Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables
We obtain the sufficient conditions of boundedness of L-index in joint variables for analytic functions in the unit ball, where L : Cⁿ → Rⁿ₊ is a continuous positive vector-function. They give an estimate of the maximum modulus of an analytic function by its minimum modulus on a skeleton in a polydi...
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1693962020-06-13T01:26:58Z Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables Bandura, A.I. Skaskiv, O.B. We obtain the sufficient conditions of boundedness of L-index in joint variables for analytic functions in the unit ball, where L : Cⁿ → Rⁿ₊ is a continuous positive vector-function. They give an estimate of the maximum modulus of an analytic function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives outside some exceptional set and the distribution of zeros. The deduced results are also new for analytic functions in the unit disc of bounded index and l-index. They generalize known results by G. H. Fricke, M. M. Sheremeta, A. D. Kuzyk, and V. O. Kushnir. 2018 Article Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables / A.I. Bandura, O.B. Skaskiv // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 177-193. — Бібліогр.: 37 назв. — англ. 1810-3200 2010 MSC. 32A10, 32A40, 32A60 http://dspace.nbuv.gov.ua/handle/123456789/169396 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We obtain the sufficient conditions of boundedness of L-index in joint variables for analytic functions in the unit ball, where L : Cⁿ → Rⁿ₊ is a continuous positive vector-function. They give an estimate of the maximum modulus of an analytic function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives outside some exceptional set and the distribution of zeros. The deduced results are also new for analytic functions in the unit disc of bounded index and l-index. They generalize known results by G. H. Fricke, M. M. Sheremeta, A. D. Kuzyk, and V. O. Kushnir. |
| format |
Article |
| author |
Bandura, A.I. Skaskiv, O.B. |
| spellingShingle |
Bandura, A.I. Skaskiv, O.B. Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables Український математичний вісник |
| author_facet |
Bandura, A.I. Skaskiv, O.B. |
| author_sort |
Bandura, A.I. |
| title |
Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables |
| title_short |
Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables |
| title_full |
Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables |
| title_fullStr |
Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables |
| title_full_unstemmed |
Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables |
| title_sort |
partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded l-index in joint variables |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169396 |
| citation_txt |
Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables / A.I. Bandura, O.B. Skaskiv // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 177-193. — Бібліогр.: 37 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT banduraai partiallogarithmicderivativesanddistributionofzerosofanalyticfunctionsintheunitballofboundedlindexinjointvariables AT skaskivob partiallogarithmicderivativesanddistributionofzerosofanalyticfunctionsintheunitballofboundedlindexinjointvariables |
| first_indexed |
2025-07-15T04:08:18Z |
| last_indexed |
2025-07-15T04:08:18Z |
| _version_ |
1837684489322496000 |
| fulltext |
Український математичний вiсник
Том 15 (2018), № 2, 177 – 193
Partial logarithmic derivatives and distribution
of zeros of analytic functions in the unit ball of
bounded L-index in joint variables
Andriy I. Bandura, Oleh B. Skaskiv
(Presented by V. Ya. Gutlyanskii)
Abstract. In this paper, we obtain sufficient conditions of bound-
edness of L-index in joint variables for analytic functions in the unit
ball, where L : Cn → Rn+ is a continuous positive vector-function. They
give an estimate of maximum modulus of analytic function by its min-
imum modulus on a skeleton in a polydisc and describe the behavior
of all partial logarithmic derivatives outside some exceptional set and
the distribution of zeros. The deduced results are also new for analytic
functions in the unit disc of bounded index and l-index. They generalize
known results of G. H. Fricke, M. M. Sheremeta, A. D. Kuzyk and V.
O. Kushnir.
2010 MSC. 32A10, 32A40, 32A60.
Key words and phrases. Analytic function, unit ball, bounded L-
index in joint variables, maximum modulus, partial derivative, minimum
modulus, distribution of zeros, skeleton of polydisc.
1. Introduction
A concept of an entire function of bounded index arose in analytic
theory of differential equations. It appeared in paper of B. Lepson [24].
An entire function f is said to be of bounded index if there exists an
integer N > 0 such 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. There was proved that
every entire solution of ordinary n-th order linear differential equation
with constant coefficients has bounded index. B. Lepson conjectured
that every entire solution of the linear differential equation of infinite or-
der with constant coefficients has the same property. In a general case
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
178 Partial logarithmic derivatives and distribution...
the hypothesis is not correct because an entire function of bounded in-
dex has exponential type [20, 32] and there exist entire solutions with
order greater than one. Therefore, it is natural to pose the same ques-
tion about l-index boundedness of entire solutions (see definition below).
This assumption has not yet been proven even in this extended formula-
tion. However, there are many papers of various authors with different
applications. Namely, theory of functions of bounded index has many
applications in value distribution theory, differential equations and its
system (see bibliography in [7, 33]). This concept is applicable as for
entire functions of one and several variables [4, 10, 22] so for analytic
functions in a domain [1, 11, 21, 23, 35]. In a comparison with traditional
approaches (for example, Wiman–Valiron’s method [15,16,31,37] or value
distribution theory [18,19,25,29,34]) it is more flexible to investigate an-
alytic solutions of ordinary and partial differential equations [2, 3, 9, 27].
