \infty$ ,
$\begin{aligned} d S_{α β, f} (r) = A_{α β, f} (r, D) + B_{α β, f} (r, D) + C_{α β, f} (r, D) + O (1) . \end{aligned}$

Proof.
Note that $C_{α β} (r, Q (\tilde{f})) = 0$ . By the definitions of $S_{α β, f} (r)$ , $A_{α β, f} (r, D)$ , $B_{α β, f} (r, D)$ and apply to Lemma 1 for $Q (\tilde{f}) ⧸ \equiv 0$ , we have
$\begin{aligned} C_{α β, f} (r, D) & = \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) [\log | Q (\tilde{f}) (t e^{i α}) | + \log | Q (\tilde{f}) (t e^{i β}) |] \frac{d t}{t} \\ + \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log | Q (\tilde{f}) (r e^{i φ}) | . \sin (ω (φ - α)) d φ + O (1) . \end{aligned}$
Hence, we get
$\begin{aligned} A_{α β, f} (r, D) + B_{α β, f} (r, D) + C_{α β, f} (r, D) \\ = \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) \log \frac{‖ \tilde{f} (t e^{i α}) ‖^{d} ‖ \tilde{f} (t e^{i β}) ‖^{d}}{| Q (\tilde{f}) (t e^{i α}) Q (\tilde{f}) (r e^{i β}) |} \frac{d t}{t} \\ + \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log \frac{‖ \tilde{f} (r e^{i φ}) ‖^{d}}{| Q (\tilde{f}) (r e^{i φ}) |} \sin (ω (φ - α)) d φ \\ + \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) [\log | Q (\tilde{f}) (t e^{i α}) | + \log | Q (\tilde{f}) (t e^{i β}) |] \frac{d t}{t} \\ + \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log | Q (\tilde{f}) (r e^{i φ}) | . \sin (ω (φ - α)) d φ + O (1) \\ = \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) \log ‖ \tilde{f} (t e^{i α}) ‖^{d} ‖ \tilde{f} (t e^{i β}) ‖^{d} \frac{d t}{t} \\ + \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log ‖ \tilde{f} (r e^{i φ}) ‖^{d} . \sin (ω (φ - α)) d φ + O (1) \\ = d S_{α β, f} (r) + O (1) . \end{aligned}$
This is conclusion of Theorem 3.

Using Theorem 3, Lemma 1, Lemma 2, some steps in [13], and the methods in proving Theorem 1.2 [1], we can get the result as follows: Theorem 4.
Assume that q, N, n are three natural numbers such that $q ⩾ N + 1$ . Let f be an algebraically nondegenerate holomorphic curve from $C$ into $P^{n} (C)$ , and $D_{1}, \dots, D_{q}$ be hypersurfaces in N-subgeneral position on $P^{n} (C)$ with $\deg Q_{j} = d_{j}$ , $j = 1, \dots, q$ . Set $d = l c m (d_{1}, \dots, d_{q})$ and $M = C_{n + d}^{n} - 1$ . If
$\begin{aligned} q > \frac{(M + 1) (2 N - n + 1)}{n + 1}, \end{aligned}$
then we have
$\begin{aligned} ‖ (q - \frac{(M + 1) (2 N - n + 1)}{n + 1}) S_{α β, f} (r) ⩽ \sum_{i = 1}^{q} d_{i}^{- 1} C_{α β, f}^{M} (r, D_{i}) + Q_{α β, f} (r), \end{aligned}$
where
$\begin{aligned} Q_{α β, f} (r) & = {\begin{cases} O (1) & a s r \to + \infty if ρ (f) < + \infty \\ O (\log r T (r, f)) & i f ρ (f) = \infty outside a set of finite measure . \end{cases} \end{aligned}$

3. Growth of holmorphic curves with radially distributed hypersurfaces

Let $f : C \to P^{n} (C)$ be a holomorphic curve and $\tilde{f} = (f_{0}, \dots, f_{n})$ be a reduced representation of f. Let D be a hypersurface with degree d which is defined by polynomial homogeneous Q with degree d. Define

\begin{aligned} δ_{f} (D) = 1 - \underset{r \to \infty}{lim sup} \frac{N_{f} (r, D)}{d T_{f} (r)}, \end{aligned}

which is called a deficiency with respective D, and if

δ_{f} (D) > 0

, then D is called a deficient hypersurface of f. Under the existence of a deficient hypersurface of f, we consider the estimation of growth of the holomorphic curve f in view of some radially distributed hypersurfaces. Since the Cartan characteristic describes the growth of holomorphic curve, then we will first give an estimate for Cartan characteristic, and the estimate can be controlled in view of the counting functions of argument distribution of preimages of some hypersurfaces, so that we obtain the estimate of growth order of a holomorphic curve in view of some radially distributed hypersurfaces.

Because the Cartan characteristic $T_{f} (r)$ of a holomorphic curve is increasing and logarithmic convex, we can consider its Pólya peak sequence. A positive increasing unbounded sequence ${r_{j}}$ is a sequence of Pólya peak order σ of $T_{f} (r)$ if there exist sequences $r_{j}^{^{'}}$ , $r_{j}^{^{″}}$ , and $ε_{j}$ such that

(i)
$r_{j}^{^{'}} \to \infty$ , $\frac{r_{j}}{r_{j}^{^{'}}} \to \infty$ , $\frac{r_{j}^{^{″}}}{r_{j}} \to \infty$ , and $ε_{j} \to 0$ as $j \to \infty$ .
(ii)
$T_{f} (r) ⩽ (1 + ε_{j}) (r / r_{j})^{σ} T_{f} (r_{j})$ , $r_{j}^{^{'}} ⩽ r ⩽ r_{j}^{^{″}}$ .

The sequence ${r_{j}}$ is called a strong Pólya peak sequence if, in addition, we have
(iii)
$T_{f} (r) / r^{σ - ε_{j}^{^{'}}} ⩽ K T_{f} (r_{j}) / r_{j}^{σ - ε_{j}^{^{'}}}$ , $1 ⩽ r ⩽ r_{j}^{^{″}}$ , for a sequence $ε_{j}^{^{'}} \to 0$ as $j \to \infty$ and a positive constant K.

