Analytic in an unit ball functions of bounded L-index in joint variables
A concept of boundedness of the L-index in joint variables (see in Bandura A. I., Bordulyak M. T., Skaskiv O. B. Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), 12–26) is generalized for analytic in a ball function. It is proved criteria of boundedness of th...
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1693102020-06-11T01:26:13Z Analytic in an unit ball functions of bounded L-index in joint variables Bandura, A.I. Skaskiv, O.B. A concept of boundedness of the L-index in joint variables (see in Bandura A. I., Bordulyak M. T., Skaskiv O. B. Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), 12–26) is generalized for analytic in a ball function. It is proved criteria of boundedness of the L-index in joint variables which describe local behavior of partial derivatives on a skeleton of a polydisc. 2017 Article Analytic in an unit ball functions of bounded L-index in joint variables / A.I. Bandura, O.B. Skaskiv // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 1-15. — Бібліогр.: 32 назв. — англ. 1810-3200 2010 MSC. 32A10, 32A22, 35G35, 32A40, 32A05, 32H50 http://dspace.nbuv.gov.ua/handle/123456789/169310 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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A concept of boundedness of the L-index in joint variables (see in Bandura A. I., Bordulyak M. T., Skaskiv O. B. Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), 12–26) is generalized for analytic in a ball function. It is proved criteria of boundedness of the L-index in joint variables which describe local behavior of partial derivatives on a skeleton of a polydisc. |
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Bandura, A.I. Skaskiv, O.B. |
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Bandura, A.I. Skaskiv, O.B. Analytic in an unit ball functions of bounded L-index in joint variables Український математичний вісник |
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Bandura, A.I. Skaskiv, O.B. |
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Analytic in an unit ball functions of bounded L-index in joint variables |
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Analytic in an unit ball functions of bounded L-index in joint variables |
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Analytic in an unit ball functions of bounded L-index in joint variables |
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Analytic in an unit ball functions of bounded L-index in joint variables |
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Analytic in an unit ball functions of bounded L-index in joint variables |
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analytic in an unit ball functions of bounded l-index in joint variables |
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Інститут прикладної математики і механіки НАН України |
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Analytic in an unit ball functions of bounded L-index in joint variables / A.I. Bandura, O.B. Skaskiv // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 1-15. — Бібліогр.: 32 назв. — англ. |
| series |
Український математичний вісник |
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AT banduraai analyticinanunitballfunctionsofboundedlindexinjointvariables AT skaskivob analyticinanunitballfunctionsofboundedlindexinjointvariables |
| first_indexed |
2025-07-15T04:03:04Z |
| last_indexed |
2025-07-15T04:03:04Z |
| _version_ |
1837684159815876608 |
| fulltext |
Український математичний вiсник
Том 14 (2017), № 1, 1 – 15
Analytic in an unit ball functions
of bounded L-index in joint variables
Andriy Bandura, Oleh Skaskiv
(Presented by V. Ya. Gutlyanskii)
Abstract. A concept of boundedness of the L-index in joint variables
(see in Bandura A. I., Bordulyak M. T., Skaskiv O. B. Sufficient condi-
tions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016),
12–26. dx.doi.org/10.15330/ms.45.1.12-26) is generalized for analytic in
a ball function. It is proved criteria of boundedness of the L-index in
joint variables which describe local behavior of partial derivatives on a
skeleton of a polydisc.
2010 MSC. 32A05, 32A10, 32A30, 32A40, 30H99.
Key words and phrases. Analytic function, ball, bounded L-index in
joint variables, maximum modulus, partial derivative.
1. Introduction
A concept of entire function of bounded index appeared in a paper
of B. Lepson [23]. An entire function f is said to be of bounded index if
there exists an integer N > 0 that
(∀z ∈ C)(∀n ∈ {0, 1, 2, . . .}) : |f (n)(z)|
n!
≤ max
{ |f (j)(z)|
j!
: 0 ≤ j ≤ N
}
.
(1.1)
The least such integer N is called the index of f.
Note that the functions from this class have interesting properties.
The concept is convenient to study the properties of entire solutions of
differential equations. In particular, if an entire solution has bounded
index then it immediately yields its growth estimates, an uniform in a
some sense distribution of its zeros, a certain regular behavior of the
solution etc.
