Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables
We investigate the slice holomorphic functions of several complex variables that have a bounded L-index in some direction and are entire on every slice {z⁰ + tb : t ∈ C} for every z⁰ ∈ Cⁿ and for a given direction b ∈ Cⁿ \ {0}. For this class of functions, we prove some criteria of boundedness of th...
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irk-123456789-1694382020-06-14T01:26:44Z Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables Bandura, A. Skaskiv, O. We investigate the slice holomorphic functions of several complex variables that have a bounded L-index in some direction and are entire on every slice {z⁰ + tb : t ∈ C} for every z⁰ ∈ Cⁿ and for a given direction b ∈ Cⁿ \ {0}. For this class of functions, we prove some criteria of boundedness of the L-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function F : Cⁿ → C is the necessary and sufficient condition for the existence of a positive continuous function L : Cⁿ → R₊ such that F has a bounded L-index in direction. The authors are thankful to Professor S. Yu. Favorov (Kharkiv) for the formulation of interesting problem. 2019 Article Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables / A. Bandura, O. Skaskiv // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 154-180. — Бібліогр.: 31 назв. — англ. 1810-3200 2010 MSC. 32A10, 32A17, 32A37, 30H99, 30A05 http://dspace.nbuv.gov.ua/handle/123456789/169438 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We investigate the slice holomorphic functions of several complex variables that have a bounded L-index in some direction and are entire on every slice {z⁰ + tb : t ∈ C} for every z⁰ ∈ Cⁿ and for a given direction b ∈ Cⁿ \ {0}. For this class of functions, we prove some criteria of boundedness of the L-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the
existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function F : Cⁿ → C is the necessary and sufficient condition for the existence of a positive continuous function L : Cⁿ → R₊ such that F has a bounded L-index in direction. |
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Bandura, A. Skaskiv, O. |
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Bandura, A. Skaskiv, O. Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables Український математичний вісник |
| author_facet |
Bandura, A. Skaskiv, O. |
| author_sort |
Bandura, A. |
| title |
Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables |
| title_short |
Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables |
| title_full |
Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables |
| title_fullStr |
Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables |
| title_full_unstemmed |
Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables |
| title_sort |
some criteria of boundedness of the l-index in direction for slice holomorphic functions of several complex variables |
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Інститут прикладної математики і механіки НАН України |
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2019 |
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http://dspace.nbuv.gov.ua/handle/123456789/169438 |
| citation_txt |
Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables / A. Bandura, O. Skaskiv // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 154-180. — Бібліогр.: 31 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT banduraa somecriteriaofboundednessofthelindexindirectionforsliceholomorphicfunctionsofseveralcomplexvariables AT skaskivo somecriteriaofboundednessofthelindexindirectionforsliceholomorphicfunctionsofseveralcomplexvariables |
| first_indexed |
2025-07-15T04:15:33Z |
| last_indexed |
2025-07-15T04:15:33Z |
| _version_ |
1837684946190204928 |
| fulltext |
Український математичний вiсник
Том 16 (2019), № 2, 154 – 180
Some criteria of boundedness of L-index
in a direction for slice holomorphic functions
of several complex variables
Andriy Bandura, Oleh Skaskiv
(Presented by I. I. Skrypnik)
The paper is devoted to the 100th anniversary
of Georgii Dmitrievich Suvorov
Abstract. In this paper, we investigate slice holomorphic functions
of several complex variables having bounded L-index in a direction, i.e.
we consider functions which are entire on every slice {z0 + tb : t ∈
C} for every z0 ∈ Cn and for a given direction b ∈ Cn \ {0}. For
this function class we prove some criteria of boundedness of L-index in
the direction describing local behavior of maximum modulus, minimum
modulus of the slice holomorphic function and providing estimates of
logarithmic derivative and distribution of zeros. Moreover, we obtain
analogs of known Hayman’s theorem and logarithmic criteria. They are
applicable in analytic theory of differential equations. We also investigate
value distribution of these functions and present existence theorem for
this function class. It indicates that bounded multiplicity of zeros for
slice holomorphic function F : Cn → C is the necessary and sufficient
condition for existence of a positive continuous function L : Cn → R+
such that F has bounded L-index in the direction.
2010 MSC. 32A10, 32A17, 32A37, 30H99, 30A05.
Key words and phrases. Bounded index, bounded L-index in direc-
tion, slice function, holomorphic function, maximum modulus, minimum
modulus, bounded l-index, existence theorem, distribution of zeros, log-
arithmic derivative, directional derivative.
1. Introduction
The paper is addendum to [1]. There was introduced a concept of L-
index boundedness in direction for slice entire functions of several complex
Received 09.05.2019
The authors are thankful to Professor S. Yu. Favorov (Kharkiv) for the formula-
tion of interesting problem.
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
A. Bandura, O. Skaskiv 155
variables. Also there was deduced some criteria of L-index boundedness
in direction. Here we present some applications of these criteria to ob-
tain more useful criteria of L- index boundedness in direction. Among
them are logarithmic criterion, analog of Hayman’s theorem and estimate
of minimum modulus. It should be note that the fist two criteria have
applications in analytic theory of differential equations. Also we con-
sider value distribution of function belonging to this function class and
prove existence theorem. It demonstrates extent of the class, i.e. slice
holomorphic functions of bounded L-index in direction.
The paper is devoted to the following problem.