Particularly, if an entire solution has bounded index [22, 32, 36] then it
immediately yields its growth estimates, an uniform distribution of its
zeros in a sense, a certain regular behavior of the solution, etc. Similar
conclusions are valid for functions of one variable which are analytic in a
domain [13,23,33,35].
To study more general entire functions, A. D. Kuzyk and M. M. She-
remeta [22] 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. In view
of results from [14] it allows to study an arbitrary entire function f with
bounded multiplicity of zeros. Besides, there are papers about bounded
l-index for analytic function of one variable [23, 35].
In a multidimensional case a situation is more difficult and inter-
esting. Recently we [11, 12] proposed approach to consider bounded
L-index in joint variables for analytic functions in a polydisc, where
L(z) = (l1(z), . . . , ln(z)), lj : Cn → R+ is a positive continuous func-
tions, j ∈ {1, . . . , n}. Although J. Gopala Krishna and S. M. Shah [21]
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 function of bounded index in a domain 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 ( [33, Th.3.3,p.71]):∫ r
0 l(t)dt→ ∞ as r → 1 (we take l(t) ≡ α1). Thus, there arises necessity
to introduce and to investigate bounded L-index in joint variables for
analytic functions in polydisc domain. Besides a polydisc, other example
of polydisc domain in Cn is a ball.
For analytic functions in the unit ball we introduced a concept of
bounded L-index in joint variables and deduced properties [1–3]. Also,
A. I. Bandura, O. B. Skaskiv 179
there was presented an application of the concept to study properties of
analytic solutions for systems of partial differential equations and to esti-
mate its growth. But in one-dimensional case there is known logarithmic
criterion of index boundedness [17,36]. It is very important to investigate
infinite products and holomorphic solutions of differential equations. The
assertion describes behavior of logarithmic derivative outside some excep-
tional set consisting with zeros of the function with some neighborhoods.
The second condition of the criterion is uniform distribution of zeros in a
sense. Above 20 years it has not been possible to obtain an analog of this
criterion for entire functions of several variables by technical difficulties.
Using other approach [4, 6, 8] we deduced the analog for a class of entire
functions of bounded L-index in a direction. But for entire functions
of bounded L-index in joint variables the analog of logarithmic criterion
remained unknown. Recently as sufficient conditions some analog of the
characterization have been obtained for the class [5].
In view of importance of logarithmic criterion for analytic functions
of one variable it is naturally to pose the following problem: What is an
analog of logarithmic criterion for analytic functions in the unit ball of
bounded L-index in joint variables?
A complete solution to this problem may give new applications of
bounded L-index in joint variables for analytic functions in the unit ball.
For example, this can be useful to investigate properties of multidimen-
sional analogs of Blaschke products. or analytic solutions of partial dif-
ferential equations system.
In this paper we will try to give some answer to the question.
2. Main definitions and notations
We need some 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 multipli-
cation, and conjugation are defined on Cn componentwise. 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),
180 Partial logarithmic derivatives and distribution...
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), 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 [23,33,35] imposed
a similar condition for a function l : D → R+ and l : G → R+, where G
is arbitrary domain in C.
An analytic function F : Bn → C is said to be of bounded L-index (in
joint variables) [1–3], 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) (for entire functions see [10,
25,30]).
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) <∞,
where
λ1,j(R) = inf
z0∈Bn
inf
{
lj(z)
lj(z0)
: z ∈ Dn
[
z0, R/L(z0)
]}
,
λ2,j(R) = sup
z0∈Bn
sup
{
lj(z)
lj(z0)
: z ∈ Dn
[
z0, R/L(z0)
]}
,
Λ1(R) = (λ1,1(R), . . . , λ1,n(R)), Λ2(R) = (λ2,1(R), . . . , λ2,n(R)).
We need the following assertion.
A. I. Bandura, O. B. Skaskiv 181
Theorem 2.1 ( [2]). Let L ∈ Q(Bn), F : Bn → C be analytic function.
If there exist R′, R′′ ∈ Rn+, R′ < R′′, |R′′| < β and p1 = p1(R
′, R′′) ≥ 1
such that for every z0 ∈ Bn
max
{
|F (z)| : z ∈ Tn
(
z0,
R′′
L(z0)
)}
≤ p1max
{
|F (z)| : z ∈ Tn
(
z0,
R′
L(z0)
)}
(2.3)
then F is of bounded L-index in joint variables.
3. Estimate maximum modulus on a skeleton in polydisc
Let ZF be a zero set of analytic in Bn function F. We denote
GR(F ) =
∪
z0∈ZF
{
z ∈ Cn : |zj − z0j | <
rj
lj(z0)
∀j ∈ {1, 2, . . . , n}
}
=
∪
z0∈ZF
Dn
(
z0,
R
L(z0)
)
.
