On Lappan's five-valued theorem for $\varphi$-normal functions in several variables
UDC 517.5 Let $\mathbb{U}^m\subset\mathbb{C}^m$ be а unit ball centered at the origin and let $\mathbb{P}^n$ be an $n$-dimensional  complex projective space with the metric $E_{\mathbb{P}^n}.$ Also, let $\varphi\colon [0,1)\rightarrow(0,\infty)$ be a smoo...
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| author | Datt, G. Datt, G. |
| author_facet | Datt, G. Datt, G. |
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UDC 517.5
Let $\mathbb{U}^m\subset\mathbb{C}^m$ be а unit ball centered at the origin and let $\mathbb{P}^n$ be an $n$-dimensional  complex projective space with the metric $E_{\mathbb{P}^n}.$ Also, let $\varphi\colon [0,1)\rightarrow(0,\infty)$ be a smoothly increasing function.  A holomorphic mapping $f\colon\mathbb{U}^m\rightarrow \mathbb{P}^n$  is called {\it $\varphi$-normal} if $({\varphi(\|z\|))}^{-1}{(E_{\mathbb{P}^n}(f(z), df(z))(\xi))}$ is bounded above for $z\in\mathbb{U}^m$ and $\xi\in\mathbb{C}^m$ such that $\|\xi\|=1,$ where $df(z)$ is the map from $T_z\big(\mathbb{U}^m\big)$ to $T_{f(z)}\big(\mathbb{P}^n\big)$ induced by $f.$ For $n=1,$ $f$ is called a $\varphi$-normal function. We present an extension of Lappan's five-valued theorem to  the class of $\varphi$-normal functions. |
| doi_str_mv | 10.37863/umzh.v74i9.6237 |
| first_indexed | 2026-03-24T03:26:39Z |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
DOI: 10.37863/umzh.v74i9.6237
UDC 517.5
G. Datt1 (Babasaheb Bhimrao Ambedkar University, Lucknow, India)
ON LAPPAN’S FIVE-VALUED THEOREM
FOR \bfitvarphi -NORMAL FUNCTIONS IN SEVERAL VARIABLES 2
ПРО П’ЯТИЗНАЧНУ ТЕОРЕМУ ЛАППАНА
ДЛЯ \bfitvarphi -НОРМАЛЬНИХ ФУНКЦIЙ КIЛЬКОХ ЗМIННИХ
Let \BbbU m \subset \BbbC m be а unit ball centered at the origin and let \BbbP n be an n-dimensional complex projective space with the
metric E\BbbP n . Also, let \varphi : [0, 1) \rightarrow (0,\infty ) be a smoothly increasing function. A holomorphic mapping f : \BbbU m \rightarrow \BbbP n is
called \varphi -normal if (\varphi (\| z\| )) - 1(E\BbbP n(f(z), df(z))(\xi )) is bounded above for z \in \BbbU m and \xi \in \BbbC m such that \| \xi \| = 1,
where df(z) is the map from Tz
\bigl(
\BbbU m
\bigr)
to Tf(z)
\bigl(
\BbbP n
\bigr)
induced by f. For n = 1, f is called a \varphi -normal function. We
present an extension of Lappan’s five-valued theorem to the class of \varphi -normal functions.
Нехай \BbbU m \subset \BbbC m — одинична куля з центром у початку координат, \BbbP n — n-вимiрний комплексний проектив-
ний простiр з метрикою E\BbbP n , а \varphi : [0, 1) \rightarrow (0,\infty ) — плавно зростаюча функцiя. Голоморфне вiдображення f :
\BbbU m \rightarrow \BbbP n називається \varphi -нормальним, якщо (\varphi (\| z\| )) - 1(E\BbbP n(f(z), df(z))(\xi )) обмежено зверху для z \in \BbbU m i
\xi \in \BbbC m так, що \| \xi \| = 1, де df(z) — вiдображення з Tz
\bigl(
\BbbU m
\bigr)
у Tf(z)
\bigl(
\BbbP n
\bigr)
, iндуковане f. При n = 1 f називається
\varphi -нормальною функцiєю. Встановлено розширення п’ятизначної теореми Лаппана на клас \varphi -нормальних функцiй.