Problem 1.1. Is it possible to deduce main facts of theory of entire
functions having bounded L-index in the direction b ∈ Cn \ {0} for func-
tions which are holomorphic on the slices {z0 + tb : t ∈ C} and are joint
continuous?
Let us introduce some notations from [1]. Let R+ = (0,+∞), R∗
+ =
[0,+∞), 0 = (0, . . . , 0), b = (b1, . . . , bn) ∈ Cn \ {0} be a given direciton,
L : Cn → R+ be a continuous function, F : Cn → C an entire function.
The slice functions on a line {z0 + tb : t ∈ C} for fixed z0 ∈ C we will
denote as gz0(t) = F (z0 + tb) and lz0(t) = L(z0 + tb).
Let H̃n
b be a class of functions which are holomorphic on every slices
{z0+ tb : t ∈ C} for each z0 ∈ Cn and let Hn
b be a class of functions from
H̃n
b which are joint continuous.
The notation ∂bF (z) stands for the derivative of the function gz(t)
at the point 0, i.e. for every p ∈ N ∂pbF (z) = g
(p)
z (0), where gz(t) =
F (z + tb) is entire function of complex variable t ∈ C for given z ∈ Cn.
In this research, we will often call this derivative as directional derivative
because if F is entire function in Cn then the derivatives of the function
gz(t) matches with directional derivatives of the function F. Together
the hypothesis on joint continuity and the hypothesis on holomorphy in
one direction do not imply holomorphy in whole n-dimensional complex
space. We give some examples to demonstrate it. For n = 2 let f :
C → C be an entire function, g : C → C be a continuous function. Then
f(z1)g(z2), f(z1)±g(z2), f(z1 ·g(z2)) are functions which are holomorphic
in the direction (1, 0) and are joint continuous in C2. Moreover, if we have
performed an affine transformation{
z1 = b2z
′
1 + b1z
′
2,
z2 = b2z
′
1 − b1z
′
2
then the appropriate new functions are also holomorphic in the direction
(b1, b2) and are joint continuous in C2, where b1 ̸= 0, b2 ̸= 0.
156 Some criteria of boundedness of L-index...
A function F ∈ Hn
b is said [1] to be of bounded L-index in the direction
b, if there exists m0 ∈ Z+ such that for all m ∈ Z+ and each z ∈ Cn
inequality
|∂mb F (z)|
m!Lm(z)
≤ max
0≤k≤m0
|∂kbF (z)|
k!Lk(z)
, (1.1)
is true. The least such integer number m0, obeying (1.1), is called the
L-index in the direction b of the function F and is denoted by Nb(F,L).
If such m0 does not exist, then we put Nb(F,L) = ∞, and the function
F is called of unbounded L-index in the direction b in this case. If
L(z) ≡ 1, then the function F is said to be of bounded index in the
direction b and Nb(F ) = Nb(F, 1) is called the index in the direction b.
For n = 1, b = 1, L(z) = l(z), z ∈ C inequality (1.1) defines a function
of bounded l-index with the l-index N(F, l) ≡ N1(F, l) [19, 20], and if in
addition l(z) ≡ 1, then we obtain a definition of index boundedness with
index N(F ) ≡ N1(F, 1) [21, 22]. It is also worth to mention paper [31],
which introduces the concept of generalized index. It is quite close to the
bounded l-index. Let Nb(F,L, z
0) stands for the L-index in the direction
b of the function F at the point z0, i.e., it is the least integer m0, for
which inequality (1.1) is satisfied at this point z = z0. By analogy, the
notation N(f, l, z0) is defined if n = 1, i.e. in the case of functions of one
variable.
Note that the positivity and continuity of the function L are weak
restrictions to deduce constructive results. Thus, we assume additional
restrictions by the function L.
Let us denote
λb(η)
= sup
z∈Cn
sup
t1,t2∈C
{
L(z + t1b)
L(z + t2b)
: |t1 − t2| ≤
η
min{L(z + t1b), L(z + t2b)}
}
.
By Qnb we denote a class of positive continuous function L : Cn → R+,
satisfying the condition
(∀η ≥ 0) : λb(η) < +∞, (1.2)
Moreover, it is sufficient to require validity of (1.2) for one value η > 0.
For a positive continuous function l(t), t ∈ C, and η > 0 we define
λ(η) ≡ λb1 (η) in the cases when b = 1, n = 1, L ≡ l. As in [26], let
Q ≡ Q1
1 be a class of positive continuous functions l(t), t ∈ C, obeying
the condition 0 < λ(η) < +∞ for all η > 0.
Besides, we denote by ⟨a, c⟩ =
n∑
j=1
ajcj the Hermitian scalar product
in Cn, where a, c ∈ Cn.
A. Bandura, O. Skaskiv 157
2. Auxiliary propositions
Let L∗(z) be a positive continuous function in Cn. We denote L ≍ L∗,
if for some θ1, θ2, 0 < θ1 ≤ θ2 < +∞, and for all z ∈ Cn the following
inequalities hold θ1L(z) ≤ L∗(z) ≤ θ2L(z).
Proposition 2.1. ([1]) Let L ∈ Qnb, L ≍ L∗. A function F ∈ H̃n
b has
bounded L∗-index in the direction b if and only if F is of bounded L-index
in the direction b.