Theorem 3.1. Let L ∈ Q(Bn), F : Bn → C be an analytic function. If
∃R ∈ Rn+ with rj ∈ (0, β
2
√
n
), ∃p2 ≥ 1 ∃Θ ∈ Rn+, 0 < Θ < R, ∃R′ > 0,
(R′ = 0 for ZF = ∅) such that ∀z0 ∈ Bn ∃R0 = R0(z0) ∈ Rn+, Θ ≤ R0 ≤
R, for which
meas
{
Tn
(
z0,
R0
L(z0)
)
∩GR′(F )
}
<
(
2π
3
)n n∏
j=1
θj
λ2,j(β)lj(z0)
(3.1)
and max
{
|F (z)| : z ∈ Tn
(
z0,
R0
L(z0)
)}
≤ p2min
{
|F (z)| : z ∈ Tn
(
z0,
R0
L(z0)
)
\GR′(F )
}
(3.2)
then the function F has bounded L-index in joint variables (meas is the
Lebesgue measure on the skeleton in the polydisc).
Proof. Denote β = β√
n
1. By Theorem 2.1 we will show that ∃p1 > 0
∀z0 ∈ Bn
max
{
|F (z)| : z ∈ Tn
(
z0,
β −R
L(z0)
)}
≤ p1max
{
|F (z)| : z ∈ Tn
(
z0,
R
L(z0)
)}
.
182 Partial logarithmic derivatives and distribution...
Denote l∗j = max
{
lj(z) : z ∈ Dn
[
z0, β
L(z0)
]}
, ρj,0 =
rj
lj(z0)
, ρj,k = ρj,0 +
k·θj
l∗j
, k ∈ N, j ∈ {1, . . . , n}. The following estimate holds
θj
l∗j
<
rj
l∗j
≤ rj
lj(z0)
=
β/
√
n
lj(z0)
− β/
√
n− rj
lj(z0)
.
Hence, there exists S∗ = (s∗1, . . . , s
∗
n) ∈ Nn independent of z0 such that
ρj,mj−1 <
β/
√
n− rj
lj(z0)
≤ ρj,mj ≤
β/
√
n
lj(z0)
for some mj = mj(z
0) ≤ s∗j because L ∈ Q(Bn). Indeed,(
β/
√
n
lj(z0)
− ρj,0
)/θj
l∗j
=
(
β/
√
n− rj
) l∗j
θjlj(z0)
=
β/
√
n− rj
θj
max
{
lj(z)
lj(z0)
: z ∈ Dn
[
z0,
β
L(z0)
]}
≤ β/
√
n− rj
θj
λ2,j(β).
Thus, s∗j =
[
β/
√
n−rj
θj
λ2,j(β)
]
, where [x] is the integer part of x ∈ R.
Let M0 = (m1, . . . ,mn) and τ
∗∗
K be such a point in Bn that
|F (τ∗∗K )| = max{|F (z)| : z ∈ Tn(z0,RK)},
where K = (k1, . . . , kn), RK = (ρ1,k1 , . . . , ρn,kn) and τ∗j,K be the in-
tersection point in C of the segment [z0j , τ
∗∗
j,K ] with |zj − z0j | = ρj,kj−1.
We construct a sequence of polydisc Dn(z0,RK) with K ≤ M0, R0 =
R/L(z0) = (ρ1,0, . . . , ρn,0) and Θ/L(z0) = (θ1/l
∗
1, . . . , θn/l
∗
n) (see Figures
1 and 2).
Denote α
(j)
K =(τ∗∗1,K , . . . , τ
∗∗
j−1,K , τ
∗
j,K , τ
∗∗
j+1,K , . . . , τ
∗∗
n,K). Hence, for ev-
ery rj > θj and K ≤ S∗ : |τ∗j,K − τ∗∗j,K | =
θj
l∗j
≤ rj
lj(α
(j)
K )
. Thus, for some
R0 = R0(α
(j)
K ) ∈ Rn+, Θ ≤ R0 ≤ R, we deduce
|F (τ∗∗K )| ≤ max
{
|F (z)| : z ∈ Tn
(
α
(j)
K ,
R0
L(α
(j)
K )
)}
≤ p2min
{
|F (z)| : z ∈ Tn
(
α
(j)
K ,
R0
L(α
(j)
K )
)
\GR′(F )
}
≤p2min
{
|F (z)| : z ∈ Tn
(
α
(j)
K ,
R0
L(α
(j)
K )
)
\GR′(F ), z ∈ Dn[z0,RK−1j ]
}
≤ p2max{|F (z)| : z ∈ Tn(z0,RK−1j )}. (3.3)
A. I. Bandura, O. B. Skaskiv 183
z
0
R0
R1
�
/L ∗
...
RM 0
� 1� � R
L (z0 )
RM 0
Figure 1
z
0
RK � 1 jRK � ∗∗
K�( j )
K
Figure 2
To deduce (3.3) we implicitly used that(
Tn
(
α
(j)
K ,
R0
L(α
(j)
K )
)
\GR′(F )
)
∩ Dn[z0,RK−1j ] ̸= ∅. (3.4)
Condition (3.1) provides (3.4). Indeed, we will find a lower estimate of n-
dimensional Lebesgue measure of the set Tn
(
α
(j)
K , R0
L(α
(j)
K )
)
∩Dn[z0,RK−1j ]
and will show that the measure is not lesser than a left-hand side of
inequality (3.1).