1. Introduction and main results. A meromorphic function f on a planar domain D \subset \BbbC is said
to be normal in D if the family \{ f \circ \tau : \tau \in \scrT \} is normal in D, where \scrT is the set of all conformal
self maps of D. A well-known result of Lehto and Virtanen [6] gives the following characterization
of normal functions: A meromorphic function f on the unit disc \BbbD \subset \BbbC is normal if and only if
\mathrm{s}\mathrm{u}\mathrm{p}z\in \BbbD
\bigl(
1 - | z| 2
\bigr)
f\#(z) < \infty , where f\# :=
| f \prime (z)|
1 + | f(z)| 2
is the spherical derivative of f. Answering
a question posed by Pommerenke [7] (Problem 3.2), Lappan gave the following known five-valued
theorem.
Result 1.1 ([5], Theorem 1). Let S be any set consisting of five distinct values in \BbbC \cup \{ \infty \} . If
f is a meromorphic function on the unit disc \BbbD such that
\mathrm{s}\mathrm{u}\mathrm{p}
\Bigl\{ \bigl(
1 - | z| 2
\bigr)
f\#(z) : z \in f - 1(S)
\Bigr\}
< \infty ,
then f is a normal function.
Lappan commented on the sharpness of the number five in the above result, he showed that five
cannot be replaced by three and that there are a few cases where five cannot be replaced by four
(see [5], Theorems 3 and 4).
1 e-mail: ggopal.datt@gmail.com.
2 The work on the initial stage of the research reported in this paper was done when the author was a postdoctoral
fellow at the Indian Institute of Science. The support by the UGC of that phase of the work, through a Dr. D. S. Kothari
Postdoctoral Fellowship (Grant no. No.F.4-2/2006 (BSR)/MA/19-20/0022), is gratefully acknowledged.
c\bigcirc G. DATT, 2022
1284 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
ON LAPPAN’S FIVE – VALUED THEOREM FOR \varphi -NORMAL FUNCTIONS IN SEVERAL VARIABLES 1285
The main aim of this work is to study the five-valued theorem of Lappan, in higher-dimensional
setting, for \varphi -normal functions, we shall formulate this work in Theorem 1.1. The notion of \varphi -
normal functions on the unit disc \BbbD \subset \BbbC was introduced by Aulaskari and Rättyä [1]. Here, \varphi
represents a smoothly increasing function, by definition: An increasing function \varphi : [0, 1) \rightarrow (0,\infty )
is called smoothly increasing if
1 \leq \varphi (r)(1 - r) \rightarrow \infty as r \rightarrow 1 - ,
and
\scrR a(z) :=
\varphi (| a+ z/\varphi (| a| )| )
\varphi (| a| )
\rightarrow 1 as | a| \rightarrow 1 -
uniformly on compact subsets of \BbbC . For given such a function \varphi , a meromorphic function f on \BbbD
is called \varphi -normal if
\| f\| \scrN \varphi := \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
f\#(z)
\varphi (| z| )
< \infty .
The class of all \varphi -normal functions is denoted by \scrN \varphi .
Aulaskari and Rättyä established the following five-valued theorem, analogous to Lappan’s five-
valued theorem, for \varphi -normal function.
Result 1.2 ([1], Theorem 9). Let f be a meromorphic function on the unit disc \BbbD \subset \BbbC and
let \varphi : [0, 1) \rightarrow (0,\infty ) be smoothly increasing. Then f \in \scrN \varphi if and only if there exists a set
S \subset \BbbC \cup \{ \infty \} consists of five distinct values such that
\mathrm{s}\mathrm{u}\mathrm{p}
\Bigl\{
f\#(z)/\varphi (| z| ) : z \in f - 1(S)
\Bigr\}
< \infty .