Theorem 2.1. ( [1]) Let L ∈ Qnb. A function F ∈ H̃n
b is of bounded
L-index in the direction b if and only if for each η > 0 there exist n0 =
n0(η) ∈ Z+ and P1 = P1(η) ≥ 1 such that for every z ∈ Cn there exists
k0 = k0(z) ∈ Z+, 0 ≤ k0 ≤ n0, and
max
{∣∣∣∂k0b F (z + tb)
∣∣∣ : |t| ≤ η
L(z)
}
≤ P1
∣∣∣∂k0b F (z)∣∣∣ . (2.1)
Theorem 2.2 ([1]). Let D be an arbitrary bounded domain in Cn, b ∈
Cn \ {0} be arbitrary direction. If L : Cn → R+ is continuous function,
F ∈ Hn
b and (∀p ∈ N) ∂pbF ∈ Hn
b and (∀z0 ∈ D) : F (z0 + tb) ̸≡ 0, then
Nb(F,L,D) <∞.
3. Estimate of maximum modulus by minimum modulus
Using Theorem 2.1, we will prove the next criterion of L-index bound-
edness in direction. Similar results was firstly deduced by G. H. Fricke [15]
for entire functions of bounded index. Further it was generalized for var-
ious classes of holomorphic functions [4, 25,26].
Theorem 3.1. Let L ∈ Qnb. A function F ∈ H̃n
b has bounded L-index
in the direction b if and only if for any r1 > 0, r2 > 0 (r1 < r2), there
exists P1 = P1(r1, r2) ≥ 1 such that for every z0 ∈ Cn
max
{
|F (z0 + tb)| : |t|= r2
L(z0)
}
≤ P1max
{
|F (z0+tb)| : |t|= r1
L(z0)
}
.
(3.1)
Proof. Our proof is based on the proof of appropriate theorem for entire
functions of bounded L-index in direction [6, 8].
Necessity. LetNb(F,L) < +∞. On the contrary, suppose that there
exist number r1 and r2, 0 < r1 < r2, such that for each P∗ ≥ 1 there
exists z∗ = z∗(P∗) ∈ Cn satisfying inequality
max
{
|F (z∗ + tb)| : |t| = r2
L(z∗)
}
>P∗max
{
|F (z∗ + tb)| : |t|= r1
L(z∗)
}
.
158 Some criteria of boundedness of L-index...
By Theorem 2.1 there exist n0 = n0(r2) ∈ Z+ and P0 = P0(r2) ≥ 1
such that for all z∗ ∈ Cn and some k0 = k0(t
∗, z∗) ∈ Z+, 0 ≤ k0 ≤ n0,
the following inequality holds
max
{∣∣∣∂k0b F (z∗ + tb)
∣∣∣ : |t| = r2/L(z
∗)
}
≤ P0|∂k0b F (z
∗)|. (3.2)
One should observe that for k0 = 0 the proof of necessity is obvious,
because (3.2) implies max
{
|F (z∗+ tb)| : |t| = r2/L(z
∗)
}
≤ P0|F (z∗)| ≤
P0max
{
|F (z∗ + tb)| : |t| = r1/L(z
∗)
}
.
Suppose that k0 > 0 and
P∗ = n0!
(
r2
r1
)n0
(
P0 +
r1
r2 − r1
)
+ 1. (3.3)
We choose t0 ∈ C such that |t0| = r1/L(z
∗) and
|F (z∗ + t0b)| = max {|F (z∗ + tb)| : |t| = r1/L(z
∗)} > 0,
and t0j ∈ C, |t0j | = r2/L(z
∗), be such that
|∂jbF (z
∗ + t0jb)| = max{|∂jbF (z
∗ + tb)| : |t|= r2/L(z
∗)},
j ∈ Z+. In the case |F (z∗+ t0b)| = 0 by uniqueness theorem for all t ∈ C
one has F (z∗ + tb) = 0. But it contradicts inequality (3). By Cauchy’s
inequality we deduce
|∂jbF (z
∗)|
j!
≤
(
L(z∗)
r1
)j
|F (z∗ + t0b)|, j ∈ Z+ (3.4)∣∣∣∂jbF (z∗ + t0jb)− ∂jbF (z
∗)
∣∣∣ = ∣∣∣∣∫ t0j
0
∂j+1
b F (z∗ + tb) dt
∣∣∣∣
≤
∣∣∣∂j+1
b F (z∗ + t0(j+1)b)
∣∣∣ r2
L(z∗)
. (3.5)
From inequalities (3.4) and (3.5) it follows that
|∂j+1b F (z∗+t0(j+1)b)| ≥
L(z∗)
r2
{
|∂jbF (z
∗+t0jb)|−|∂jbF (z
∗)|
}
≥ L(z∗+t∗b)
r2
∣∣∣∂jbF (z∗ + t0jb)
∣∣∣− j!Lj+1(z∗)
r2(r1)j
|F (z∗ + t0b)|,
A. Bandura, O. Skaskiv 159
j ∈ Z+. Hence, for k0 ≥ 1 we obtain
|∂k0b F (z
∗ + t0k0b)| ≥
L(z∗)
r2
|∂k0−1
b F (z∗ + t0(k0−1)b)|
−(k0−1)!Lk0(z∗)
r2(r1)k0−1
|F (z∗+t0b)|≥ . . .≥
Lk0(z∗)
(r2)k0
|F (z∗+t00b)|
−
(
0!
(r2)k0
+
1!
(r2)k0−1r1
+ . . .+
(k0−1)!
r2(r1)k0−1
)
Lk0(z∗)|F (z∗+t0b)|
=
Lk0(z∗)
(r2)k0
|F (z∗ + t0b)|
|F (z∗ + t00b)|
|F (z∗ + t0b)|
−
k0−1∑
j=0
j!