Received 27.03.2017
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
2 Analytic in an unit ball functions of bounded...
Afterwards, S. Shah [28] and W. Hayman [19] independently proved
that every entire function of bounded index is a function of exponential
type. Namely, its growth is at most the first order and normal type.
To study more general entire functions, A. D. Kuzyk and M. M. She-
remeta [21] introduced a boundedness of the l-index, replacing |f (p)(z)|
p! on
|f (p)(z)|
p!lp(|z|) in (1.1), where l : R+ → R+ is a continuous function. It allows to
consider an arbitrary entire function f with bounded multiplicity of zeros.
Because for the function f there exists a positive continuous function l(z)
such that f is of bounded l-index [14]. Besides, there are papers where
the definition of bounded l-index is generalizing for analytic function of
one variable [22,30].
In a multidimensional case a situation is more difficult and interesting.
Recently we with N. V. Petrechko [12,13] proposed approach to consider
bounded L-index in joint variables for analytic in a polydisc functions,
where L(z) = (l1(z), . . . , ln(z)), lj : Cn → R+ is a positive continuous
functions, j ∈ {1, . . . , n}. Although J. Gopala Krishna and S.M. Shah [20]
introduced an analytic in a domain (a nonempty connected open set)
Ω ⊂ Cn (n ∈ N) function of bounded index for α = (α1, . . . , αn) ∈ Rn+.
But analytic in a domain function of bounded index by Krishna and Shah
is an entire function. It follows from necessary condition of the l-index
boundedness for analytic in the unit disc function ( [29, Th.3.3,p.71]):∫ r
0 l(t)dt → ∞ as r → 1 (we take l(t) ≡ α1). Thus, there arises neces-
sity to introduce and to investigate bounded L-index in joint variables
for analytic in polydisc domain functions. Above-mentioned paper [12]
is devoted analytic in a polydisc functions. Besides a polydisc, other
example of polydisc domain in Cn is a ball. There are two known mono-
graphs [26, 32] about spaces of holomorphic functions in the unit ball of
Cn : Bergman spaces, Hardy spaces, Besov spaces, Lipschitz spaces, the
Bloch space, etc. It shows the relevance of research of properties of holo-
morphic function in the unit ball. In this paper we will introduce and
study analytic in a ball functions of bounded L-index in joint variables.
Of course, there are wide bibliography about entire functions of boun-
ded L-index in joint variables [9–11,15–18,24,25].
Note that there exists other approach to consider bounded index in Cn
— so-called functions of bounded L-index in direction (see [1–7]), where
L : Cn → R+ is a positive continuous function.
A. Bandura, O. Skaskiv 3
2. Main definitions and notations
We need standard notations. Denote R+ = [0,+∞), 0 = (0, . . . , 0) ∈
Rn+, 1 = (1, . . . , 1) ∈ Rn+, 1j = (0, . . . , 0, 1︸︷︷︸
j−th place
, 0, . . . , 0) ∈ Rn+,
R = (r1, . . . , rn) ∈ Rn+, z = (z1, . . . , zn) ∈ Cn, |z| =
√∑n
j=1 |zj |2.
For A = (a1, . . . , an) ∈ Rn, B = (b1, . . . , bn) ∈ Rn we will use for-
mal notations without violation of the existence of these expressions
AB = (a1b1, · · · , anbn), A/B = (a1/b1, . . . , an/bn), A
B = ab11 a
b2
2 · . . . ·abnn ,
∥A∥ = a1 + · · · + an, and the notation A < B means that aj < bj ,
j ∈ {1, . . . , n}; the relation A ≤ B is defined similarly. For K =
(k1, . . . , kn) ∈ Zn+ denote K! = k1! · . . . · kn!. Addition, scalar multi-
plication, and conjugation are defined on Cn componentwise. For z ∈ Cn
and w ∈ Cn we define
⟨z, w⟩ = z1w1 + · · ·+ znwn,
where wk is the complex conjugate of wk. The polydisc {z ∈ Cn : |zj −
z0j | < rj , j = 1, . . . , n} is denoted by Dn(z0, R), its skeleton {z ∈ Cn :
|zj − z0j | = rj , j = 1, . . . , n} is denoted by Tn(z0, R), and the closed
polydisc {z ∈ Cn : |zj − z0j | ≤ rj , j = 1, . . . , n} is denoted by Dn[z0, R],
Dn = Dn(0,1), D = {z ∈ C : |z| < 1}. The open ball {z ∈ Cn : |z−z0| <
r} is denoted by Bn(z0, r), its boundary is a sphere Sn(z0, r) = {z ∈ Cn :
|z − z0| = r}, the closed ball {z ∈ Cn : |z − z0| ≤ r} is denoted by
Bn[z0, r], Bn = Bn(0, 1), D = B1 = {z ∈ C : |z| < 1}.