T. V. Tan and N. V. Thin [8] established the following result wherein the set E, appeared in the
statement of Result 1.2, consists of only four distinct values and still yields the same conclusion as
in Result 1.2.
Result 1.3 ([8], Theorem 4). Let f be a meromorphic function on the unit disc \BbbD and let \varphi :
[0, 1) \rightarrow (0,\infty ) be smoothly increasing. Assume that there is a subset S \subset \BbbC \cup \{ \infty \} containing four
distinct values such that
\mathrm{s}\mathrm{u}\mathrm{p}
z\in f - 1(S)
f\#(z)
\varphi (| z| )
< \infty and \mathrm{s}\mathrm{u}\mathrm{p}
z\in f - 1(S\setminus \{ \infty \} )
(f \prime )\#(z) < \infty .
Then f is \varphi -normal.
Hu and Thin, recently, extended the concept of \varphi -normal function to higher dimensional settings
in [4]. Let \BbbU m :=
\bigl\{
z \in \BbbC m : \| z\| < 1
\bigr\}
be the unit ball centered at the origin, and \BbbP n be the
n-dimensional complex projective space. Let \mathrm{H}\mathrm{o}\mathrm{l}
\bigl(
\BbbU m,\BbbP n
\bigr)
denote the set of holomorphic mappings
from \BbbU m to \BbbP n. When n = 1, \mathrm{H}\mathrm{o}\mathrm{l}
\bigl(
\BbbU m,\BbbP 1
\bigr)
is just the set of meromorphic functions on \BbbU m.
According to Hu – Thin: Let \varphi : [0, 1) \rightarrow (0,\infty ) be an increasing function that satisfies the following
properties:
\varphi (r)(1 - r) \geq 1 for all r \in [0, 1) (1.1)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1286 G. DATT
and
\scrR a(z) :=
\varphi (\| a+ z/\varphi (\| a\| )\| )
\varphi (\| a\| )
\rightarrow 1 as \| a\| \rightarrow 1 - (1.2)
uniformly on compact subsets of \BbbC m. Here \| \cdot \| denotes the Euclidean norm in \BbbC m. For given such
a function \varphi , an element f \in \mathrm{H}\mathrm{o}\mathrm{l}
\bigl(
\BbbU m,\BbbP n
\bigr)
is called \varphi -normal if
\| f\| \scrN \varphi := \mathrm{s}\mathrm{u}\mathrm{p}
\| \xi \| =1, z\in \BbbU m
E\BbbP n(f(z), df(z)(\xi ))
\varphi (\| z\| )
< \infty , (1.3)
where E\BbbP n(f(z), df(z)(\xi )) is the norm, associated with the Fubini – Study metric ds2\BbbP n on \BbbP n, at
f(z) in the direction of the vector df(z)(\xi ) \in Tf(z)
\bigl(
\BbbP n
\bigr)
, where Tf(z)
\bigl(
\BbbP n
\bigr)
is the holomorphic
tangent space to \BbbP n at f(z). Note here that df(z) is the mapping from Tz
\bigl(
\BbbU m
\bigr)
to Tf(z)
\bigl(
\BbbP n
\bigr)
induced by f. We shall defer the explanation of E\BbbP n(., .) to the Section 2.
Hu and Thin found the following analogue of Lappan’s five-valued theorem.
Result 1.4 ([4], Theorem 2.5). Let f \in \mathrm{H}\mathrm{o}\mathrm{l}
\bigl(
\BbbU m,\BbbP 1
\bigr)
and let \varphi : [0, 1) \rightarrow \BbbR + be an increasing
function satisfying (1.1) and (1.2). Then f \in \scrN \varphi if and only if there exists a set S of five distinct
values in \BbbC \cup \{ \infty \} such that
\mathrm{s}\mathrm{u}\mathrm{p}
\| \xi \| =1, z\in f - 1(S)
E\BbbP 1(f(z), df(z)(\xi ))
\varphi (\| z\| )
< \infty .