(
r2
r1
)j . (3.6)
From (3) one has |F (z∗ + t00b)|/|F (z∗ + t0b)| > P∗. Besides, the follow-
ing inequality is true
k0−1∑
j=0
j!
(
r2
r1
)j
≤ k0!
(
(r2/r1)
k0 − 1
r2/r1 − 1
)
≤ n0!
r1
r2 − r1
(
r2
r1
)n0
.
Applying (3.3), we get
|F (z∗+t00b)|
|F (z∗+t0b)|
−
k0−1∑
j=0
j!
rj2
rj1
>P∗−
n0!r1
r2 − r1
(
r2
r1
)n0
= n0!
(
r2
r1
)n0
P0 + 1.
In view of (3.2) and (3.4), from (3.6) it follows that
∣∣∣∂k0b F (z∗+t0k0b)∣∣∣> Lk0(z∗)
(r2)k0
(
P∗ − n0!
r1
r2 − r1
(
r2
r1
)n0
)(
r1
L(z∗)
)k0
×
|∂k0b F (z
∗)|
k0!
≥
(
r1
r2
)n0
(
P∗−n0!
r1
r2−r1
(
r2
r1
)n0
)
|∂k0b F (z
∗+t0k0b)|
n0!P0
.
Hence, P∗ < n0!
(
r2
r1
)n0
(
P0 +
r1
r2−r1
)
, and it contradicts (3.3).
Sufficiency. Choose any r1 ∈ (0, 1) and r2 ∈ (1,+∞). For given
z0 ∈ Cn we develop the function F (z0 + tb) in power series by powers t
F (z0 + tb)=
∞∑
m=0
bm(z
0)tm, bm(z
0)=
∂mb F (z
0)
m!
in the disc {t : |t| ≤ β/L(z0)}. For r > 0 we define Mb(r, z
0, F ) =
max{|F (z0 + tb)| : |t| = r}, µb(r, z0, F ) = max{|bm(z0)|rm : m ≥ 0},
νb(r, z
0, F ) = max{|bm(z0)|rm : |bm(z0)|rm = µb(r, z
0, F )}.
160 Some criteria of boundedness of L-index...
By Cauchy’s inequality µb(r, z
0, F )≤Mb(r, z
0, F ). But for r =
1
L(z0)
one has
Mb(r1r, z
0, F )≤
∞∑
m=0
|bm(z0)|rmrm1 ≤µb(r, z0, F )
∞∑
m=0
rm1 =
µb(r, z
0, F )
1− r1
and, applying monotonicity of νb(r, z
0, F ) in r, we get
lnµb(r2r, z
0, F )−lnµb(r, z
0, F )=
∫ r2r
r
νb(t, z
0, F )
t
dt≥νb(r, z0, F ) ln r2.
Hence, it follows that
νb(r, z
0, F ) ≤ 1
ln r2
(lnµb(r2r, z
0, F )− lnµb(r, z
0, F ))
≤ 1
ln r2
{lnMb(r2r, z
0, F )− ln((1− r1)Mb(r1r, z
0, F ))}
= − ln(1− r1)
ln r2
+
1
ln r2
{lnMb(r2r, z
0, F )− lnMb(r1r, z
0, F ))} (3.7)
Let Nb(F,L, z
0) be the L-index in direction of the function F at
the point z0, i.e, Nb(F,L, z
0) is the least number m0, for which in-
equality (1.1) holds at the point z = z0. Obviously that Nb(F,L, z
0) ≤
νb(1/L(z
0), z0, F ) = νb(r, z
0, F ). But inequality (3.1) can be rewritten
in the following form Mb
(
r2
L(z0)
, z0, F
)
≤ P1(r1, r2)Mb
(
r1
L(z0)
, z0, F
)
.
Thus, from (3.7) one has Nb(F,L, z
0) ≤ − ln(1−r1)
ln r2
+ lnP1(r1,r2)
ln r2
for ev-
ery z0 ∈ Cn, that is Nb(F,L) ≤ − ln(1−r1)
ln r2
+ lnP1(r1,r2)
ln r2
. Theorem 3.1 is
proved.
In view of proof of sufficiency in Theorem 3.1 the following lemma is
valid.
Lemma 3.1. Let L ∈ Qnb, F ∈ H̃n
b. If there exist numbers r1 and r2,
0 < r1 < 1 < r2, and P1 ≥ 1 such that for every z0 ∈ Bn inequality (3.1)
holds then the function F is of bounded L-index in the direction b.
We can relax sufficient conditions of Lemma 3.1, replacing the condi-
tion 0 < r1 < 1 < r2 < +∞ by 0 < r1 < r2 < +∞.
Proposition 3.1. Let L ∈ Qnb, F ∈ H̃n
b. If there exist r1 and r2, 0 <
r1 < r2 < +∞, and P1 ≥ 1 such that for all z0 ∈ Cn inequality (3.1)
holds, then the function has F bounded L-index in the direction b.
A. Bandura, O. Skaskiv 161
Proof. Our proof uses the idea of A. D. Kuzyk and M. M. Sheremeta [20].
They proposed the method to investigate of l-index boundedness of entire
solutions of linear differential equations.