For K = (k1, . . . , kn) ∈ Zn+ and the partial derivatives of an analytic
in Bn function F (z) = F (z1, . . . , zn) we use the notation
F (K)(z) =
∂∥K∥F
∂zK
=
∂k1+···+knf
∂zk11 . . . ∂zknn
.
Let L(z) = (l1(z), . . . , ln(z)), where lj(z) : Bn → R+ is a continuous
function such that
(∀z ∈ Bn) : lj(z) > β/(1− |z|), j ∈ {1, . . . , n}, (2.1)
where β >
√
n is a some constant.
S. N. Strochyk, M. M. Sheremeta, V. O. Kushnir [29, 30] imposed a
similar condition for a function l : D → R+ and l : G → R+, where G is
arbitrary domain in C.
Remark 2.1. Note that if R ∈ Rn+, |R| ≤ β, z0 ∈ Bn and
4 Analytic in an unit ball functions of bounded...
z ∈ Dn[z0, R/L(z0)] then z ∈ Bn. Indeed, we have
|z| ≤ |z − z0|+ |z0| ≤
√√√√ n∑
j=1
r2j
l2j (z
0)
+ |z0| <
√√√√ n∑
j=1
r2j
β2
(1− |z0|)2 + |z0|
=
(1− |z0|)
β
√√√√ n∑
j=1
r2j + |z0| ≤ (1− |z0|)
β
β + |z0| = 1.
An analytic function F : Bn → C is said to be of bounded L-index (in
joint variables), if there exists n0 ∈ Z+ such that for all z ∈ Bn and for
all J ∈ Zn+
|F (J)(z)|
J !LJ(z)
≤ max
{
|F (K)(z)|
K!LK(z)
: K ∈ Zn+, ∥K∥ ≤ n0
}
. (2.2)
The least such integer n0 is called the L-index in joint variables of the
function F and is denoted by N(F,L,Bn) (see [9]– [16]).
By Q(Bn) we denote the class of functions L, which satisfy (2.1) and
the following condition
(∀R ∈ Rn+, |R| ≤ β, j ∈ {1, . . . , n}) : 0 < λ1,j(R) ≤ λ2,j(R) <∞, (2.3)
where
λ1,j(R) = inf
z0∈Bn
inf
{
lj(z)
lj(z0)
: z ∈ Dn
[
z0, R/L(z0)
]}
, (2.4)
λ2,j(R) = sup
z0∈Bn
sup
{
lj(z)
lj(z0)
: z ∈ Dn
[
z0, R/L(z0)
]}
. (2.5)
Λ1(R) = (λ1,1(R), . . . , λ1,n(R)), Λ2(R) = (λ2,1(R), . . . , λ2,n(R)).
It is not difficult to verify that the class Q(Bn) can be defined as
following: for every j∈{1, . . . , n}
sup
z,w∈Bn
{
lj(z)
lj(w)
: |zk − wk| ≤
rk
min{lk(z), lk(w)}
, k ∈ {1, . . . , n}
}
<∞,
(2.6)
i. e. conditions (2.3) and (2.6) are equivalent.
Example 2.1. The function F (z) = exp{ 1
(1−z1)(1−z2)} has bounded L-
index in joint variables with L(z) =
(
1
(1−|z1|)2(1−|z|) ,
1
(1−|z|)(1−|z2|)2
)
and
N(F,L,Bn) = 0.
A. Bandura, O. Skaskiv 5
3. Local behavior of derivatives of function of bounded
L-index in joint variables
The following theorem is basic in theory of functions of bounded in-
dex. It was necessary to prove more efficient criteria of index boundedness
which describe a behavior of maximum modulus on a disc or a behavior
of logarithmic derivative (see [1, 6, 22,29]).