Now the following natural question arises:
Whether the cardinality of S in the statement of Result 1.4 can be reduced and yet yield the same
conclusion?
Motivated by the Result 1.3 we establish the following theorem wherein we reduce the cardinality
of S but we impose extra conditions, analogous to the conditions in the statement of Result 1.3.
Theorem 1.1. Let f \in \mathrm{H}\mathrm{o}\mathrm{l}
\bigl(
\BbbU m,\BbbP 1
\bigr)
and let \varphi : [0, 1) \rightarrow \BbbR + be an increasing function satis-
fying (1.1) and (1.2). Suppose that there exists a set S with four distinct points in \BbbC \cup \{ \infty \} such
that
\mathrm{s}\mathrm{u}\mathrm{p}
\| \xi \| =1, z\in f - 1(S)
E\BbbP 1(f(z), df(z)(\xi ))
\varphi (\| z\| )
< \infty (1.4)
and
\mathrm{s}\mathrm{u}\mathrm{p}
\| \xi \| =1, z\in f - 1(S\setminus \{ \infty \} )
E\BbbP 1
\Bigl( \sum m
l=1
fzl(z), d
\sum m
l=1
fzl(z)(\xi )
\Bigr)
(\varphi (\| z\| ))2
< \infty , (1.5)
where fzl(z) =
\partial f
\partial zl
, l = 1, . . . ,m. Then f \in \scrN \varphi .
We end this section with a brief explanation of some common notations.
1.1. Some notations. We fix the following notation, which we shall use without any further
clarification.
As in the discussion above, \| \cdot \| will denote the Euclidean norm. Expressions like “unit vector”
will be with reference to this norm.
We shall denote the standard Hermitian product in \BbbC n by \langle \cdot , \cdot \rangle .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
ON LAPPAN’S FIVE – VALUED THEOREM FOR \varphi -NORMAL FUNCTIONS IN SEVERAL VARIABLES 1287
2. Basic notions. This section is devoted to elaborating upon concepts and terms that mentioned
in Section 1, and to introducing certain notions that we shall need in our proofs.
The n-dimensional complex projective space \BbbP n is the space
\bigl(
\BbbC n+1 \setminus \{ 0\}
\bigr)
/ \sim , where (a0, . . .
. . . , an) \sim (b0, . . . , bn) if and only if (a0, . . . , an) = \lambda (b0, . . . , bn) for some \lambda \in \BbbC \setminus \{ 0\} . In other
words, \BbbP n is the set of all complex lines, passing through the origin in \BbbC n+1. A point in \BbbP n is an
equivalence class of some (a0, . . . , an) \in \BbbC m \setminus \{ 0\} which we denote by [a0 : \cdot \cdot \cdot : an], and it is
called the homogeneous coordinate of the point. When n = 1, the projective space \BbbP 1 is identified
with the extended complex plane \BbbC \cup \{ \infty \} .
Let us also revisit to the notion of holomorphic mappings alluded to in Section 1. Let D \subset \BbbC m
be a domain, and f : D \rightarrow \BbbP n be a holomorphic mapping. Fixing a system of homogeneous
coordinates on \BbbP n, for each a \in D, we have a holomorphic map \widetilde f(z) := (f0(z), . . . , fn(z)) on
some neighborhood U of a such that \{ z \in U | f0(z) = . . . = fn(z) = 0\} = \varnothing and f(z) =
= [f0(z) : \cdot \cdot \cdot : fn(z)] for each z \in U. We shall call any such holomorphic map \widetilde f : U \rightarrow \BbbC n+1 a
reduced representation of f on U. A holomorphic mapping f : D \rightarrow \BbbP 1 is called a meromorphic
function on D.