From (iii), we have $lim inf_{j \to \infty} \frac{\log T_{f} (r_{j})}{\log r_{j}} ⩾ σ$ . If $T_{f} (r)$ has the lower order $τ < \infty$ and order $0 < ρ ⩽ \infty$ , then for a finite positive number σ with $τ ⩽ σ ⩽ ρ$ and a set E of positive numbers with finite logarithmic measure, there must be a sequence of strong Pólya peak ${r_{j}}$ of order σ of $T_{f} (r_{j})$ outside E (see Theorem 1.1.3 in [18]). Consider p pairs of real numbers ${α_{j}, β_{j}}$ such that
$\begin{aligned} - π ⩽ α_{1} < β_{1} ⩽ α_{2} < β_{2} ⩽ \dots ⩽ α_{p} < β_{p} ⩽ π, α_{p + 1} = α_{1} + 2 π, \end{aligned}$
(3.1)
and we denote $D = D (α_{1}, β_{1}, \dots, α_{p}, β_{p})$ by the corresponding ray system $\arg z = α_{j}$ , $β_{j} (1 ⩽ j ⩽ p)$ . For the system $D$ , define
$\begin{aligned} ω (D) = max {ω_{j} = \frac{π}{β_{j} - α_{j}} : 1 ⩽ j ⩽ p} \end{aligned}$
and $W_{D} (r, f, H) = max {r^{ω_{j}} B_{α_{j} β_{j}, f} (r, H) : 1 ⩽ j ⩽ p}$ .

With above notations, Zheng proved the results as follows:
Theorem 5 [19]

Let $f : C \to P^{n} (C)$ be a transcendental holomorphic curve on $C$ of lower order $τ < \infty$ and order $0 < ρ ⩽ \infty$ . Let H be a hyperplane and $δ = δ_{f} (H) > 0$ . Assume that for a $σ > 0$ with $τ ⩽ σ ⩽ ρ$ ,

\begin{aligned} \sum_{j = 1}^{p} (α_{j + 1} - β_{j}) < \frac{4}{σ} \arcsin \sqrt{\frac{δ}{2}} . \end{aligned}

(3.2)

Then for any sequence of Pólya peaks

{r_{j}}

of order σ, we have

\begin{aligned} T_{f} (r_{j}) ⩽ K W_{D} (r_{j}, f, H) \end{aligned}

(3.3)

for some positive K only depending on

{r_{j}}

independent of j.

Theorem 6 [19]

Let $f : C \to P^{n} (C)$ be an m-nondegenerate transcendental holomorphic curve on $C$ of finite lower order τ. Let $H, H_{0}, H_{1}, \dots, H_{q}$ with $q = 2 n - m$ be $q + 2$ hyperplanes in general position, and $δ = δ_{f} (H) > 0$ . Assume that for a ray system $D (α_{1}, β_{1}, \dots, α_{p}, β_{p})$ , ( 3.2 ) holds with $σ = max {ω (D), τ}$ and

\begin{aligned} \sum_{k = 0}^{q} N_{α_{j} β_{j}, f} (r, H_{k}) = o (T_{f} (r)), 1 ⩽ j ⩽ p, \end{aligned}

where

N_{α_{j} β_{j}, f} (r, H_{k})

is counting function of f with respective to

H_{k}

\bar{Ω} (α_{j}, β_{j})

for

j = 1, \dots, p, 0 ⩽ k ⩽ q

. Then

ρ (f) ⩽ ω (D)

Note that Zheng [19] stated Theorem 5 and Theorem 6 for holomorphic curve. However in the proof of spread lemma, we need $\log r = o (T_{f} (r))$ , then f must be a transcendental holomorphic curve. Motivate from the Theorem 5 and Theorem 6, we prove two following results:

Theorem 7.
Let $f : C \to P^{n} (C)$ be a transcendental holomorphic curve on $C$ of lower order $τ < \infty$ and order $0 < ρ ⩽ \infty$ . Let D be a small moving hypersurface with respect to f and $δ = δ_{f} (D) > 0$ . Assume that for a $σ > 0$ with $τ ⩽ σ ⩽ ρ$ ,
$\begin{aligned} \sum_{j = 1}^{p} (α_{j + 1} - β_{j}) < \frac{4}{σ} \arcsin \sqrt{\frac{δ}{2}} . \end{aligned}$
(3.4)
Then for any sequence of Pólya peaks ${r_{m}}$ of order σ, we have
$\begin{aligned} T_{f} (r_{m}) ⩽ K_{} W_{D} (r_{m}, f, D) \end{aligned}$
(3.5)
for some positive* $K_{}$ only depending on* ${r_{m}}$ independent of m, where $W_{D} (r, f, D) = max {r^{ω_{j}} B_{α_{j} β_{j}, f} (r, D) : 1 ⩽ j ⩽ p}$ .
Theorem 8.
Assume that q, N, n are three natural numbers such that $q ⩾ N + 1$ , and $f : C \to P^{n} (C)$ be a nondegenerate algebraically transcendental holomorphic curve of finite lower order τ. Let $D, D_{1}, \dots, D_{q}$ be $q + 1$ $(q > \frac{(M + 1) (2 N - n + 1)}{n + 1})$ hypersurfaces in N-subgeneral position in $P^{n} (C)$ , and $δ = δ_{f} (D) > 0$ . Assume that for a ray system $D (α_{1}, β_{1}, \dots, α_{p}, β_{p})$ , ( 3.4 ) holds with $σ = max {ω (D), τ}$ and
$\begin{aligned} \sum_{k = 1}^{q} \frac{N_{α_{j} β_{j}, f} (r, D_{k})}{\deg D_{k}} = o (T_{f} (r)), 1 ⩽ j ⩽ p, \end{aligned}$
where $N_{α_{j} β_{j}, f} (r, D_{k})$ is counting function of f with respective to $D_{k}$ in $\bar{Ω} (α_{j}, β_{j})$ for $j = 1, \dots, p, 1 ⩽ k ⩽ q$ . Then $ρ (f) ⩽ ω (D)$ .