The set Tn
(
α
(j)
K , R0
L(α
(j)
K )
)
∩Dn[z0,RK−1j ] is the Cartesian product of
the following arcs on circles: for every m ∈ {1, . . . , n}, m ̸= j (see Figure
3){
zm ∈ C : |zm − τ∗∗m,K | = r0m
lm(α
(j)
K )
}∩{
zm ∈ C : |zm − z0m| ≤ ρm,km
}
and for m = j (see Figure 4){
zj ∈ C : |zj − τ∗j,K | =
r0j
lj(α
(j)
K )
}∩{
zj ∈ C : |zj − z0j | ≤ ρj,kj−1
}
.
It is easy to prove that the length of arc equals
2r0m
lm(α
(j)
K )
· arccos r0m
2lm(α
(j)
K )ρm,km
for m ̸= j (3.5)
184 Partial logarithmic derivatives and distribution...
z
0
m
�
m ,km
� ∗∗m ,K
r 0
Figure 3 with r0=
r 0m
lm (
� ( j )
K
)
z
0
j
�
j ,k j
� 1� j,
k
j
� ∗∗j ,K� ∗j ,K
r 0
Figure 4 with r0=
r 0j
l j (
� ( j )
K
)
and
2r0j
lj(α
(j)
K )
· arccos
r0j
2lj(α
(j)
K )ρj,kj−1
for m = j. (3.6)
But for m ̸= j r0m
lm(α
(j)
K )
≤ ρm,km and
r0j
lj(α
(j)
K )
≤ ρj,kj−1 then the argument
in arccosine from (3.6) and (3.5) does not exceed 1
2 . This means that the
length of arc is not lesser than
2r0m
lm(α
(j)
K )
arccos
1
2
≥ 2θmπ
3lm(z0)λ2,m(β)
for every m ∈ {1, 2, . . . , n},
because L∈Q(Bn). Accordingly, the measure of the set Tn
(
α
(j)
K , R0
L(α
(j)
K )
)
∩
Dn[z0,RK−1j ] on the skeleton of polydisc is always not lesser than∏n
m=1
2θmπ
3lm(z0)λ2,m(β)
. Assuming a strict inequality in (3.1), we deduce that
(3.4) is valid.
Applying (3.3) mj-th times in every variable zj , we obtain
max
{
|F (z)| : z ∈ Tn
(
z0,
β −R
L(z0)
)}
≤ max{|F (z)| : z ∈ Tn(z0,RM0)}
≤ p2max{|F (z)| : z ∈ Tn(z0,RM0−1n)}
≤ pmn2 max{|F (z)| : z ∈ Tn(z0,RM0−mn1n)} ≤ . . . ≤
≤ pmn+1
2 max{|F (z)| : z ∈ Tn(z0,RM0−mn1n−1n−1)}
≤ p
mn+mn−1
2 max{|F (z)| : z ∈ Tn(z0,RM0−mn1n−mn−11n−1)} . . .
≤ p
∥M0∥
2 max{|F (z)| : z ∈ Tn(z0,R0)}
≤ p
∥S∗∥
2 max
{
|F (z)| : z ∈ Tn
(
z0,
R
L(z0)
)}
.
By Theorem 2.1 the function F has bounded L-index in joint variables.
A. I. Bandura, O. B. Skaskiv 185
Let us to denote c(z′, r) = {z ∈ D : |z − z′| = r
l(z′)}. For n = 1
Theorem 3.1 implies the following corollary.
Corollary 3.1. Let l ∈ Q(D), f : D → C be an analytic function.
If ∃r ∈ (0, β/2), ∃r′ ≥ 0, ∃p2 ≥ 1 ∃θ ∈ (0, r), such that ∀z0 ∈ D
∃r0 = r0(z0) ∈ [θ; r], and meas
{
c(z0, r0) ∩GR′(F )
}
< 2πθ
3l(z0)λ2(2r+2)
and
max
{
|f(z)| : z ∈ c(z0, r0)
}
≤ p2min
{
|f(z)| : z ∈ c(z0, r0) \Gr′(f)
}
(3.7)
then the function f has bounded l-index (here meas means the Lebesgue
measure on the circle).
In a some sense, this corollary is new even for analytic functions of
one variable because the circle c(z0, r0) can contain zeros of the function
f. Meanwhile, in corresponding theorems from [23,33] the circle c(z0, r0)
is chosen such that f(z) ̸= 0 for all z ∈ c(z0, r0).
4. Behavior of partial logarithmic derivatives
Theorem 4.1. Let L ∈ Q(Bn). If an analytic function F : Bn → C
satisfies the following conditions:
1) for every R ∈ Rn+, |R| ≤ β, there exists p1 = p1(R) > 0 such that
for all z ∈ Bn \GR(F ) and for all j ∈ {1, . . . , n}
1
|F (z)|
∣∣∣∣∂F (z)∂zj
∣∣∣∣ ≤ p1lj(z), (4.1)
2) for every R ∈ Rn+, |R| ≤ β, and R′ ≥ 0 there exists p2=p2(R,R
′)≥
1 that for all z0 ∈ Bn such that Tn(z0, R
L(z0))\GR′(F ) =
∪
iCi ̸= ∅,
where the sets Ci are connected disjoint sets, and either
a) max
i
min
z∈Ci
|F (z)| ≤ p2min
i
min
z∈Ci
|F (z)|, or
b) max
i
max
z∈Ci
|F (z)| ≤ p2min
i
max
z∈Ci
|F (z)|, or
c) |F (z∗)| = maximaxz∈Ci |F (z)|, |F (z∗∗)| = miniminz∈Ci |F (z)|,
and z∗, z∗∗ belong to the same set Ci0
3) for every R ∈ Rn+, |R| ≤ β, there exists Θ, R′ ∈ Rn+, 0 < θj <
2rj
2+3λ2,j(β)
, such that for all z ∈ Bn
meas {GR′(F ) ∩ Dn [z,R/L(z)]} <
(
2π
3
)n n∏
j=1
θj(rj − θj)
λ2,j(β)l2j (z)
,
(4.2)