We now recall the meaning of the object E\BbbP n that alluded to in (1.3). First, we see it in the
general case: Let M be a complete Hermitian complex manifold of dimension n with a Hermitian
metric
ds2M =
n\sum
i, k=1
hi k(p)dzid\=zk,
where z = (z1, . . . , zn) are local coordinates in a neighborhood of a point p \in M. Then ds2M
reduces to a norm on the holomorphic tangent space Tp(M) of M at p in the direction of the vector
\xi \in Tp(M)
EM (p, \xi ) =
\bigl(
ds2M (\xi , \=\xi )
\bigr) 1/2
, \xi \in Tp(M),
which further defines the distance between points p, q \in M by
dM (p, q) := \mathrm{i}\mathrm{n}\mathrm{f}
\gamma
1\int
0
EM
\bigl(
\gamma (t), \gamma \prime (t)
\bigr)
dt,
where the infimum is taken over all parametric curves \gamma : [0, 1] \rightarrow M satisfying \gamma (0) = p and \gamma (1) =
= q. When M = \BbbP n, the Hermitian metric has a nice description, let us recall briefly: Fixing a system
of homogeneous coordinates \xi = [\xi 0 : . . . : \xi n], the Hermitian metric is given in homogeneous
coordinates by
ds2\BbbP n =
\langle d\xi , d\xi \rangle \langle \xi , \xi \rangle - | \langle \xi , d\xi \rangle | 2
\langle \xi , \xi \rangle 2
.
This is called the Fubini – Study metric in homogeneous coordinates on \BbbP n. In the local coordinates
on the open sets Ul := \{ [\xi 0 : . . . : \xi n] \in \BbbP n : \xi l \not = 0\} , l = 0, . . . , n, of the standard covering of \BbbP n,
we can rewrite the expression. Suppose that the coordinates are \zeta 1 = \xi 1/\xi 0, . . . , \zeta n = \xi n/\xi 0, with
\zeta = (\zeta 1, . . . , \zeta n) in the open set U0
\sim = \BbbC n. Then
ds2\BbbP n =
\bigl(
1 + \| \zeta \| 2
\bigr)
\| d\zeta \| 2 - | \langle \zeta , d\zeta \rangle | 2\bigl(
1 + \| \zeta \| 2
\bigr) 2 .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1288 G. DATT
When n = 1, this is the well-known spherical metric on \BbbP 1 \sim = \BbbC \cup \{ \infty \} , and in this case for a
holomorphic map f : \BbbC \rightarrow \BbbP n, E\BbbP 1(f, df) is the spherical derivative f\# of f, which is defined
as f\# :=
| f \prime |
1 + | f | 2
. We could also derive the spherical derivative from the differential form of
type (1, 1) associated with the Fubini – Study metric. Recently, Dovbush [2] proved a version of
Zalcman’s lemma in several variables using the spherical derivative derived from the associated
differential form.
3. Essential lemmas. One of the well-known result in the theory of normality is the rescaling
lemma of Zalcman. We shall use the following Zalcman type rescaling result in order to prove
Theorem 1.1. Loosely speaking, it says that a non-\varphi -normal map can be rescaled at a small scale to
obtain a non-constant holomorphic map g : \BbbC \rightarrow \BbbP n in the limit.
Lemma 3.1 ([4], Theorem 2.4). A holomorphic mapping f \in \mathrm{H}\mathrm{o}\mathrm{l}
\bigl(
\BbbU m,\BbbP n
\bigr)
is not \varphi -normal if
and only if there exist
(a) a compact subset K0 \subset \BbbU m;
(b) points zj \in \BbbU m such that \| zj\| \rightarrow 1 - ;
(c) points z\ast j \in K0 such that wj := zj + z\ast j /\varphi (\| zj\| ) \in \BbbU m for j sufficiently large;
(d) positive numbers \rho j with \rho j \rightarrow 0+;
(e) Euclidean unit vectors \xi j \in \BbbC m with \| \xi j\| = 1
such that the sequence gj(\zeta ) := f(wj + (\rho j/\varphi (\| zj\| ))\xi j\zeta ), where \zeta \in \BbbC satisfies
(wj + (\rho j/\varphi (\| zj\| ))\xi j\zeta ) \in \BbbU m,
converges uniformly on compact subsets of \BbbC to a non-constant holomorphic mapping g : \BbbC \rightarrow \BbbP n.