4. Uniqueness of holomorphic curves in angular domain

The uniqueness of holomorphic curve on complex plane is investigated by many authors. So far, there is not any result about uniqueness theorem of holomorphic curve on angular domain by inverse image of hypersurfaces. In this section, using the spread relation for holomorphic curve (Theorem 11) and Theorem 4, we obtain the following results:

Theorem 9.
Let q, N, n be positive integers such that $q ⩾ N + 1$ , and f and g be nondegenerate algebraically transcendental holomorphic curves from $C$ into $P^{n} (C)$ and f has lower order $0 < τ < \infty$ . Let $D_{1}, \dots, D_{q}$ be q ( $q > \frac{(M + 1) (2 N - n + 1)}{n + 1} + \frac{N M}{δ}$ , $δ = min {d_{1}, \dots, d_{q}}$ , where $d_{1}, \dots, d_{q}$ are degree of $D_{1}, \dots, D_{q}$ respectively) hypersurfaces in N-subgeneral position in $P^{n} (C)$ , and D be a small moving hypersurface of f such that $δ = δ_{f} (D) > 0$ . Assume that for a ray system $D (α_{1}, β_{1}, \dots, α_{p}, β_{p})$ , ( 3.4 ) holds with $σ = max {ω (D), τ}$ and three following conditions hold: (i)
$f^{- 1} (D_{j}) \cap X = g^{- 1} (D_{j}) \cap X$ , $j = 1, \dots, q$ , where
$\begin{aligned} X = ⋃_{k = 1}^{p} {z : α_{k} ⩽ \arg z ⩽ β_{k}}; \end{aligned}$

(ii)
$f (z) = g (z)$ for all $z \in ⋃_{j = 1}^{q} (f^{- 1} (D_{j}) \cap X)$ ;
(iii)
$\log T_{f} (r) = O (\log T_{g} (r))$ and $\log \log T_{f} (r) = o (\log r) as r \to \infty$ .
If $ω (D) < ρ (f)$ , then $f \equiv g$ .
Remark 10.
In the Theorem 9, the condition $\log \log T_{f} (r) = o (\log r) as r \to \infty$ is a technique condition. About the uniqueness of meromorphic functions on an angular domain share sets, the authors [8] required that $lim inf_{n \to \infty} \frac{\log \log T_{f} (r_{n})}{\log r_{n}} = 0$ for Pólya peak sequence. However, it is not true, we only have
$\begin{aligned} \underset{n \to \infty}{lim inf} \frac{\log \log T_{f} (r_{n})}{\log r_{n}} ⩾ 0. \end{aligned}$
Let f be a holomorphic curve such that $T_{f} (r) = a r (1 + o (1)) (a > 0)$ when $r ≫ 0$ , then $τ (f) = ρ (f) = 1$ and $\log \log T_{f} (r) = o (\log r)$ as $r \to \infty$ .

5. Some preliminaries in angular Nevanlinna theory for meromorphic functions

First, we remind some definitions which is contained the book of A. A. Goldberg and I. V. Ostrovskii [4]. We consider the set

\begin{aligned} Ω (α, β; r) = Ω (α, β) \cap {1 < | z | < r} . \end{aligned}

Let f be a meromorphic function on the angle

\bar{Ω} (α, β; r)

0 < β - α ⩽ 2 π

1 ⩽ r < \infty

. We recall that

\begin{aligned} A_{α β} (r, f) & = \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) [\log^{+} | f (t e^{i α}) | + \log^{+} | f (t e^{i β}) |] \frac{d t}{t}; \\ B_{α β} (r, f) & = \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log^{+} | f (r e^{i φ}) | . \sin (ω (φ - α)) d φ; \\ C_{α β} (r, f) & = 2 ω \int_{1}^{r} c_{α β} (r, f) (\frac{1}{t^{ω}} + \frac{t^{ω}}{r^{2 ω}}) \frac{d t}{t} \\ = 2 \sum_{1 ⩽ ρ_{n} ⩽ r, α ⩽ ψ_{n} ⩽ β} (\frac{1}{{ρ_{n}}^{ω}} - \frac{{ρ_{n}}^{ω}}{r^{2 ω}}) \sin ω (ψ_{n} - α), \end{aligned}

where

\begin{aligned} c_{α β} (r, f) = \sum_{1 < ρ_{n} ⩽ r, α ⩽ ψ_{n} ⩽ β} \sin (ω (ψ_{n} - α)), \end{aligned}

and

{ρ_{n} e^{i ψ_{n}}}

are poles of

f (z)

counted according with multiplicity. We denote

S_{α β} (r, f)

by the angular Nevanlinna characteristics on

\bar{Ω} (α, β; r)

and defined as following:

\begin{aligned} S_{α β} (r, f) = A_{α β} (r, f) + B_{α β} (r, f) + C_{α β} (r, f) . \end{aligned}

In order to prove theorems, we need the following lemmas.

Lemma 1 [4]

(Carleman formula) Let f be a nonconstant meromorphic function in $\bar{Ω} (α, β; r)$ . Then

\begin{aligned} C_{α β} (r, \frac{1}{f}) - C_{α β} (r, f) & = \frac{ω}{π} \int_{1}^{r} (\frac{1}{t^{ω}} - \frac{t^{ω}}{r^{2 ω}}) [\log | f (t e^{i α}) | + \log | f (t e^{i β}) |] \frac{d t}{t} \\ + \frac{2 ω}{π r^{ω}} \int_{α}^{β} \log | f (r e^{i φ}) | . \sin (ω (φ - α)) d φ + O (1) . \end{aligned}

Some time we denote $C_{α β, f} (r, 0)$ instead of $C_{α β} (r, \frac{1}{f})$ .

Lemma 2 [4]

Let k be a natural number and f be nonconstant meromorphic function on $C$ . Then we have the estimate

\begin{aligned} A_{α β} (r, \frac{f^{(k)}}{f}) + B_{α β} (r, \frac{f^{(k)}}{f}) \\ = {\begin{cases} O (1) & a s r \to + \infty if ρ (f) < + \infty \\ O (\log r T (r, f)) & i f ρ (f) = \infty outside a set of finite measure . \end{cases} \end{aligned}

From Lemma 2, we have

\begin{aligned} A_{α β} (r, \frac{f^{(k)}}{f}) + B_{α β} (r, \frac{f^{(k)}}{f}) = o (T (r, f)) \end{aligned}

(5.1)

for all r large enough outside a set of finite measure.