186 Partial logarithmic derivatives and distribution...
then F has bounded L-index in joint variables (here meas is 2n-dimen-
sional the Lebesgue measure).
Proof. Let z0 ∈ Bn. In view of Theorem 3.1, we need to prove that
meas
{
Tn
(
z0,
R0
L(z0)
)
∩GR′(F )
}
<
(
2π
3
)n n∏
j=1
θj
λ2,j(β)lj(z0)
for some R0 = R0(z0).
Let dS = ds1 · . . . · dsn, S = (s1, . . . , sn), ωz be a volume measure in
R2n. By the Fubini–Tonelli theorem we have∫
Dn[z0,R]
u(z)dωz=
∫ r1
0
. . .
∫ rn
0
s1 . . . sn
(∫ 2π
0
. . .
∫ 2π
0
u(z0 + SeiΘ)dθ1. . .dθn
)
ds1. . .dsn =
∫ R
0
(∫ 2π
0
. . .
∫ 2π
0
u(z0 + SeiΘ)d(s1θ1) . . . d(snθn)
)
dS,
where u is measurable function. Let u(z) = χF (z) be a characteristic
function of the set GR′(F ) for the function F. We substitute R/L(z0)
instead R. Hence,
meas
{
Dn
[
z0,
R
L(z0)
]
∩GR′(F )
}
=
∫
Dn[z0,R/L(z0)]
χF (z)dωz
=
∫ R/L(z0)
0
∫
Tn(z0,S/L(z0))
χF (z)dµzdS
=
∫ R/L(z0)
0
meas
{
Tn(z0, S) ∩GR′(F )
}
dS, (4.3)
where µz is the measure on the skeleton of polydisc in Cn.
Combining (4.2) and (4.3), we obtain∫ R/L(z0)
0
meas
{
Tn(z0, S) ∩GR′(F )
}
dS
= meas
{
Dn
[
z0,
R
L(z0)
]
∩GR′(F )
}
<
(
2π
3
)n n∏
j=1
θj(rj − θj)
λ2,j(β)l2j (z
0)
. (4.4)
Besides, we have∫ Θ/L(z0)
0
meas
{
Tn(z0, S) ∩GR′(F )
}
dS
= meas
{
Dn
[
z0,
Θ
L(z0)
]
∩GR′(F )
}
≤πn
n∏
j=1
θ2j
l2j (z
0)
.
A. I. Bandura, O. B. Skaskiv 187
Thus, the following difference is positive(
2π
3
)n n∏
j=1
θj(rj − θj)
λ2,j(β)l2j (z
0)
−
∫ Θ/L(z0)
0
meas
{
Tn(z0, S) ∩GR′(F )
}
dS
≥
(
2π
3
)n n∏
j=1
θj(rj − θj)
λ2,j(β)l2j (z
0)
− πn
n∏
j=1
θ2j
l2j (z
0)
= πn
n∏
j=1
θj
l2j (z
0)
2rj − θj(2 + 3λ2,j(β))
3λ2,j(β)
> 0
because θj <
2rj
2+3λ2,j(β)
. From (4.4) it follows that
∫ R/L(z0)
Θ/L(z0)
meas
{
Tn(z0, S) ∩GR′(F )
}
dS <
(
2π
3
)n n∏
j=1
θj(rj − θj)
λ2,j(β)l2j (z
0)
−
∫ Θ/L(z0)
0
meas
{
Tn(z0, S) ∩GR′(F )
}
dS≤
(
2π
3
)n n∏
j=1
θj(rj − θj)
λ2,j(β)l2j (z
0)
.
(4.5)
By mean value theorem there exists R0 = R0(z0) with r0j ∈ [θj , rj ] such
that ∫ R/L(z0)
Θ/L(z0)
meas{Tn(z0, S)∩GR′(F )}dS
= meas{Tn(z0, R0/L(z0))∩GR′(F )}
n∏
j=1
rj−θj
lj(z0)
.
Hence, in view of (4.5) we obtain a desired inequality
meas
{
Tn(z0, R0/L(z0)) ∩GR′(F )
}
<
(
2π
3
)n n∏
j=1
θj
λ2,j(β)lj(z0)
.