We shall also use the first and the second fundamental theorem of Nevanlinna theory in the proof
of Theorem 1.1. We refer the monograph by Hayman [3] for detailed study of Nevanlinna theory.
First fundamental theorem. Let f(z) be a non-constant meromorphic in the complex plane \BbbC .
Then
N(r, 1/f) \leq T (r, f) +O(1),
where T (r, f) and N(r, f) are the characteristic function and the counting function, respectively, of
Nevanlinna theory.
Second fundamental theorem. Let f(z) be a non-constant meromorphic in the complex plane
\BbbC . If ak \in \BbbC \cup \{ \infty \} , k = 1, . . . , q, q \geq 3, are distinct complex numbers, then
(q - 2)T (r, f) \leq
q\sum
k=1
N
\biggl(
r,
1
f - ak
\biggr)
+ o(T (r, f)),
where T (r, f) and N(r, f) are the characteristic function and the counting function (ignoring mul-
tiplicities), respectively, of Nevanlinna theory.
4. Proof of Theorem 1.1. Suppose, on the contrary, that, under conditions (1.4) and (1.5),
f \not \in \scrN \varphi . Then by Lemma 3.1, there exist
(a) a compact subset K0 \subset \BbbU m ;
(b) points zj \in \BbbU m such that \| zj\| \rightarrow 1 - ;
(c) points z\ast j \in K0 such that wj := zj + z\ast j /\varphi (\| zj\| ) \in \BbbU m for j >> 1;
(d) positive numbers \rho j with \rho j \rightarrow 0+ ;
(e) Euclidean vectors \xi j \in \BbbC m with \| \xi j\| = 1
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
ON LAPPAN’S FIVE – VALUED THEOREM FOR \varphi -NORMAL FUNCTIONS IN SEVERAL VARIABLES 1289
such that the sequence gj(\zeta ) := f(wj + (\rho j/\varphi (\| zj\| ))\xi j\zeta ), where \zeta \in \BbbC satisfies
(wj + (\rho j/\varphi (\| zj\| ))\xi j\zeta ) \in \BbbU m,
converges uniformly on compact subsets of \BbbC to a non-constant holomorphic mapping g : \BbbC \rightarrow \BbbP n.
From the proof of Result 1.4 (see [4], Proof of Theorem 2.5) we have:
( \star ) For all a \in S each zero of g(\zeta ) - a has multiplicity at least 2.
We now aim for the following claim: For a \in S \setminus \{ \infty \} , each zero of g(\zeta ) - a has multiplicity
at least 3.
Suppose that a0 \in S \setminus \{ \infty \} and \zeta 0 \in \BbbC such that g(\zeta 0) - a0 = 0. By Hurwitz’s theorem,
there exists a sequence \zeta 0 j \rightarrow \zeta 0 such that gj(\zeta 0 j) = f(wj + (\rho j/\varphi (\| zj\| ))\xi j\zeta 0 j) = a0 for all j
sufficiently large. We write w0
j := wj + (\rho j/\varphi (\| zj\| ))\xi j\zeta 0 j . From ( \star ), we have g\prime (\zeta 0) = 0, and
condition (1.5) shows that there exists a constant M > 0 such that, for j \in \BbbN ,
1\Bigl(
\varphi
\bigl(
\| w0
j\|
\bigr) \Bigr) 2 E\BbbP 1
\Biggl(
m\sum
l=1
fzl
\bigl(
w0
j
\bigr)
, d
m\sum
l=1
fzl
\bigl(
w0
j
\bigr)
(\xi j)
\Biggr)
\leq M. (4.1)
Set \xi j := (\xi 1 j , . . . , \xi mj), then, for j \in \BbbN ,
g\prime j(\zeta ) =
\rho j
\varphi (\| zj\| )
m\sum
l=1
\xi l jfzl(wj + (\rho j/\varphi (\| zj\| ))\xi j\zeta ).