6. Proofs of Theorem 7 and Theorem 8

Let $f : C \to P^{n} (C)$ be a transcendental holomorphic curve of lower order $τ (0 < τ < \infty)$ and $\tilde{f} = (f_{0}, f_{1}, \dots, f_{n}) : C \to C^{n + 1} ∖ {0}$ its reduced representation. Suppose that D is a moving hypersurface with degree d which is defined by the moving polynomial homogeneous with degree d as follows

\begin{aligned} Q (x) = Q (x_{0}, \dots, x_{n}) = \sum_{j = 0}^{n_{d}} a_{k} (z) x_{0}^{i_{j 0}} \dots x_{n}^{i_{j n}} \end{aligned}

such that

a_{0}, \dots, a_{n_{d}}

are holomorphic function without common zero, where

x = (x_{0}, \dots, x_{n})

i_{j 0} + \dots + i_{j n} = d

for

j = 0, \dots, n_{d}

n_{d} = (\frac{n + d}{n}) - 1

. We denote by

a = (a_{0} : \dots : a_{n_{d}}) : C \to P^{n_{d}} (C)

the holomorphic curve associated with D (or with Q), and denote

\tilde{a} = (a_{0}, \dots, a_{n_{d}})

by the reduced representation of a. The moving hypersurface D is called small with respective to f if

T_{a} (r) = o (T_{f} (r))

r \to \infty

. We denote

N_{Q (\tilde{f})} (r, 0)

(or

N_{f} (r, D)

) by the counting function of zeros of

Q (\tilde{f})

. Then the defect of f with respect the small moving hypersurface D is defined by

\begin{aligned} δ_{f} (D) = 1 - \underset{r \to \infty}{lim sup} \frac{N_{Q (\tilde{f})} (r, 0)}{d (T_{f} (r) + T_{a} (r))} = 1 - \underset{r \to \infty}{lim sup} \frac{N_{Q (\tilde{f})} (r, 0)}{d T_{f} (r)} \end{aligned}

since

T_{a} (r) = o (T_{f} (r))

r \to \infty

Let D be a small moving hypersurface with respect to f, we put

\begin{aligned} ‖ f (z), a (z) ‖ = \frac{| Q (\tilde{f} (z)) |}{‖ \tilde{f} (z) ‖^{d} . ‖ a (z) ‖}, \end{aligned}

where

‖ \tilde{f} (z) ‖ = max {f_{0} (z), \dots, f_{n} (z)}

and

‖ a (z) ‖ = \sum_{i = 0}^{n_{d}} | a_{i} (z) |

. We consider Pólya peak sequence

{r_{j}}

of order σ of

T_{f} (r)

. It means that

{r_{j}}

is positive sequence and there exist sequences

r_{j}^{^{'}}

r_{j}^{^{″}}

, and

ε_{j}

such that

(i)
$r_{j}^{^{'}} \to \infty$ , $\frac{r_{j}}{r_{j}^{^{'}}} \to \infty$ , $\frac{r_{j}^{^{″}}}{r_{j}} \to \infty$ , and $ε_{j} \to 0$ as $j \to \infty$ .
(ii)
$T_{f} (r) ⩽ (1 + ε_{j}) (r / r_{j})^{σ} T_{f} (r_{j})$ , $r_{j}^{^{'}} ⩽ r ⩽ r_{j}^{^{″}}$ .
Let $Λ (r)$ be a positive function satisfying
$\begin{aligned} Λ (r) = o (T_{f} (r)), r \to \infty . \end{aligned}$
(6.1)
Put
$\begin{aligned} E_{Λ} (r, a) \equiv E_{Λ} (r, a; f) = {θ : \log ‖ f (r e^{i θ}), a (r e^{i θ}) ‖ < - Λ (r)} \subset [- π, π) \end{aligned}$
and let
$\begin{aligned} μ_{Λ} (a) = \underset{m \to \infty}{lim inf} meas E_{Λ} (r_{m}, a) . \end{aligned}$
Then we define
$\begin{aligned} μ (a) = inf_{Λ} μ_{Λ} (a), \end{aligned}$
where the infimum is taken over all functions Λ satisfying (6.1). With above notations, we obtain the following result which extends the result due to Niino [11]:
Theorem 11.
Let $f : C \to P^{n} (C)$ be a transcendental holomorphic curve of lower order $τ (0 < τ < \infty)$ and order $0 < ρ ⩽ \infty$ . Let D be a small moving hypersurface with degree d of f and $a : C \to P^{n_{d}} (C)$ is its associated holomorphic curve such that $f (C) ⧸ \subset D$ and $δ_{f} (D) > 0$ . Then for any sequence ${r_{j}}$ of Pólya peaks of order $β > 0$ $(τ ⩽ β ⩽ ρ)$ for f and any positive function $Λ (r)$ with $Λ (r) \to 0$ as $r \to \infty$ , we have
$\begin{aligned} \underset{j \to \infty}{lim inf} E_{Λ} (r_{j}, a) ⩾ min {2 π, \frac{4}{β} \sin^{- 1} (δ_{f} (D) / 2)^{1 / 2}} . \end{aligned}$
Here $\sin^{- 1} x = \arcsin x$ , $x \in [0, \frac{1}{\sqrt{2}}]$ .