Theorem 3.1. Let L ∈ Q(Bn). An analytic in Bn function F has
bounded L-index in joint variables if and only if for each R ∈ Rn+,
|R| ≤ β, there exist n0 ∈ Z+, p0 > 0 such that for every z0 ∈ Bn
there exists K0 ∈ Zn+, ∥K0∥ ≤ n0, and
max
{
|F (K)(z)|
K!LK(z)
: ∥K∥ ≤ n0, z ∈ Dn
[
z0, R/L(z0)
]}
≤ p0
|F (K0)(z0)|
K0!LK0(z0)
. (3.1)
Proof. Let F be an analytic function of bounded L-index in joint vari-
ables with N = N(F,L,Bn) <∞. For every R ∈ Rn+, |R| ≤ β, we put
q = q(R) = [2(N + 1)∥R∥
n∏
j=1
(λ1,j(R))
−N (λ2,j(R))
N+1] + 1
where [x] is the entire part of the real number x, i.e. it is a floor function.
For p ∈ {0, . . . , q} and z0 ∈ Bn we denote
Sp(z
0, R) = max
{
|F (K)(z)|
K!LK(z)
: ∥K∥ ≤ N, z ∈ Dn
[
z0,
pR
qL(z0)
]}
,
S∗
p(z
0, R) = max
{
|F (K)(z)|
K!LK(z0)
: ∥K∥ ≤ N, z ∈ Dn
[
z0,
pR
qL(z0)
]}
.
Using (2.4) and Dn
[
z0, pR
qL(z0)
]
⊂ Dn
[
z0, R
L(z0)
]
, we have
Sp(z
0, R) = max
{
|F (K)(z)|
K!LK(z)
LK(z0)
LK(z0)
: ∥K∥ ≤N, z ∈Dn
[
z0,
pR
qL(z0)
]}
≤ S∗
p(z
0, R)max
n∏
j=1
lNj (z0)
lNj (z)
: z ∈ Dn
[
z0,
pR
qL(z0)
]
≤ S∗
p(z
0, R)
n∏
j=1
(λ1,j(R))
−N .
6 Analytic in an unit ball functions of bounded...
and, using (2.5), we obtain
S∗
p(z
0, R) = max
{
|F (K)(z)|
K!LK(z)
LK(z)
LK(z0)
: ∥K∥≤N, z ∈ Dn
[
z0,
pR
qL(z0)
]}
≤ max
{
|F (K)(z)|
K!LK(z)
(Λ2(R))
K : ∥K∥ ≤N, z ∈ Dn
[
z0,
pR
qL(z0)
]}
≤ Sp(z
0, R)
n∏
j=1
(λ2,j(R))
N . (3.2)
Let K(p) ∈ Zn+ with ∥K(p)∥ ≤ N and z(p) ∈ Dn
[
z0, pR
qL(z0)
]
be such that
S∗
p(z
0, R) =
|F (K(p))(z(p))|
K(p)!LK
(p)
(z0)
(3.3)
Since by the maximum principle z(p) ∈ Tn(z0, pR
qL(z0)
), we have z(p) ̸= z0.
We choose z̃
(p)
j = z0j +
p−1
p (z
(p)
j − z0j ). Then for every j ∈ {1, . . . , n} we
have
|z̃(p)j − z0j | =
p− 1
p
|z(p)j − z0j | =
p− 1
p
prj
qlj(z0)
, (3.4)
|z̃(p)j − z
(p)
j | = |z0j +
p− 1
p
(z
(p)
j − z0j )− z
(p)
j | = 1
p
|z0j − z
(p)
j |
=
1
p
prj
qlj(z0)
=
rj
qlj(z0)
. (3.5)
From (3.4) we obtain z̃(p) ∈ Dn
[
z0, (p−1)R
q(R)L(z0)
]
and
S∗
p−1(z
0, R) ≥ |F (K(p))(z̃(p))|
K(p)!LK
(p)
(z0)
.