Inequality (3.1) for 0 < r1 < r2 < +∞ implies
max
{
|F (z0 + tb)| : |t| = 2r2
r1 + r2
r1 + r2
2L(z0)
}
≤ P1max
{
|F (z0 + tb)| : |t| = 2r1
r1 + r2
r1 + r2
2L(z0)
}
.
Defining L∗(z) = 2L(z)
r1+r2
, we obtain
max
{
|F (z0 + tb)| : |t| = 2r2
(r1 + r2)L∗(z0)
}
≤P1max
{
|F (z0 + tb)| : |t| = 2r1
(r1 + r2)L∗(z0)
}
,
where 0 < 2r1
r1+r2
< 1 < 2r2
r1+r2
< +∞. This means that F has bounded
L∗-index in the direction b. And by Proposition 2.1 the function F has
bounded L-index in the direction b.
The following theorem gives estimate of maximum modulus by min-
imum modulus. It was firstly obtained by G. H. Fricke [15] for entire
functions of bounded index.
Theorem 3.2. Let L ∈ Qnb. If the function F ∈ H̃n
b is of bounded L-
index in the direction b then for each R > 0 there exist P2(R) ≥ 1 and
η(R) ∈ (0, R) such that for every z0 ∈ Cn and some r = r(z0) ∈ [η(R), R]
the inequality holds
max{|F (z0+tb)| : |t|=r/L(z0)}≤P2min{|F (z0+tb)| : |t|=r/L(z0)}.
(3.8)
Proof. Our proof is based on the proof of appropriate theorem for entire
functions of bounded L-index in direction [6].
Let Nb(F,L) = N < +∞ and R ≥ 0. Put
R0 = 1, r0 =
R
8(R+ 1)
, Rj =
Rj−1
4N
rNj−1, rj =
1
8
Rj(j = 1, 2, . . . , N).
Let z0 ∈ Cn and N0 = Nb(z
0, L, F ) be the L-index in the direction b of
the function F at the point z0, i.e., Nb(z
0, L, F ) is the least number m0,
for which inequality (1.1) holds for z = z0. The maximum in right-hand
162 Some criteria of boundedness of L-index...
side (1.1) is attained at m0. But 0 ≤ N0 ≤ N. For given z0 ∈ Cn the
function F (z0 + tb) can be developed in power series by powers t
F (z0+tb)=
∞∑
m=0
bm(z
0)tm, bm(z
0) =
∂mb F (z
0)
m!
.
Put am(z
0) = |bm(z0)|
Lm(z0)
=
|∂mb F (z0)|
m!Lm(z0)
. For every m ∈ Z+ the inequality
aN0(z
0) ≥ am(z
0) = R0am(z
0) is true. Then there exists the least num-
ber n0 ∈ {0, 1, . . . , N0} such that for all m∈Z+ an0(z
0)≥am(z0)RN0−n0 .
Thus, an0(z
0) ≥ aN0(z
0)RN0−n0 and aj(z
0) < aN0(z
0)RN0−j for j < n0,
because if aj0(z
0) ≥ aN0(z
0)RN0−j0 for some j0 < n0 then aj0(z
0) ≥
am(z
0)RN0−j0 for all m ∈ Z+, but it contradicts the choice of n0. From
inequalities aj(z
0) < aN0(z
0)RN0−j (j < n0) and am(z
0) ≤ aN0(z
0)
(m > n0) for t ∈ Sz0 and |t| = 1
L(z0)
rN0−n0 we deduce
|F (z0 + tb)| =
= |bn0(z
0)tn0 +
∑
m̸=n0
bm(z
0)tm| ≥ |bn0(z
0)||t|n0 −
∑
m̸=n0
|bm(z0)||t|m
=an0(z
0)rn0
N0−n0
−
∑
m̸=0
am(z
0)rmN0−n0
=an0(z
0)rn0
N0−n0
−
∑
j<n0
aj(z
0)rjN0−n0
−
∑
m>n0
am(z
0)rmN0−n0
≥aN0(z
0)RN0−n0r
n0
N0−n0
−
∑
j<n0
aN0(z
0)RN0−jr
j
N0−n0
−
∑
m>n0
aN0(z
0)rmN0−n0
≥ aN0(z
0)RN0−n0r
n0
N0−n0
− n0aN0(z
0)RN0−n0+1
−aN0(z
0)rn0+1
N0−n0
1
1− rN0−n0
=aN0(z
0)
(
RN0−n0r
n0
N0−n0
− n0
4N
RN0−n0r
N
N0−n0
−rn0
N0−n0
rN0−n0
1− rN0−n0
)
≥ aN0(z
0)
(
RN0−n0r
n0
N0−n0
− 1
4
RN0−n0r
n0
N0−n0
−1
4
RN0−n0r
n0
N0−n0
)
=
1
2
aN0(z
0)RN0−n0r
n0
N0−n0
. (3.9)
Besides, for t ∈ C we have
|F (z0 + tb)| ≤
+∞∑
m=0
|bm(z0)||t|m =
∞∑
m=0
am(z
0)rmN0−n0
≤ aN0(z
0)
+∞∑
m=0
rmN0−n0
=
aN0(z
0)
1− rN0−n0
≤ aN0(z
0)
1− 1/8
=
8
7
aN0(z
0). (3.10)
A. Bandura, O. Skaskiv 163
From (3.9) and (3.10) it follows
max
{
|F (z0 + tb)| : |t| = rN0−n0/L(z
0)
}
≤ 8
7
aN0(z
0)
≤ 16
7
1
RN0−n0
r−n0
N0−n0
min
{
|F (z0 + tb)| : |t| = rN0−n0
L(z0)
}
≤ 16
7
1
RN
r−NN min
{
|F (z0 + tb)| : |t| = rN0−n0/L(z
0)
}
,
i.e., (3.8) holds with P2(R) =
16
7RNrNN
, η(R) = rN =
1
8RN
and r =
rN0−n0 . Theorem is proved.