Note that
E\BbbP 1
\bigl(
g\prime j(\zeta 0 j), dg
\prime
j(\zeta 0 j)(1)
\bigr)
=
=
\rho 2j
(\varphi (\| zj\| ))2
m\sum
l=1
| \xi l j | E\BbbP 1
\bigl(
fzl
\bigl(
w0
j
\bigr)
, dfzl
\bigl(
w0
j
\bigr)
(\xi j)
\bigr)
\leq
\leq
\rho 2j
(\varphi (\| zj\| ))2
E\BbbP 1
\Biggl(
m\sum
l=1
fzl
\bigl(
w0
j
\bigr)
, d
m\sum
l=1
fzl
\bigl(
w0
j
\bigr)
(\xi j)
\Biggr)
\leq
\leq \rho 2j
(\varphi (\| w0
j\| ))2
(\varphi (\| zj\| ))2
M. (4.2)
The last inequality follows from (4.1). Hence by (1.2) and (4.2), we get
E\BbbP 1
\bigl(
g\prime (\zeta 0), dg
\prime (\zeta 0)(1)
\bigr)
= \mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
E\BbbP 1
\bigl(
g\prime j(\zeta 0 j), dg
\prime
j(\zeta 0 j)(1)
\bigr)
= 0.
Therefore, g\prime \prime (\zeta 0) = 0. Hence, for any a \in S \setminus \{ \infty \} , each zero of g(\zeta ) - a has multiplicity at least
3. We write S = \{ a1, a2, a3, a4\} , where a1, a2, a3 are finite and a4 is either finite or infinite. By
the first and second fundamental theorem of Nevanlinna theory, we have
2T (r, g) \leq
4\sum
k=1
N(r, 1/g - ak) + o(T (r, g)) \leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1290 G. DATT
\leq 1
3
3\sum
k=1
N(r, 1/g - ak) +
1
2
N(r, 1/g - a4) + o(T (r, g)) \leq
\leq 3
2
T (r, g) + o(T (r, g)),
for all r \in [1,\infty ) excluding a set of finite Lebesgue measure. This shows that g is constant, which
is a contradiction.
References
1. R. Aulaskari, J. Rättyä, Properties of meromorphic \varphi -normal function, Michigan Math. J., 60, 93 – 111 (2011).
2. P. V. Dovbush, Zalcman’s lemma in \BbbC n, Complex Var. and Elliptic Equat., 65, № 5, 796 – 800 (2020).
3. W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford (1964).
4. P. C. Hu, N. V. Thin, Generalizations of Montel’s normal criterion and Lappan’s five-valued theorem to holomorphic
curves, Complex Var. and Elliptic Equat., 65, 525 – 543 (2020).