From (3.3) it follows that
0 ≤ S∗
p(z
0, R)− S∗
p−1(z
0, R) ≤ |F (K(p))(z(p))| − |F (K(p))(z̃(p))|
K(p)!LK
(p)
(z0)
=
1
K(p)!LK
(p)
(z0)
∫ 1
0
d
dt
|F (K(p))(z̃(p) + t(z(p) − z̃(p)))|dt
≤ 1
K(p)!LK
(p)
(z0)
×
∫ 1
0
n∑
j=1
|z(p)j − z̃
(p)
j |
∣∣∣F (K(p)+1j)(z̃(p)+t(z(p) − z̃(p)))
∣∣∣ dt
A. Bandura, O. Skaskiv 7
=
1
K(p)!LK
(p)
(z0)
×
n∑
j=1
|z(p)j − z̃
(p)
j |
∣∣∣F (K(p)+1j)(z̃(p) + t∗(z(p) − z̃(p)))
∣∣∣ , (3.6)
where 0 ≤ t∗ ≤ 1, z̃(p) + t∗(z(p) − z̃(p)) ∈ Dn(z0, pR
qL(z0)
). For z ∈
Dn(z0, pR
qL(z0)
) and J ∈ Zn+, ∥J∥ ≤ N + 1 we have
|F (J)(z)|LJ(z)
J !LJ(z0)LJ(z)
≤ (Λ2(R))
J max
{
|F (K)(z)|
K!LK(z)
: ∥K∥ ≤ N
}
≤
n∏
j=1
(λ2,j(R))
N+1(λ1,j(R))
−N max
{
|F (K)(z)|
K!LK(z0)
: ∥K∥ ≤ N
}
≤
n∏
j=1
(λ2,j(R))
N+1(λ1,j(R))
−NS∗
p(z
0, R).
From (3.6) and (3.5) we obtain
0 ≤ S∗
p(z
0, R)− S∗
p−1(z
0, R) ≤
n∏
j=1
(λ2,j(R))
N+1(λ1,j(R))
−N
× S∗
p(z
0, R)
n∑
j=1
(k
(p)
j + 1)lj(z
0)|z(p)j − z̃
(p)
j |
=
n∏
j=1
(λ2,j(R))
N+1(λ1,j(R))
−N S
∗
p(z
0, R)
q(R)
n∑
j=1
(k
(p)
j + 1)rj
≤
n∏
j=1
(λ2,j(R))
N+1(λ1,j(R))
−N S
∗
p(z
0, R)
q(R)
(N + 1)∥R∥ ≤ 1
2
S∗
p(z
0, R).
This inequality implies S∗
p(z
0, R) ≤ 2S∗
p−1(z
0, R), and in view of inequa-
lity (3.2) we have
Sp(z
0, R) ≤ 2
n∏
j=1
(λ1,j(R))
−NS∗
p−1(z
0, R)
≤ 2
n∏
j=1
(λ1,j(R))
−N (λ2,j(R))
NSp−1(z
0, R)
Therefore,
max
{
|F (K)(z)|
K!LK(z)
: ∥K∥ ≤ N, z ∈ Dn
[
z0,
pR
qL(z0)
]}
= Sq(z
0, R)
8 Analytic in an unit ball functions of bounded...
≤ 2
n∏
j=1
(λ1,j(R))
−N (λ2,j(R))
NSq−1(z
0, R) ≤ . . .
≤ (2
n∏
j=1
(λ1,j(R))
−N (λ2,j(R))
N )qS0(z
0, R)
= (2
n∏
j=1
(λ1,j(R))
−N (λ2,j(R))
N )q
× max
{
|F (K)(z0)|
K!LK(z0)
: ∥K∥ ≤ N
}
. (3.7)
From (3.7) we obtain inequality (3.1) with
p0 = (2
n∏
j=1
(λ1,j(R))
−N (λ2,j(R))
N )q
and some K0 with ∥K0∥ ≤ N . The necessity of condition (3.1) is proved.
Now we prove the sufficiency. Suppose that for every R ∈ Rn+, |R| ≤
β, there exist n0 ∈ Z+, p0 > 1 such that for all z0 ∈ Bn and some
K0 ∈ Zn+, ∥K0∥ ≤ n0, the inequality (3.1) holds.
We write Cauchy’s formula as following ∀z0 ∈ Bn ∀k ∈ Zn+ ∀S ∈ Zn+
F (K+S)(z0)
S!
=
1
(2πi)n
∫
Tn
(
z0, R
L(z0)
) F (K)(z)
(z − z0)S+1
dz.
Therefore, applying (3.1), we have
|F (K+S)(z0)|
S!
≤ 1
(2π)n
∫
Tn
(
z0, R
L(z0)
) |F (K)(z)|
|z − z0|S+1
|dz|
≤
∫
Tn
(
z0, R
L(z0)
) |F (K)(z)| LS+1(z0)
(2π)nRS+1
|dz|
≤
∫
Tn
(
z0, R
L(z0)
) |F (K0)(z0)|
×
K!p0
∏n
j=1 λ
n0
2,j(R)L
S+K+1(z0)
(2π)nK0!RS+1LK0(z0)
|dz|
= |F (K0)(z0)|
K!p0
∏n
j=1 λ
n0
2,j(R)L
S+K(z0)
K0!RSLK0(z0)
.