From Theorem 11, we get $μ (a) ⩾ min {2 π, \frac{4}{β} \sin^{- 1} (δ_{f} (D) / 2)^{1 / 2}}$ .
Proof.
The map f has the lower order $τ < \infty$ and order $0 < ρ ⩽ \infty$ , then for a finite positive number σ with $τ ⩽ σ ⩽ ρ$ and a set E of positive numbers with finite logarithmic measure, then there exists a sequence of strong Pólya peak ${r_{j}}$ of order σ of $T_{f} (r_{j})$ outside E (see Theorem 1.1.3 in [18]). The proof of Theorem 11 are the same the proof of Theorem 2 in [11]. We only recall some steps. Note that $τ ⩽ β ⩽ ρ$ , $β < \infty$ . We may assume that $‖ f (0) ‖ = 1$ and $‖ a (0) ‖ = 1$ . We have
$\begin{aligned} E_{Λ} (r, a) = {θ : d \log ‖ \tilde{f} (r e^{i θ}) ‖ + \log ‖ a (r e^{i θ}) ‖ - \log | F (r e^{i θ}) | > Λ (r)}, \end{aligned}$
where $F (z) = Q (\tilde{f})$ . Define the entire function $h (z)$ by $h (z) = c z^{- k} F (z)$ , where k is a nonnegative integer, c is a nonzero constant, and $h (0) = 1$ . Then, we have
$\begin{aligned} N_{F} (r, 0) = N_{h} (r, 0) + k \log r . \end{aligned}$
We put
$\begin{aligned} G (z) & = d \log ‖ \tilde{f} (z) ‖ + \log ‖ a (z) ‖ - \log | h (z) | \\ Λ_{1} (r) & = Λ (r) + k \log r - \log | c | \end{aligned}$
and $E_{Λ_{1}} (r, G) = {θ : G (r e^{i θ}) > Λ_{1} (r)}$ . Then we deduce that $E_{Λ_{1}} (r, G) = E_{Λ} (r, a)$ and $Λ_{1} (r) = o (T_{f} (r))$ as $r \to \infty$ . Set $T (r) = T_{f} (r) + T_{a} (r)$ . Since a is a small mapping of f, then we get
$\begin{aligned} δ_{f} (D) = 1 - \underset{r \to \infty}{lim sup} \frac{N_{h} (r, 0)}{d T (r)} . \end{aligned}$
Therefore, in order to prove Theorem 11, it is sufficient to prove that
$\begin{aligned} \underset{j \to \infty}{lim inf} meas E_{Λ_{1}} (r_{j}, G) ⩾ min {2 π, \frac{4}{β} \sin^{- 1} (δ_{f} (D) / 2)^{1 / 2}} . \end{aligned}$
The remain proofs are the same in [11], pages 103 to 104, we omit it at here.
Proofs of Theorem 7. Proofs of Theorem 7.

Suppose that (3.5) does not hold. Then there exists a subsequence of ${r_{m}}$ , still denoted by ${r_{m}}$ such that $W_{D} (r_{m}, f, D) = o (T (r_{m}, f))$ . Set

\begin{aligned} Γ (r) = \frac{W_{D} (r_{m}, f, D)}{T (r_{m}, f)}, r_{m - 1} < r ⩽ r_{m}, \end{aligned}

and

Λ (r) = \sqrt{Γ (r)}

. Obiviously,

Λ (r) \to 0

r \to \infty

. For a positive function

Λ (r)

satisfying

Λ (r) \to 0

r \to \infty

, we denote

\begin{aligned} E_{Λ} (r, a) = {θ \in [- π, π) : \log ‖ f (r e^{i θ}), a (r e^{i θ}) ‖ < - Λ (r) T_{f} (r)} . \end{aligned}

(6.2)

Taking an

ε > 0

such that

\begin{aligned} \sum_{j = 1}^{p} (α_{j + 1} - β_{j} + 2 ε) + 2 ε < \frac{4}{σ + 2 ε} \arcsin \sqrt{δ_{f} (D) / 2} . \\ K := meas (E_{Λ} (r_{m}, a) \cap (⋃_{j = 1}^{p} (α_{j} + ε, β_{j} + ε)) . \end{aligned}

Then we get

\begin{aligned} K & ⩾ meas (E_{Λ} (r_{m}, a) - meas ([- π, π) ∖ ⋃_{j = 1}^{p} (α_{j} + ε, β_{j} - ε)) \\ = meas (E_{Λ} (r_{m}, a)) - meas (⋃_{j = 1}^{p} (β_{j} - ε, α_{j + 1} + ε)) \\ = meas (E_{Λ} (r_{m}, a)) - \sum_{j = 1}^{p} (α_{j + 1} - β_{j} + 2 ε) > ε > 0. \end{aligned}

It is easy to see that there exists a

j_{0}

such that for all m large enough, we have

\begin{aligned} meas (E_{Λ} (r_{m}, a) \cap (α_{j_{0}} + ε, β_{j_{0}} - ε)) ⩾ \frac{K}{p} ⩾ \frac{ε}{p} . \end{aligned}

(6.3)

Set

E_{m} = E_{Λ} (r_{m}, a) \cap (α_{j_{0}} + ε, β_{j_{0}} - ε)

. Then from (6.3), we get

\begin{aligned} \int_{α_{j_{0}} + ε}^{β_{j_{0}} - ε} \frac{‖ \tilde{f} (r_{m} e^{i θ}) ‖^{d} ‖ a (r_{m} e^{i θ}) ‖}{Q (\tilde{f} (r_{m} e^{i θ}))} d θ & ⩾ \int_{E_{m}} \frac{‖ \tilde{f} (r_{m} e^{i θ}) ‖^{d} ‖ a (r_{m} e^{i θ}) ‖}{Q (\tilde{f} (r_{m} e^{i θ}))} d θ \\ ⩾ \frac{ε}{p} Λ (r_{m}) T_{f} (r_{m}) . \end{aligned}

(6.4)

By the definition of

B_{α β, f} (r, D)

, we have

\begin{aligned} r_{m}^{j_{0}} B_{α_{j_{0}} β_{j_{0}}, f} (r_{m}, D) = \frac{2 ω_{j_{0}}}{π} \int_{α_{j_{0}}}^{β_{j_{0}}} \log \frac{‖ \tilde{f} (r_{m} e^{i θ}) ‖^{d} ‖ a (r_{m} e^{i θ}) ‖}{| Q (\tilde{f}) (r_{m} e^{i θ}) |} \sin (ω_{j_{0}} (θ - α_{j_{0}})) d θ . \end{aligned}

Then from (6.4), for

ε > 0

small enough, we see

\begin{aligned} W_{D} (r_{m}, f, D) & ⩾ \frac{2 ω_{j_{0}}}{π} \int_{α_{j_{0}}}^{β_{j_{0}}} \log \frac{‖ \tilde{f} (r_{m} e^{i θ}) ‖^{d} ‖ a (r_{m} e^{i θ}) ‖}{| Q (\tilde{f}) (r_{m} e^{i θ}) |} \sin (ω_{j_{0}} (θ - α_{j_{0}})) d θ \\ ⩾ \frac{2 ω_{j_{0}} \sin ω_{j_{0}} ε}{π} \int_{α_{j_{0}} + ε}^{β_{j_{0}} - ε} \log \frac{‖ \tilde{f} (r_{m} e^{i θ}) ‖^{d} ‖ a (r_{m} e^{i θ}) ‖}{| Q (\tilde{f}) (r_{m} e^{i θ}) |} d θ \\ ⩾ \frac{2 ε Λ (r_{m}) T_{f} (r_{m}) ω_{j_{0}} \sin ω_{j_{0}} ε}{p π} . \end{aligned}

(6.5)

Note that

\begin{aligned} W_{D} (r_{m}, f, D) = Λ^{2} (r_{m}) T_{f} (r_{m}) . \end{aligned}

Thus, we have

\begin{aligned} 0 < \frac{2 ε ω_{j_{0}} \sin ω_{j_{0}} ε}{p π} ⩽ Λ (r_{m}) \to 0 as m \to \infty, \end{aligned}

which is a contradiction and we finish the proof of Theorem 7.