Clearly, that for every point z0 ∈ Bn we have Tn
(
z0, R0
L(z0)
)
\ ZF =∪
iC
′
i, where C
′
i are connected disjoint sets, C ′
i ⊃ Ci and Ci is defined
in condition 2). Without loss of generality we assume that two any
points from C ′
i can be connected by a segment of line lying inside in
C ′
i. Otherwise we can split C ′
i by the sets with the property. Let z∗ ∈
Tn(z0, R/L(z0)) be such that |F (z∗)| = max
{
|F (z)| : z ∈ Tn
(
z0, R0
L(z0)
)}
.
Then there exists i0 that z∗ ∈ C ′
i0
. Let z∗∗ ∈ Ci0 ⊂ C ′
i0
be such
188 Partial logarithmic derivatives and distribution...
that |F (z∗∗)| = minz∈Ci0 |F (z)|. We connect the points z∗ and z∗∗ by
a piecewise-analytic curve z = z(t) = (z1(t), . . . , zn(t)), t ∈ [0; 1]. The
curve is chosen such that
∫ 1
0 |z′j(t)|dt ≤
2πrj
lj(z0)
. Integrating from z∗ to z∗∗,
we obtain
ln
∣∣∣∣ F (z∗)F (z∗∗)
∣∣∣∣≤ ∣∣∣∣ln F (z∗)
F (z∗∗)
∣∣∣∣=
∣∣∣∣∣
∫ z∗
z∗∗
d lnF (z)
∣∣∣∣∣=
∣∣∣∣∣∣
∫ z∗
z∗∗
n∑
j=1
1
F (z)
∂F (z)
∂zj
dzj
∣∣∣∣∣∣
≤
∣∣∣∣∣∣
∫ z∗
z∗∗
n∑
j=1
p1lj(z)|dzj |
∣∣∣∣∣∣≤
n∑
j=1
p1lj(z
0)λ2,j(R)
2πrj
lj(z0)
=2p1π
n∑
j=1
rjλ2,j(R).
Hence,
max
{
|F (z)| : z ∈ Tn
(
z0,
R0
L(z0)
)}
= |F (z∗)|
≤exp{2p1π
n∑
j=1
rjλ2,j(R)}|F (z∗∗)|=exp{2p1π
n∑
j=1
rjλ2,j(R)} min
z∈Ci0
|F (z)|
≤exp{2p1π
n∑
j=1
rjλ2,j(R)}p2min
i
min
z∈Ci
|F (z)|
=exp{2p1π
n∑
j=1
rjλ2,j(R)}p2min
{
|F (z)| : z ∈ Tn
(
z0,
R0
L(z0)
)
\GR′(F )
}
.
By Theorem 3.1 the function F has bounded L-index in joint variables.
Let us to denote ∆ as Laplace operator. We will consider ∆ln |F | as
generalized function. Using some known results from potential theory,
we can rewrite Theorem (4.1) in the following way
Theorem 4.2. Let L ∈ Q(Bn). If an analytic function F : Bn → C
satisfies the following conditions
1) for every R ∈ Rn+, |R| ≤ β, there exists p1 = p1(R) > 0 such that
for all z ∈ Bn \GR(F ) and for every j ∈ {1, . . . , n}∣∣∣∣∂ lnF (z)∂zj
∣∣∣∣ ≤ p1lj(z). (4.6)
2) for every R ∈ Rn+, |R| ≤ β, and R′ > 0 there exists p2=p2(R,R
′)≥
1 such that for all z0 ∈ Cn such that Tn(z0, R
L(z0)) \ GR′(F ) =∪
iCi ̸= ∅, where the sets Ci are connected disjoint sets, and either
A. I. Bandura, O. B. Skaskiv 189
a) max
i
min
z∈Ci
|F (z)| ≤ p2min
i
min
z∈Ci
|F (z)|, or
b) max
i
max
z∈Ci
|F (z)| ≤ p2min
i
max
z∈Ci
|F (z)|, or
c) |F (z∗)| = maximaxz∈Ci |F (z)|, |F (z∗∗)| = miniminz∈Ci |F (z)|,
and z∗, z∗∗ belong to the same set Ci0
3) for every R ∈ Rn+, |R| ≤ β, there exists Θ ∈ Rn+, 0 < θj <
2rj
2+3λ2,j(β)
, such that for all z ∈ Bn
∫
Dn[z,R/L(z)]
∆ln |F |dV2n ≤ 2π
(
2
3
)n n∏
j=1
θj(rj − θj)λ
2
1,j(R)
r2jλ2,j(β)
.
then F has bounded L-index in joint variables.
Proof. L. I. Ronkin [28, p. 230] deduced the following formula for entire
function: ∫
Dn[0,R∗]
∆ln |F |dV2n = 2π
∫
ZF∩Dn[0,R∗]
γF (z)dV2n−2,
where γF (z) is a multiplicity of zero point of the function F at point z,
R∗ ∈ Rn+ is arbitrary radius. Let χF (z) be a characteristic function of
zero set of F. Then χF (z) ≤ γF (z). Hence,
V2n
(
GR′(F )
∩
Dn(z,R/L(z))
)
≤
∫
z0∈ZF∩Dn(z,R/L(z))
V2n
(
Dn(z0, R/L(z0))
)
dV2n−2
≤ max
{
V2n
(
Dn(z0, R/L(z0))
)
: z0 ∈ ZF ∩ Dn(z,R/L(z))
}
×
∫
z0∈ZF∩Dn(z,R/L(z))
γF (z)dV2n−2
= max
{
V2n
(
Dn(z0, R/L(z0))
)
: z0 ∈ ZF ∩ Dn(z,R/L(z))
}
× 1
2π
∫
Dn[z, R
L(z)
]