This implies
|F (K+S)(z0)|
(K + S)!LS+K(z0)
≤
∏n
j=1 λ
n0
2,j(R)p0K!S!
(K + S)!RS
|F (K0)(z0)|
K0!LK0(z0)
. (3.8)
A. Bandura, O. Skaskiv 9
Obviously, that
K!S!
(K + S)!
=
s1!
(k1 + 1) · . . . · (k1 + s1)
· · · sn!
(kn + 1) · . . . · (kn + sn)
≤ 1.
We choose rj ∈ (1, β/
√
n], j ∈ {1, . . . , n}. Then |R| =
√∑n
j=1 r
2
j ≤ β.
Hence,
p0
∏n
j=1 λ
n0
2,j(R)
RS
→ 0 as ∥S∥ → +∞. Thus, there exists s0 such that
for all S ∈ Zn+ with ∥S∥ ≥ s0 the inequality holds
p0K!S!
∏n
j=1 λ
n0
2,j(R)
(K + S)!RS
≤ 1.
Inequality (3.8) yields |F (K+S)(z0)|
(K+S)!LK+S(z0)
≤ |F (K0)(z0)|
K0!LK0 (z0)
. This means that for
every j ∈ Zn+
|F (J)(z0)|
J !LJ(z0)
≤ max
{
|F (K)(z0)|
K!LK(z0)
: ∥K∥ ≤ s0 + n0
}
where s0 and n0 are independent of z0. Therefore, the analytic in Bn
function F has bounded L-index in joint variables with N(F,L,Bn) ≤
s0 + n0.
Theorem 3.2. Let L ∈ Q(Bn). In order that an analytic in Bn function
F be of bounded L-index in joint variables it is necessary that for every
R ∈ Rn+, |R| ≤ β, ∃n0 ∈ Z+ ∃p ≥ 1 ∀z0 ∈ Bn ∃K0 ∈ Zn+, ∥K0∥ ≤ n0,
and
max
{
|F (K0)(z)| : z ∈ Dn
[
z0, R/L(z0)
]}
≤ p|F (K0)(z0)| (3.9)
and it is sufficient that for every R ∈ Rn+, |R| ≤ β, ∃n0 ∈ Z+ ∃p ≥ 1
∀z0 ∈ Bn ∀j ∈ {1, . . . , n} ∃K0
j = (0, . . . , 0, k0j︸︷︷︸
j-th place
, 0, . . . , 0) such that
k0j ≤ n0 and
max
{
|F (K0
j )(z)| : z ∈ Dn
[
z0, R/L(z0)
]}
≤ p|F (K0
j )(z0)|. (3.10)
Proof. Proof of Theorem 3.1 implies that the inequality (3.1) is true for
some K0. Therefore, we have
p0
K0!
|F (K0)(z0)|
LK0(z0)
≥ max
{
|F (K0)(z)|
K0!LK0(z)
: z ∈ Dn
[
z0, R/L(z0)
]}
10 Analytic in an unit ball functions of bounded...
= max
{
|F (K0)(z)|
K0!
LK
0
(z0)
LK0(z0)LK0(z)
: z ∈ Dn
[
z0, R/L(z0)
]}
≥ max
{
|F (K0)(z)|
K0!
∏n
j=1 (λ2,j(R))
−n0
LK0(z0)
: z ∈ Dn
[
z0, R/L(z0)
]}
.
This inequality implies
p0
∏n
j=1(λ2,j(R))
n0
K0!
|F (K0)(z0)|
LK0(z0)
≥ max
{
|F (K0)(z)|
K0!LK0(z0)
: z ∈ Dn
[
z0, R/L(z0)
]}
. (3.11)
From (3.11) we obtain inequality (3.9) with p = p0
∏n
j=1 (λ2,j(R))
n0 . The
necessity of condition (3.9) is proved.
Now we prove the sufficiency of (3.10). Suppose that for every R ∈
Rn+, |R| ≤ β, ∃n0 ∈ Z+, p > 1 such that ∀z0 ∈ Bn and some K0
J ∈ Zn+
with k0j ≤ n0 the inequality (3.10) holds.