Now, we prove Theorem 8. First, we need some lemmas as follows: Lemma 3.

Let $f : C \to P^{n} (C)$ be an algebraically nondegenerate transcendental holomorphic curve and $N ⩾ n$ is a integer. Let $D, D_{1}, \dots, D_{q} (q > \frac{(M + 1) (2 N - n + 1)}{n + 1})$ be hypersurfaces in N-subgeneral position in $P^{n} (C)$ . Let $d, d_{1}, \dots, d_{q}$ be degree of $D, D_{1}, \dots, D_{q}$ respectively and $δ = min {d_{1}, \dots, d_{q}}$ . Then we have

\begin{aligned} (q - \frac{(M + 1) (2 N - n + 1)}{n + 1}) B_{α β, f} (r, D) \\ ⩽ \frac{M}{δ} \sum_{j = 1}^{q} C_{α β, f}^{1} (r, D_{j}) + O (\log r + \log T_{f} (r)) \end{aligned}

holds for all r large enough outside a set with finite measure.

Proof.

By using Theorem 4, we have

\begin{aligned} (q + 1 - \frac{(M + 1) (2 N - n + 1)}{n + 1}) S_{α β, f} (r) \\ ⩽ d^{- 1} C_{α β, f}^{M} (r, D) + \sum_{i = 1}^{q} d_{i}^{- 1} C_{α β, f}^{M} (r, D_{i}) + O (\log r + \log T_{f} (r)) \\ ⩽ S_{α β, f} (r) + \frac{M}{δ} \sum_{i = 1}^{q} C_{α β, f}^{1} (r, D_{i}) + O (\log r + \log T_{f} (r)) . \end{aligned}

Then we get

\begin{aligned} (q - \frac{(M + 1) (2 N - n + 1)}{n + 1}) S_{α β, f} (r) \\ ⩽ \frac{M}{δ} \sum_{i = 1}^{q} C_{α β, f}^{1} (r, D_{i}) + O (\log r + \log T_{f} (r)) . \end{aligned}

Hence

\begin{aligned} (q - \frac{(M + 1) (2 N - n + 1)}{n + 1}) B_{α β, f} (r, D) \\ ⩽ \frac{M}{δ} \sum_{i = 1}^{q} C_{α β, f}^{1} (r, D_{i}) + O (\log r + \log T_{f} (r)) . ◼ \end{aligned}

Lemma 4 [19]

Let f be holomorphic curve on $\bar{Ω} (α, β)$ and D be a hypersurfaces defined the zeros of homogeneous polynomial Q such that $Q (\tilde{f}) ⧸ \equiv 0$ . Then we have

\begin{aligned} r^{ω} C_{α β, f} (r, D) ⩽ 4 ω N (r) + 2 ω^{2} r^{ω} \int_{1}^{r} \frac{N (t)}{t^{1 + ω}} d t, \end{aligned}

where

N (t) = N_{α β, f} (r, D)

is the counting function the zeros of

Q (\tilde{f})

\bar{Ω} (α, β) \cap {z : 1 ⩽ | z | ⩽ r}

. The inequality also holds with

C_{α β, f}^{1} (r, D)

and

N_{α β, f}^{1} (r, D)

instead of

C_{α β, f} (r, D)

and

N_{α β, f} (r, D)

Proof of Theorem 8. Proof of Theorem 8.

Using Lemma 3 and Lemma 4, we prove Theorem 8 as follows. Suppose that $ρ = ρ (f) > ω (D)$ . Note that $ρ (f) ⩾ τ (f) = τ$ , then $ρ > σ = max {ω (D), τ}$ . Choose $ε > 0$ such that $ρ > σ + ε$ and (3.4) holds for $σ + ε$ . By Theorem 7, we see that (3.5) holds for some constant $K_{*}$ . Now we estimate $W_{D} (r_{m}, D, f)$ . Combine Lemma 3 and Lemma 4, we get

\begin{aligned} W_{D} (r_{m}, D, f) & ⩽ C_{*} max_{1 ⩽ j ⩽ p} {N_{α_{j} β_{j}, f} (r_{m}) + r_{m}^{ω_{j}} \int_{1}^{r_{m}} \frac{N_{α_{j} β_{j}} (t)}{t^{ω_{j} + 1}} d t} \\ + O (r_{m}^{ω (D)} \log r_{m} T_{f} (r_{m})), \end{aligned}

where

N_{α_{j} β_{j}} (t) = \sum_{k = 1}^{q} N_{α_{j} β_{j}, f} (t, D_{k})

and

C_{*} > 0

is a suitable constant. By arguments as [18], from (3.5), we get

\begin{aligned} W_{D} (r_{m}, D, f) = o (T_{f} (r_{m})) + O (r_{m}^{ω (D)} \log r_{m} T_{f} (r_{m})) \end{aligned}

and we deduce

\begin{aligned} σ + ε ⩽ \underset{m \to \infty}{lim inf} \frac{\log T_{f} (r_{m})}{\log r_{m}} ⩽ ω (D) ⩽ σ . \end{aligned}

It is a contradiction and then

ρ (f) ⩽ ω (D)

7. Proofs of Theorem 9

Proof of Theorem 9. Proof of Theorem 9.