∆ln |F |dV2n
190 Partial logarithmic derivatives and distribution...
≤ max
{
πn
n∏
j=1
r2j
l2j (z0)
: z0 ∈ ZF ∩ Dn(z,R/L(z))
}
×
(
2
3
)n n∏
j=1
θj(rj − θj)λ
2
1,j(R)
r2jλ2,j(β)
≤ πn
n∏
j=1
r2j
λ21,j(R)l
2
j (z)
(
2
3
)n n∏
j=1
θj(rj − θj)λ
2
1,j(R)
r2jλ2,j(β)
=
(
2π
3
)n n∏
j=1
θj(rj − θj)
λ2,j(β)l2j (z)
i.e. inequality (4.2) holds.
For n = 1 Theorem 4.1 implies the following corollary.
Corollary 4.1. Let l ∈ Q(D), f : D → C be an analytic function,
n(r, z0, f) be a number of zeros of the f in the disc |z− z0| ≤ r
l(z0)
. If the
function f satisfies the following conditions
1) for every r ∈ (0, β) there exists p1 = p1(r) > 0 such that for all
z ∈ D \Gr(f) ∣∣∣∣f ′(z)f(z)
∣∣∣∣ ≤ p1l(z),
2) for every r ∈ (0, β) and r′ ≥ 0 exists p2 = p2(r, r
′) ≥ 1 that
for all z0 ∈ D such that {z ∈ D : |z − z0| = r
l(z0)
} \ Gr′(f) =∪
iCi ̸= ∅, where the sets Ci are connected disjoint sets, and ei-
ther a) max
i
min
z∈Ci
|f(z)| ≤ p2min
i
min
z∈Ci
|f(z)|, or b) max
i
max
z∈Ci
|f(z)| ≤
p2min
i
max
z∈Ci
|f(z)|, or
c) |f(z∗)| = maximaxz∈Ci |f(z)|, |f(z∗∗)| = miniminz∈Ci |f(z)|,
and z∗, z∗∗ belong to the same set Ci0
3) for every r ∈ (0, β) there exist θ ∈ (0, 2r
2+3λ2(β)
), r′ > 0 such that
for all z ∈ D
n(r, z, f) <
2
3
θ(r − θ)
λ2(β)l2(z)r′2
then f has bounded l-index.
References
[1] A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in
joint variables, J. Math. Sci., 227 (2017), No. 1, 1–12, doi:10.1007/s10958-017-
3570-6.
A. I. Bandura, O. B. Skaskiv 191
[2] A. Bandura, O. Skaskiv, Sufficient conditions of boundedness of L-index and
analog of Hayman’s Theorem for analytic functions in a ball, Stud. Univ. Babeş-
Bolyai Math. 63 (2018), No. 4, 483–501.
[3] A. I. Bandura, O. B. Skaskiv, Analytic functions in the unit ball of bounded L-
index: asymptotic and local properties, Mat. Stud. 48 (2017), No. 1, 37–73, doi:
10.15330/ms.48.1.37-73.
[4] A. I. Bandura, O. B. Skaskiv, Entire functions of bounded L-index in direction,
Mat. Stud., 27 (2007), No. 1, 30–52 (in Ukrainian).
[5] A. Bandura, O. Skaskiv, Entire functions of bounded L-index: its zeros and behav-
ior of partial logarithmic derivatives, J. Complex Analysis, 2017 (2017), Article
ID 3253095, 10 p., doi:10.1155/2017/3253095.
[6] A. I. Bandura, Some improvements of criteria of L-index boundedness in direc-
tion, Mat. Stud. 47 (2017), No. 1, 27–32, doi: 10.15330/ms.47.1.27-32.
[7] A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index,
Lviv: Publisher I. E. Chyzhykov, 2016, 128 p. http://chyslo.com.ua.
[8] A. I. Bandura, O. B. Skaskiv, Directional logarithmic derivative and the distribu-
tion of zeros of an entire function of bounded L-index along the direction, Ukrain.
Mat. J., 69 (2017), No. 1, 500–508, doi:10.1007/s11253-017-1377-8.
[9] A. Bandura, O. Skaskiv, P. Filevych, Properties of entire solutions of some
linear PDE’s, J. Appl. Math. Comput. Mech., 16 (2017), No. 2, 17–28,
doi:10.17512/jamcm.2017.2.02.
[10] A. I. Bandura, M. T. Bordulyak, O. B. Skaskiv, Sufficient conditions of bound-
edness of L-index in joint variables, Mat. Stud., 45 (2016), No. 1, 12–26,
doi:10.15330/ms.45.1.12-26.
[11] A. I. Bandura, N. V. Petrechko, O. B. Skaskiv, Maximum modulus in a bidisc
of analytic functions of bounded L-index and an analogue of Hayman’s theorem,
Mathematica Bohemica, doi: 10.21136/MB.2017.0110-16 (in print).