We write Cauchy’s formula as following ∀z0 ∈ Bn ∀S ∈ Zn+
F (K0
J+S)(z0)
S!
=
1
(2πi)n
∫
Tn(z0,R/L(z0))
F (K0
J )(z)
(z − z0)S+1
dz.
This yields
|F (K0
j+S)(z0)|
S!
≤ 1
(2π)n
∫
Tn(z0,R/L(z0))
|F (K0
j )(z)|
|z − z0|S+1
|dz|
≤ 1
(2π)n
max{|F (K0
j )(z)| : z ∈ Dn
[
z0, R/L(z0)
]
}L
S+1(z0)
RS+1
×
∫
Tn(z0,R/L(z0))
|dz|
= max{|F (K0
j )(z)| : z ∈ Dn
[
z0, R/L(z0)
]
}L
S(z0)
RS
.
Now we put R = ( β√
n
, . . . , β√
n
) and use (3.10)
|F (K0
j+S)(z0)|
S!
≤ LS(z0)
(β/
√
n)
∥S∥ max{|F (K0
j )(z)| : z ∈ Dn
[
z0, R/L(z0)
]
}
≤ pLS(z0)
(β/
√
n)∥S∥
|F (K0
j )(z0)|. (3.12)
A. Bandura, O. Skaskiv 11
We choose S ∈ Zn+ such that ∥S∥ ≥ s0, where
p
(β/
√
n)s0
≤ 1. Therefore,
(3.12) implies that for all j ∈ {1, . . . , n} and k0j ≤ n0
|F (K0
j+S)(z0)|
LK
0
j+S(z0)(K0
j +S)!
≤ p
(β/
√
n)
∥S∥
S!K0
j !
(S +K0
j )!
|F (K0
j )(z0)|
LK
0
j (z0)K0
j !
≤ |F (K0
j )(z0)|
LK
0
j (z0)K0
j !
.
Consequently, N(F,L,Bn) ≤ n0 + s0.
Remark 3.1. Inequality (3.9) is necessary and sufficient condition of
boundedness of l-index for functions of one variable [22, 29, 31]. But it
is unknown whether this condition is sufficient condition of boundedness
of L-index in joint variables. Our restrictions (3.10) are corresponding
multidimensional sufficient conditions.
Lemma 3.1. Let L1, L2 ∈ Q(Bn) and for every z ∈ Bn L1(z) ≤ L2(z).
If analytic in Bn function F has bounded L1-index in joint variables
then F is of bounded L2-index in joint variables and N(F,L2,Bn) ≤
nN(F,L1,Bn).
Proof. Let N(F,L1,Bn) = n0. Using (2.2) we deduce
|F (J)(z)|
J !LJ2 (z)
=
LJ1 (z)
LJ2 (z)
|F (J)(z)|
J !LJ1 (z)
≤ LJ1 (z)
LJ2 (z)
max
{
|F (K)(z)|
K!LK1 (z)
: K ∈ Zn+, ∥K∥ ≤ n0
}
≤ LJ1 (z)
LJ2 (z)
max
{
LK2 (z)
LK1 (z)
|F (K)(z)|
K!LK2 (z)
: K ∈ Zn+, ∥K∥ ≤ n0
}
≤ max
∥K∥≤n0
(
L1(z)
L2(z)
)J−K
max
{
|F (K)(z)|
K!LK2 (z)
: K ∈ Zn+, ∥K∥ ≤ n0
}
.
Since L1(z) ≤ L2(z) it means that for all ∥J∥ ≥ nn0
|F (J)(z)|
J !LJ2 (z)
≤ max
{
|F (K)(z)|
K!LK2 (z)
: K ∈ Zn+, ∥K∥ ≤ n0
}
.
Thus, F has bounded L2-index in joint variables and N(F,L2,Bn) ≤
nN(F,L1,Bn).
Denote L̃(z) = (l̃1(z), . . . , l̃n(z)). The notation L ≍ L̃ means that
there exist Θ1 = (θ1,j , . . . , θ1,n) ∈ Rn+, Θ2 = (θ2,j , . . . , θ2,n) ∈ Rn+ such
that ∀z ∈ Bn θ1,j l̃j(z) ≤ lj(z) ≤ θ2,j l̃j(z) for each j ∈ {1, . . . , n}.