We suppose that $f ⧸ \equiv g$ , then there are two numbers $α, β \in {0, \dots, n}$ , $α \neq β$ such that $f_{α} g_{β} ⧸ \equiv f_{β} g_{α}$ . Assume that $z_{0} \in C$ is a zero of $Q (\tilde{f})$ in X, from the condition $f (z) = g (z)$ when $z \in ⋃_{i = 1}^{q} (f^{- 1} (D_{i}) \cap X)$ , we get $f (z_{0}) = g (z_{0})$ . This implies $z_{0}$ is a zero of $\frac{f_{α}}{f_{β}} - \frac{g_{α}}{g_{β}}$ . Since $D_{1}, \dots, D_{q}$ are in N-subgeneral position, then there are at most N hypersurfaces intersecting at $f (z_{0})$ . Therefore, we have $\sum_{i = 0}^{q} C_{α_{k} β_{k}, f}^{1} (r, D_{i}) ⩽ N C_{α_{k} β_{k}, \frac{f_{α}}{f_{β}} - \frac{g_{α}}{g_{β}}} (r, 0)$ . Apply to Theorem 4 for some $ε_{0} > 0$ small enough, we obtain

\begin{aligned} (q - \frac{(M + 1) (2 N - n + 1)}{n + 1}) S_{α_{k} β_{k}, f} (r) \\ ⩽ \sum_{i = 1}^{q} d_{i}^{- 1} C_{α_{k} β_{k}, f}^{M} (r, D_{i}) + O (\log r T_{f} (r)) \\ ⩽ \frac{M}{δ} \sum_{i = 0}^{q} C_{α_{k} β_{k}, f}^{M} (r, D_{i}) + O (\log r T_{f} (r)) \\ ⩽ \frac{N M}{δ} C_{α_{k} β_{k}, \frac{f_{α}}{f_{β}} - \frac{g_{α}}{g_{β}}} (r, 0) + O (\log r T_{f} (r)) \\ ⩽ \frac{N M}{δ} (S_{α_{k} β_{k}, f} (r) + S_{α_{k} β_{k}, g} (r)) + O (\log r T_{f} (r)) \end{aligned}

(7.1)

for all

k = 1, \dots, p

. Similarly, we also have

\begin{aligned} (q - \frac{(M + 1) (2 N - n + 1)}{n + 1}) S_{α_{k} β_{k}, g} (r) & ⩽ \sum_{i = 1}^{q} d_{i}^{- 1} C_{α_{k} β_{k}, g}^{M} (r, D_{i}) + O (\log r T_{g} (r)) \\ ⩽ \frac{M}{δ} \sum_{i = 0}^{q} C_{α_{k} β_{k}, g}^{M} (r, D_{i}) + O (\log r T_{g} (r)) \\ ⩽ \frac{N M}{δ} C_{α_{k} β_{k}, \frac{f_{α}}{f_{β}} - \frac{g_{α}}{g_{β}}} (r, 0) + O (\log r T_{g} (r)) \\ ⩽ \frac{N M}{δ} (S_{α_{k} β_{k}, f} (r) + S_{α_{k} β_{k}, g} (r)) + O (\log r T_{g} (r)) \end{aligned}

(7.2)

for all

k = 1, \dots, p

, where

δ = min {d_{0}, \dots, d_{q}}

. Since

\log T_{f} (r) = O (\log T_{g} (r))

, combine (7.1) and (7.2), we deduce

\begin{aligned} (q - \frac{(M + 1) (2 N - n + 1)}{n + 1} - \frac{N M}{δ}) (S_{α_{k} β_{k}, f} (r) + S_{α_{k} β_{k}, g} (r)) \\ ⩽ O (\log r T_{f} (r)) . \end{aligned}

(7.3)

Note that

ρ (f) > ω (D)

, we need to treat two cases as follows:

Case I. $ρ (f) > τ$ . Then $ρ (f) > σ = max {ω (D), τ} > τ$ . We can take $ε > 0$ such that

\begin{aligned} \sum_{j = 1}^{p} (α_{j + 1} - β_{j} + 2 ε) + 2 ε < \frac{4}{σ + 2 ε} \sin^{- 1} (δ_{f} (D) / 2)^{1 / 2}, \end{aligned}

where

α_{p + 1} = α_{1} + 2 π

. By Theorem 11 to f gives the existence of the Pólya peak sequence

{r_{n}}

of order

σ + 2 ε

of f outside a set with finite measure. For n large enough, we have

\begin{aligned} meas E_{Λ} (r_{n}, a) > \frac{4}{σ + 2 ε} \sin^{- 1} (δ_{f} (D) / 2)^{1 / 2} - ε . \end{aligned}

For n large enough, set

\begin{aligned} K := meas (E_{Λ} (r_{n}, a) \cap (⋃_{j = 1}^{p} (α_{j} + ε, β_{j} + ε)) . \end{aligned}

Then we get

\begin{aligned} K & ⩾ meas (E_{Λ} (r_{n}, a) - meas ([- π, π) ∖ ⋃_{j = 1}^{p} (α_{j} + ε, β_{j} - ε)) \\ = meas (E_{Λ} (r_{n}, a)) - meas (⋃_{j = 1}^{p} (β_{j} - ε, α_{j + 1} + ε)) \\ = meas (E_{Λ} (r_{n}, a)) - \sum_{j = 1}^{p} (α_{j + 1} - β_{j} + 2 ε) > ε > 0. \end{aligned}

It is easy to see that there exists a

j_{0}

such that for all n large enough, we have

\begin{aligned} meas (E_{Λ} (r_{n}, a) \cap (α_{j_{0}} + ε, β_{j_{0}} - ε)) ⩾ \frac{K}{p} . \end{aligned}

(7.4)

Set

E_{n} = E_{Λ} (r_{n}, a) \cap (α_{j_{0}} + ε, β_{j_{0}} - ε)

and in (6.2), choose

Λ (r) = \frac{1}{\log r}

with

Λ (r) \to 0

r \to \infty

. Then from (7.4), we get

\begin{aligned} \int_{α_{j_{0}} + ε}^{β_{j_{0}} - ε} \frac{‖ \tilde{f} (r e^{i θ}) ‖^{d} ‖ a (r e^{i θ}) ‖}{Q (\tilde{f} (r e^{i θ}))} d θ & ⩾ \int_{E_{n}} \frac{‖ \tilde{f} (r e^{i θ}) ‖^{d} ‖ a (r e^{i θ}) ‖}{Q (\tilde{f} (r e^{i θ}))} d θ \\ ⩾ \frac{K}{p} \frac{T_{f} (r_{n})}{\log r_{n}} . \end{aligned}

(7.5)