Below we will prove the sufficint conditions which are symmetric to
necessary conditions from Theorem 3.2
Theorem 3.3. Let L ∈ Qnb, F ∈ H̃n
b. If there exist R > 0, P2 ≥ 1
and η ∈ (0, R) such that for all z0 ∈ Cn and some r = r(z0) ∈ [η,R]
inequality (3.8) is valid, then the function F has bounded L-index in the
direction b.
Proof. Directly this proposition was unknown for entire functions of
bounded index, i.e. for functions of one variable. Firstly, it was ob-
tained for entire functions of bounded L-index in direction in [10,11]. In
view of Proposition 3.1 it is sufficient to show that there exists P1 such
that for all z0 ∈ Cn
max
{
|F (z0 + tb)| : |t| = (R+ 1)/L(z0)
}
≤ P1max
{
|F (z0 + tb)| : |t| = R/L(z0)
}
. (3.11)
Suppose that there exist R > 0, P2 ≥ 1 and η ∈
(
0, R
)
such that for all
z0 ∈ Cn and some r = r(z0) ∈
[
η,R
]
one has
max
{
|F (z0+tb)| : |t|=r/L(z0)
}
≤P2min
{
|F (z0 + tb)| : |t|=r/L(z0)
}
.
Denote L∗ = max
{
L(z0 + tb) : |t| ≤ (2R+ 2)/L(z0)
}
, ρ0 = R/L(z0),
ρk = ρ0 + kη/L∗, k ∈ Z+. We have
η
L∗ <
R
L∗ ≤ R
L(z0)
<
2R+ 2
L(z0)
− R+ 1
L(z0)
.
Then there exists n∗ ∈ N, independent of z0 such that
ρp−1 <
R+ 1
L(z0)
≤ ρp ≤
2R+ 2
L(z0)
,
164 Some criteria of boundedness of L-index...
for some p = p(z0) ≤ n∗, because L ∈ Qnb. Indeed,(
2R+ 2
L(z0)
− ρ0
)/( η
L∗
)
=
(R+ 2)L∗
ηL(z0)
=
R+ 2
η
max
{
L(z0 + tb)
L(z0)
: |t| ≤ 2R+ 2
L(z0)
}
≤ R+ 2
η
λb(2R+ 2).
Therefore, n∗ =
[
R+2
η λb(2R+ 2)
]
, where [a] is integer part of number
a ∈ R. Let |F (z0 + t∗∗k b)| = max{|F (z0 + tb)| : t ∈ ck}, ck = {t ∈ C :
|t| = ρk}, and t∗k be an intersection point of the segment [0, t∗∗k ] with the
circle ck−1. Then for every r > η and for each k ≤ n∗ the inequality
holds |t∗∗k − t∗k| =
η
L∗ ≤ r
L(z0+t∗kb)
. Thus, for some r = r(z0+ t∗kb) ∈ [η,R]
we deduce
|F (z0 + t∗∗k b)| ≤ max
{
|F (z0 + tb)| : |t− t∗k| = r/L(z0 + t∗kb)
}
≤ P2min
{
|F (z0 + tb)| : |t− t∗k| = r/L(z0 + t∗kb)
}
≤ P2min
{
|F (z0 + tb)| : |t− t∗k| = r/L(z0 + t∗kb), |t− t0| ≤ ρk−1
}
≤ P2max{|F (z0 + tb)| : t ∈ ck−1}.
Hence,
max
{
|F (z0 + tb)| : |t| = (R+ 1)/L(z0)
}
≤ max{|F (z0 + tb)| : t ∈ cp} ≤ P2max{|F (z0 + tb)| : t ∈ cp−1}
≤ . . . ≤ (P2)
pmax{|F (z0 + tb)| : t ∈ c0}
≤ (P2)
n∗
max
{
|F (z0 + tb)| : |t| = R/L(z0)
}
.
We obtained (3.11) with P1 = (P2)
n∗
. Theorem 3.3 is proved.
4. Estimate of directional logarithmic derivative
In this section we deduce analog of logarithmic criterion for function
from the class H̃n
b. The one-dimensional analog of the criterion is efficient
to investigate boundedness of l-index of infinite products [12, 28, 30]. As
necessary conditions the criterion was obtained by G. H. Fricke [14, 15]
for entire functions of bounded index.
Below we prove the criterion of L-index boundedness in direction,
which describes behavior of directional logarithmic derivative and distri-
bution of zeros. We need additional denotations.
Denote
Gr(F ) := Gb
r (F ) :=
∪
z : F (z)=0
{z + tb : |t| < r/L(z)}, (4.1)
A. Bandura, O. Skaskiv 165
where a0k are zeros of the function F (z0 + tb) for given z0 ∈ Cn.
By n
(
r, z0, 1/F
)
=
∑
|a0k|≤r
1 we denote counting function of zeros a0k.