12 Analytic in an unit ball functions of bounded...
Theorem 3.3. Let L ∈ Q(Bn), L ≍ L̃, β|Θ1| >
√
n. An analytic in Bn
function F has bounded L̃-index in joint variables if and only if it has
bounded L-index.
Proof. It is easy to prove that if L ∈ Q(Bn) and L ≍ L̃ then L̃ ∈ Q(Bn).
Let N(F, L̃,Bn) = ñ0 < +∞. Then by Theorem 3.1 for every R̃ =
(r̃1, . . . , r̃n) ∈ Rn+, |R| ≤ β, there exists p̃ ≥ 1 such that for each z0 ∈ Bn
and some K0 with ∥K0∥ ≤ ñ0, the inequality (3.1) holds with L̃ and R̃
instead of L and R. Hence
p̃
K0!
|F (K0)(z0)|
LK0(z0)
=
p̃
K0!
ΘK0
2 |F (K0)(z0)|
ΘK0
2 LK0(z0)
≥ p̃
K0!
|F (K0)(z0)|
ΘK0
2 L̃K0(z0)
≥ 1
ΘK0
2
max
{
|F (K)(z)|
K!L̃K(z)
: ∥K∥ ≤ ñ0, z ∈ Dn
[
z0, R̃/L̃(z)
]}
≥ 1
ΘK0
2
max
{
ΘK
1 |F (K)(z)|
K!LK(z)
: ∥K∥ ≤ ñ0, z ∈ Dn
[
z0,Θ1R̃/L(z)
]}
≥
min0≤∥K∥≤n0
{ΘK
1 }
ΘK0
2
× max
{
|F (K)(z)|
K!LK(z)
: ∥K∥ ≤ ñ0, z∈Dn
[
z0,Θ1R̃/L̃(z)
]}
.
In view of Theorem 3.1 we obtain that function F has bounded L-index
in joint variables.
Theorem 3.4. Let L ∈ Q(Bn). An analytic in Bn function F has
bounded L-index in joint variables if and only if there exist R ∈ Rn+,
with |R| ≤ β, n0 ∈ Z+, p0 > 1 such that for each z0 ∈ Bn and for some
K0 ∈ Zn+ with ∥K0∥ ≤ n0 the inequality (3.1) holds.
Proof. The necessity of this theorem follows from the necessity of The-
orem 3.1. We prove the sufficiency. The proof of Theorem 3.1 with
R = ( β√
n
, . . . , β√
n
) implies that N(F,L,Bn) < +∞.
Let L∗(z) = R0L(z)
R , R0 = ( β√
n
, . . . , β√
n
). In general case from validity
of (3.1) for F, L and R = (r1, . . . , rn) with |R| ≤ β, R ̸= R0, we obtain
max
{
|F (K)(z)|
K!(L∗(z0))K
: ∥K∥ ≤ n0, z ∈ Dn
[
z0, R0/L
∗(z0)
]}
= max
{
|F (K)(z)|
K!(R0L(z)/R)K
: ∥K∥ ≤ n0, z ∈ Dn
[
z0, R0/(R0L(z)/R)
]}
A. Bandura, O. Skaskiv 13
≤ max
{
n∥K∥/2|F (K)(z)|
K!LK(z)
: ∥K∥ ≤ n0, z∈Dn
[
z0, R/L(z0)
]}
≤ p0
K0!
nn0/2|F (K0)(z0)|
LK0(z0)
=
nn0/2(β/
√
n)∥K
0∥p0
RK0K0!
|F (K0)(z0)|
(R0L(z)/R)K
0
< nn0/2p0 max
∥K0∥≤n0
(β/
√
n)∥K
0∥
RK0
|F (K0)(z0)|
K0!(L∗(z))K0 .
i. e. (3.1) holds for F, L∗ and R0 = ( β√
n
, . . . , β√
n
). As above we apply
Theorem 3.1 to the function F (z) and L∗(z) = R0L(z)/R. This implies
that F is of bounded L∗-index in joint variables. Therefore, by Theorem
3.3 the function F has bounded L-index in joint variables.
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A. Bandura, O. Skaskiv 15
Contact information
Andriy Bandura Ivano-Frankivsk National
Technical University of Oil and Gas,
Ivano-Frankivsk, Ukraine
E-Mail: andriykopanytsia@gmail.com
Oleh Skaskiv Ivan Franko National University of Lviv,
Lviv, Ukraine
E-Mail: olskask@gmail.com
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