[12] A. I. Bandura, N. V. Petrechko, Properties of power series of analytic in a bidisc
functions of bounded L-index in joint variables, Carpathian Math. Publ., 9 (2017),
No. 1, 6–12, doi:10.15330/cmp.9.1.6-12.
[13] T. O. Banakh, V. O. Kushnir, On growth and distribution of zeros of analytic
functions of bounded l-index in arbitrary domains, Mat. Stud., 14 (2000), No. 2,
165–170.
[14] M. T. Bordulyak, A proof of Sheremeta conjecture concerning entire function of
bounded l-index, Mat. Stud., 11 (1999), No. 2, 108–110.
[15] I. E. Chyzhykov, N. S. Semochko, Generalization of the Wiman-Valiron method
for fractional derivatives, Int. J. Appl. Math. 29 (2016), No. 2, 19–30.
[16] P. C. Fenton, Wiman-Valiron theory in several variables, Annales Academiae
Scientiarum Fennicae Mathematica, 38 (2013), No. 1, 29–47.
192 Partial logarithmic derivatives and distribution...
[17] G. H. Fricke, Entire functions of locally slow growth, J. Anal. Math., 28 (1975),
No. 1, 101–122.
[18] G. H. Fricke, S. M. Shah, On bounded value distribution and bounded index,
Nonlinear Anal. 2 (1978), No. 4, 423–435.
[19] G. H. Fricke, A note on bounded index and bounded value distribution, Indian J.
Pure Appl. Math. 11 (1980), No. 4, 428–432.
[20] W. K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44
(1973), No. 1, 117–137.
[21] G. J. Krishna, S. M. Shah, Functions of bounded indices in one and several com-
plex variables, In: Mathematical essays dedicated to A.J. Macintyre, Ohio Univ.
Press, Athens, Ohio, 1970, 223–235.
[22] A. D. Kuzyk, M. M. Sheremeta, Entire functions of bounded l-distribution of
values, Math. Notes. 39 (1986), No. 1, 3–8, doi: 10.1007/BF01647624.
[23] V. O. Kushnir, M. M. Sheremeta, Analytic functions of bounded l-index, Mat.
Stud., 12 (1999), No. 1, 59–66.
[24] B. Lepson, Differential equations of infinite order, hyperdirichlet series and ana-
lytic in Bn functions of bounded index, Proc. Sympos. Pure Math., Amer. Math.
Soc.: Providence, Rhode Island, 2 (1968), 298–307.
[25] F. Nuray, R. F. Patterson, Multivalence of bivariate functions of bounded index,
Le Matematiche, 70 (2015), No. 2, 225–233, doi: 10.4418/2015.70.2.14.
[26] R. Patterson, F. Nuray, A characterization of holomorphic bivariate functions of
bounded index, Mathematica Slovaca, 67 (2017), No. 3, 731–736, doi: 10.1515/ms-
2017-0005.
[27] N. Petrechko, Bounded L-index in joint variables and analytic solutions of some
systems of PDE’s in bidisc, Visn. Lviv Univ. Ser. Mech. Math., Issue 83 (2017),
100–108.
[28] L. I. Ronkin, Introduction to theory of entire functions of several variables. AMS,
Translations of mathematical monographs, V. 44, (1974).
[29] R. Roy, S. M. Shah, Functions of bounded index, bounded value distribution and
v-bounded index, Nonlinear Anal. 11 (1987), 1383–1390.
[30] M. Salmassi, Functions of bounded indices in several variables, Indian J. Math.,
31 (1989), No. 3, 249–257.
[31] T. M. Salo, O. B. Skaskiv, O. M. Trakalo, On the best possible description of ex-
ceptional set in Wiman-Valiron theory for entire function, Mat.Stud., 16 (2001),
No. 2, 131–140.
[32] S. M. Shah, Entire function of bounded index, Lect. Notes in Math., 599 (1977),
117–145.
A. I. Bandura, O. B. Skaskiv 193
[33] M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers,
1999, 141 p.
[34] M. N. Sheremeta, An l-index and an l-distribution of the values of entire func-
tions, Soviet Math. (Iz. VUZ) 34 (1990), No. 2, 115–117.
[35] S. N. Strochyk, M. M. Sheremeta, Analytic in the unit disc functions of bounded
index, Dopov. Akad. Nauk Ukr., (1993), No. 1, 19–22 (Ukrainian).
[36] M. N. Sheremeta, A. D. Kuzyk, Logarithmic derivative and zeros of an en-
tire function of bounded l-index, Sib. Math. J., 33 (1992), No. 2, 304–312, doi:
10.1007/BF00971102.
[37] Sh. Strelitz, Asymptotic properties of entire transcendental solutions of algebraic
differential equations, Contemp. Math. 25 (1983), 171–214.
Contact information
Andriy Ivanovych
Bandura
Ivano-Frankivsk National Technical
University of Oil and Gas,
Ivano-Frankivsk, Ukraine
E-Mail: andriykopanytsia@gmail.com
Oleh Bohdanovych
Skaskiv
Ivan Franko National University of Lviv,
Lviv, Ukraine
E-Mail: olskask@gmail.com
CoverUMB_V15_N2.pdf
Страница 1
Страница 2
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