Theorem 4.1. Let F ∈ H̃n
b, L ∈ Qnb. If the function F has bounded
L-index in the direction b, then
1) for each r > 0 there exists P = P (r) > 0 such that for every
z ∈ Cn\Gb
r (F ) ∣∣∣∣∂bF (z)F (z)
∣∣∣∣ ≤ PL(z); (4.2)
2) for any r > 0 there exists ñ(r) ∈ Z+ such that for all z0 ∈ Cn such
that F (z0 + tb) ̸≡ 0 one has
n
(
r
L(z0)
, z0,
1
F
)
≤ ñ(r). (4.3)
Proof. Out proof is based on the proof of appropriate proposition for
entire functions of bounded L-index in direction [6, 8].
Firstly, we will show that the condition “F (z) is of bounded L-index
in the direction” implies that for every z0 ∈ Cn\Gb
r (F ) (r > 0) and for
each ãk = z0 + a0kb one has
|z0 − ãk| >
r|b|
2L(z̃0)λb (r)
. (4.4)
On the contrary, suppose that there exist z0 ∈ Cn\Gb
r (F ) and ã
k = z0+
a0kb such that |z0− ãk| ≤
r|b|
2L(z̃0)λb (r)
≤ r|b|
2L(z0)
<
r|b|
L(z0)
. Hence, |a0k| <
r
L(z0)
. But for λb2 the following estimates hold L(ãk) ≤ λb (r)L(z
0). Then
|z0 − ãk| = |b| · |a0k| ≤ r|b|
2L(ãk)
, i.e., |a0k| ≤ r
2L(ãk)
. This contradicts
z̃0 ∈ Cn\Gb
r (F ).
Put in Theorem 3.2 R =
r
2λb (r)
. Then there exist P2 ≥ 1 and η ∈
(0, R) such that for every z̃0 = z0 ∈ Cn and for some r∗ ∈ [η,R] inequality
(3.8) holds with r∗ instead r. Therefore, by Cauchy’s inequality
|∂bF (z0)| ≤
L(z0)
r∗
max
{
|F (z0 + tb) : |t| = r∗/L(z0)
}
≤ P2
L(z0)
η
min{|F (z0 + tb)| : |t| = r∗/L(z0)}. (4.5)
166 Some criteria of boundedness of L-index...
In view of (4.4) for every point z0 + t0b ∈ Cn\Gb
r (F ) the following
set
{
z0 + tb : |t| ≤ r
2λb2 (r)L(z
0)
}
does not contain zeros of the function
F (z0+ tb). Therefore, applying the maximum modulus principle to 1/F
as a function of variable t, we have
|F (z0)| ≥ min
{
|F (z0 + tb)| : |t| = r∗/L(z0)
}
(4.6)
From inequalities (4.5) and (4.6) it follows (4.2) with P =
P2
η
.
Now we will prove that for a function F of bounded L-index in the
direction b there exists P3 > 0 such that for every z0 ∈ Cn (F (z0+ tb) ̸≡
0), r ∈ (0, 1]
n(r/L(z0), z0, 1/F )min
{
|F (z0+tb)| : |t| = r/L(z0)
}
≤ P3max
{
|F (z0 + tb)| : |t| = 1/L(z0)
}
. (4.7)
Applying Cauchy’s inequality and Theorem 3.1 for all t on the circle
|t| = 1
L(z0)
, we have
∣∣∣∂bF (z0+tb)∣∣∣≤ L(z0)
r
max
{
|F (z0 + θb)| : |θ−t|= r
L(z0)
}
≤ L(z0)
r
max
{
|F (z0 + tb)| : |t| = r + 1
L(z0)
}
≤ P1(1, r + 1)
r
L(z0)max
{
|F (z0+tb)| : |t| = 1
L(z0)
}
. (4.8)
If F (z0 + tb) ̸= 0 on the circle
{
t ∈ C : |t| = r/L(z0)
}
, then
n
(
r
L(z0)
, z0,
1
F
)
=
∣∣∣∣ 1
2π i
∫
|t|= r
L(z0)
∂bF (z
0 + tb)
F (z0 + tb)
dt
∣∣∣∣
≤
max
{∣∣∂bF (z0 + tb)
∣∣ : |t| = r/L(z0)
}
min {|F (z0 + tb)| : |t| = r/L(z0)}
r
L(z0)
. (4.9)
From (4.8) and (4.9) we deduce
n
(
r/L(z0), z0, 1/F
)
min
{
|F (z0 + tb)| : |t| = r/L(z0)
}
≤ r
L(z0)
max
{
|∂bF (z0 + tb)| : |t| = r/L(z0)
}
≤ 1
L(z0)
max
{
|∂bF (z0 + tb)| : |t| = 1/L(z0)
}
≤ P1(1, r + 1)/(r)max
{
|F (z0 + tb)| : |t| = 1/L(z0)
}
.
A. Bandura, O. Skaskiv 167
Thus, we obtained (4.7) with P3 =
P1(1, r + 1)
r
. If the function F (z0+tb)
has zeros on the circle {t ∈ C : |t| = r/L(z0)}, then inequality (4.7) is
obvious.
Set R = 1 in Theorem 3.2. Then there exist P2 = P2(1) ≥ 1 and
η ∈ (0, 1) such that for every z0 ∈ Cn and some r∗ = r∗(z0, t0) ∈ [η, 1]
max
{
|F (z0+tb)| : |t| = r∗
L(z0)
}
≤P2min
{
|F (z0+tb)| : |t|= r∗
L(z0)
}
.