5. P. Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helv., 49, 492 – 495 (1974).
6. O. Lehto, K. L. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math., 97, 47 – 65 (1957).
7. Ch. Pommerenke, Problems in complex function theory, Bull. London Math. Soc., 4, 354 – 366 (1972).
8. T. V. Tan, N. V. Thin, On Lappan’s five-point theorem, Comput. Methods and Funct. Theory, 17, 47 – 63 (2017).
Received 21.07.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
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| id | umjimathkievua-article-6237 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:39Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/26/e987350c25214fde70f08c11bb2d9626.pdf |
| spelling | umjimathkievua-article-62372022-12-15T17:03:17Z On Lappan's five-valued theorem for $\varphi$-normal functions in several variables On Lappan's five-valued theorem for $\varphi$-normal functions in several variables Datt, G. Datt, G. Complex projective space, holomorphic mapping, normal functions, $\varphi$-normal function 30D45, 32A18 UDC 517.5 Let $\mathbb{U}^m\subset\mathbb{C}^m$ be а unit ball centered at the origin and let $\mathbb{P}^n$ be an&nbsp;$n$-dimensional&nbsp; complex projective space with the metric $E_{\mathbb{P}^n}.$&nbsp;Also, let $\varphi\colon [0,1)\rightarrow(0,\infty)$ be a smoothly increasing function.&nbsp;&nbsp;A holomorphic mapping $f\colon\mathbb{U}^m\rightarrow \mathbb{P}^n$&nbsp; is called {\it $\varphi$-normal} if $({\varphi(\|z\|))}^{-1}{(E_{\mathbb{P}^n}(f(z), df(z))(\xi))}$ is bounded above for $z\in\mathbb{U}^m$ and $\xi\in\mathbb{C}^m$ such that $\|\xi\|=1,$ where $df(z)$ is the map from $T_z\big(\mathbb{U}^m\big)$ to $T_{f(z)}\big(\mathbb{P}^n\big)$ induced by $f.$&nbsp;For $n=1,$ $f$ is called a $\varphi$-normal function.&nbsp;We present an extension of Lappan's five-valued theorem to&nbsp;&nbsp;the class of $\varphi$-normal functions. УДК 517.5 Про п'ятизначну теорему Лаппана для $\varphi$-нормальних функцій кількох змінних Нехай $\mathbb{U}^m\subset\mathbb{C}^m$ — одинична куля з центром у початку координат, $\mathbb{P}^n$ — $n$-вимірний комплексний проективний простір з метрикою $E_{\mathbb{P}^n},$&nbsp;а $\varphi\colon [0,1)\rightarrow(0,\infty)$ — плавно зростаюча функція.&nbsp;&nbsp;Голоморфне відображення $f\!:\mathbb{U}^m\rightarrow\mathbb{P}^n$ називається {\it $\varphi$-нормальним}, якщо $({\varphi(\|z\|))}^{-1}{(E_{\mathbb{P}^n}(f(z),df(z))(\xi))}$ обмежено зверху для&nbsp;$z\in\mathbb{U}^m$ і $\xi\in\mathbb{C}^m$ так, що&nbsp;$\|\xi\|=1,$ де $df(z)$ — відображення з&nbsp;$T_z\big(\mathbb{U}^m\big)$ у&nbsp;$T_{f(z)}\big(\mathbb{P}^n\big)$, індуковане $f.$&nbsp;При $n=1$ $f$ називається $\varphi$-нормальною функцією.&nbsp;Встановлено розширення п'ятизначної теореми Лаппана на клас $\varphi$-нормальних функцій. Institute of Mathematics, NAS of Ukraine 2022-11-08 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6237 10.37863/umzh.v74i9.6237 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 9 (2022); 1284 - 1290 Український математичний журнал; Том 74 № 9 (2022); 1284 - 1290 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6237/9303 Copyright (c) 2022 Gopal Datt |
| spellingShingle | Datt, G. Datt, G. On Lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| title | On Lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| title_alt | On Lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| title_full | On Lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| title_fullStr | On Lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| title_full_unstemmed | On Lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| title_short | On Lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| title_sort | on lappan's five-valued theorem for $\varphi$-normal functions in several variables |
| topic_facet | Complex projective space holomorphic mapping normal functions $\varphi$-normal function 30D45 32A18 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6237 |
| work_keys_str_mv | AT dattg onlappan039sfivevaluedtheoremforvarphinormalfunctionsinseveralvariables AT dattg onlappan039sfivevaluedtheoremforvarphinormalfunctionsinseveralvariables |