By the definition of

B_{α β, f} (r, D)

and (7.5), for

ε > 0

small enough, we see

\begin{aligned} B_{α_{j_{0}} β_{j_{0}}, f} (r_{n}, D) & = \frac{2 ω_{j_{0}}}{π r_{n}^{ω_{j_{0}}}} \int_{α_{j_{0}}}^{β_{j_{0}}} \log \frac{‖ \tilde{f} (r_{n} e^{i θ}) ‖^{d} ‖ a (r_{n} e^{i θ}) ‖}{| Q (\tilde{f}) (r_{n} e^{i θ}) |} \sin (ω_{j_{0}} (θ - α_{j_{0}})) d θ \\ ⩾ \frac{2 ω_{j_{0}} \sin ω_{j_{0}} ε}{π r_{n}^{ω_{j_{0}}}} \int_{α_{j_{0}} + ε}^{β_{j_{0}} - ε} \log \frac{‖ \tilde{f} (r_{n} e^{i θ}) ‖^{d} ‖ a (r_{n} e^{i θ}) ‖}{| Q (\tilde{f}) (r_{n} e^{i θ}) |} d θ \\ ⩾ \frac{2 K T_{f} (r_{n}) ω_{j_{0}} \sin ω_{j_{0}} ε}{p π r_{n}^{ω_{j_{0}}} \log r_{n}} . \end{aligned}

(7.6)

By the assumption

q > \frac{(M + 1) (2 N - n + 1)}{n + 1} + \frac{N M}{δ}

, from (7.3) and (7.6), we obtain

\begin{aligned} T_{f} (r_{n}) ⩽ \frac{p π r_{n}^{ω_{j_{0}}} \log r_{n}}{2 K ω_{j_{0}} \sin ω_{j_{0}} ε} O (\log r_{n} T_{f} (r_{n})) . \end{aligned}

(7.7)

Note that

lim inf_{n \to \infty} \frac{\log \log T_{f} (r_{n})}{\log r_{n}} = 0

since

\log \log T_{f} (r) = o (\log r)

r \to \infty

. By property of Pólya peak sequence

{r_{n}}

and (7.7), we have

\begin{aligned} σ + 2 ε ⩽ \underset{n \to \infty}{lim inf} \frac{\log T_{f} (r_{n})}{\log r_{n}} ⩽ ω_{j_{0}} ⩽ ω (D) < σ, \end{aligned}

which is impossible. Hence

f \equiv g

Case II. $ρ (f) = τ$ . By the argument as Case I with all the $σ + 2 ε$ is replaced by $σ = τ$ , we can derive

\begin{aligned} ρ (f) = τ ⩽ max {ω (D), τ} = σ ⩽ ω (D) < ρ (f) . \end{aligned}

This is impossible. Hence

f \equiv g

Footnotes

Acknowledgements

The research results are supported by the Ministry of Education and Training of Vietnam under the project with the name The Nevanlinna Cartan’s second main theorem and some applications and grant number B2024-TNA-19.

Data sharing statements

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

D.P.

Quang

S.D.

Thai

D.D.

, The second main theorem for meromorphic mappings into a complex projective space, Acta Math VietNam. 38 (2013), 187–205. doi:10.1007/s40306-013-0011-6.

Ciechanowicz

Marchenko

I.I.

, On deviations and strong asymptotic functions of meromorphic functions of finite lower order, J. Math. Anal. Appl. 382 (2011), 383–398. doi:10.1016/j.jmaa.2011.04.050.

Fedorov

M.A.

Grishin

A.F.

, Some questions of Nevanlinna theory for the complex half plane, Math. Phys. Anal. Geom. 1 (1998), 223–271.

Goldberg

A.A.

Ostroskii

, Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs, Vol. 236, 2008.

Hayman

Yang

, Growth and values of functiona regular in an angle, Proc. Lond. Math. Soc. 44(3) (1982), 193–214. doi:10.1112/plms/s3-44.2.193.

Kowalshki

Marchenko

, On deviations and spreads of entire curves, Ann. Polon. Math. 123 (2019), 345–368.

Krutin

V.I.

, On the magnitudes of the positive deviations and the defects of entire curves of finite lower order, Math. USSR Izvestija. 13(2) (1979), 307–334. doi:10.1070/IM1979v013n02ABEH002045.

Lin

Mori

Tohge

, Uniqueness theorems in an angular domain, Tohoku Math. J. 58 (2006), 509–527.

Marchenko

I.I.

, On the magnitudes of deviations and spreads of meromorphic functions of finite lower order, Sbornik: Mathematics. 186(3) (1995), 391–408. doi:10.1070/SM1995v186n03ABEH000023.

10.

Marchenko

I.I.

, Baernstein’s star-function, maximum modulus points and a problem of Erdös, Annales Fennici Mathematici. 47 (2022), 181–202. doi:10.54330/afm.112881.

11.

Niino

, General defect relations of holomorphic curves, Trans. Amer. Math. Soc. 289(1) (1985), 99–113. doi:10.1090/S0002-9947-1985-0779054-5.

12.

Petrenko

V.P.

, The growth of integral curves of finite order, Math. USSR Sbornik. 26(4) (1975), 427–448. doi:10.1070/SM1975v026n04ABEH002489.

13.

Quang

S.D.

, Value distribution theory on angular domains for homolorphic mappings and arbitrary families of moving hypersurfaces, Bull. Math. Science. 13(2) (2023), 2250008, 27 pages. doi:10.1142/S1664360722500084.

14.

Tsuji

, Potential Theory in Modern Function Theory, Maruzen Co. LTD, Tokyo, 1959.

15.

Wang

, The evolution of immersed locally convex plane curves driven by anisotropic curvature flow, Adv. Nonlinear Anal. 12(1) (2023), 117–131. doi:10.1515/anona-2022-0245.

16.

, Deviations and spreads of holomorphic curves of finite lower order, Houston. J. Math. 45 (2019), 395–411.

17.

Xuan

, On the growth of holomorphic curve, Kodai Math J. 37 (2014), 285–302.

18.

Zheng

, Value Distribution of Meromorphic Functions, Springer-Verlag, Berlin Heidelberg, 2011.

19.

Zheng

, Value distribution of holomorphic curves on an angular domain, Michigan Math. J. 64 (2015), 849–879. doi:10.1307/mmj/1447878034.