Next, by Theorem 3.1 there exists P1 ≥ 1 such that for all z0 ∈ Bn
max
{
|F (z0 + tb)| : |t| = 1/L(z0)
}
≤ P1(1, η)max
{
|F (z0 + tb)| : |t| = η/L(z0)
}
≤ P1(1, η)max
{
|F (z0 + tb)| : |t| = r∗/L(z0)
}
≤ P1(1, η)P2min
{
|F (z0 + tb)| : |t| = r∗/L(z0)
}
.
Taking into account (4.7), we obtain
n
(
r∗/L(z0), z0, 1/F
)
min
{
|F (z0 + tb)| : |t| = r∗/L(z0)
}
≤ P3P1(1, η)P2min
{
|F (z0 + tb)| : |t| = r∗/L(z0)
}
,
i.e., n
(
r∗
L(z0)
, z0, 1
F
)
≤ P1(1, η)P2P3. Hence,
n
(
r∗
L(z0)
, z0,
1
F
)
≤ P4 = P1(1, η)P2P3 =
P1(1, η)P2(1)P1(1, r + 1)
r
.
If r ∈ (0, η], then property (4.3) is proved.
Let r > η and L∗ = max
{
L(z0 + tb) : |t| = r
L(z0)
}
. Using properties
Qnb, we have L∗ ≤ λb(r)L(z
0). Put ρ = η
L(z0)λb(r)
, R = r
L(z0)
. We can
cover every set K = {z0 + tb : |t| ≤ R} by a finite number m = m(r)
of closed sets Kj = {z0 + tb : |t − tj | ≤ ρ}, where tj ∈ K. Since
η
λb(r)L(z0)
≤ η
L∗ ≤ η
L(z0+tjb)
every set Kj contains at least [P4] zeros of
the function F (z0+tb). Therefore, n
(
r
L(z0)
, z0, 1/F
)
≤ ñ(r) = [P4]m(r)
and property (4.3) is proved.
By nz0(r, F ) = nb
(
r, z0, 1/F
)
:=
∑
|a0k|≤r
1 we denote counting func-
tion of zeros a0k for the slice function F (z0 + tb) in the disc {t ∈ C : |t| ≤
r}. If for given z0 ∈ Cn and for all t ∈ C F (z0 + tb) ≡ 0, then we put
nz0(r) = −1. Denote n(r) = supz∈Cn nz(r/L(z)).
168 Some criteria of boundedness of L-index...
Theorem 4.2. Let L ∈ Qnb, F ∈ H̃n
b. If the following conditions are
satisfied
1) there exists r1 > 0 such that n(r1) ∈ [−1;∞);
2) there exist r2 ∈> 0, P > 0 such that 2r2 · n(r1) < r1/λb(r1) and
for all z ∈ Cn\Gr2(F ) inequality (4.2) is true;
then the function F has bounded L-index in the direction b.
Proof. Analog of the proposition was firstly deduced for entire functions
of bounded L-index in direction [10,11]. Suppose that conditions 1) and
2) are true.
At first, we consider the case n(r1) ∈ {−1; 0}. Then in the best case
the function F can only identically equals zero on the complex line z∗+tb
for some z∗ ∈ Cn, i.e., F (z∗+tb) ≡ 0. For all points lying on such complex
lines inequality (3.8) is obvious.
Let z0 ∈ Cn \ Gr2 . For any points t1 and t2 such that |tj | = r2
L(z0)
,
j ∈ {1, 2}, one has
ln
∣∣∣∣F (z0 + t2b)
F (z0 + t1b)
∣∣∣∣ ≤ ∫ t2
t1
∣∣∣∂bF (z0 + tb)
F (z0 + tb)
∣∣∣|dt|
≤ P
∫ t2
t1
L(z0 + tb)|dt| ≤ Pλb (r2)L(z
0)
πr2
L(z0)
≤ πr2Pλb (r2)
(we also use that L ∈ Qnb). Hence,
max
{
|F (z0+tb)| : |t| = r2
L(z0)
}
≤P2min
{
|F (z0+tb)| : |t| = r1
L(z0)
}
,
where P2 = exp {πr2 Pλ2 (r2)} . Therefore, by Theorem 3.3 the function
F has bounded L-index in the direction b.
Let r1 > 0 be a such that n(r1) ∈ [1;∞) and 2n(r1)r2 < r1/λb(r1).
Put c = r1
2r2λb(r1)
− n(r1) > 0. Clearly, r2= r1/(2(n(r1)+c)λb(r1)).
Under condition 1) each set K =
{
z0+tb : |t| ≤ r1
L(z0)
}
has no more
n(r1) zeros of the function F, where F (z0 + tb) ̸≡ 0.
Under condition 2) there exists P > 0 such that |∂bF (z)
F (z) | ≤ PL(z) for
every z ∈ Cn\Gr2 , i.e., for all z ∈ K, lying outside the sets{
z0 + tb : |t− a0k| <
r2
L(z0 + a0kb)
}
,
where a0k ∈ K are zeros of the slice function F (z0+tb) ̸≡ 0. By definition
λb we obtain
L(z0)/λb(r1) ≤ L(z0 + a0kb).
A. Bandura, O. Skaskiv 169
Then |∂bF (z)
F (z) | ≤ PL(z) for every point z ∈ Cn, lying outside union of the
sets
c0k =
{
z0 + tb : |t− a0k| ≤
r2λb(r1)
L(z0)
=
r1
2(n(r1) + c)L(z0)
}
.
The total sum of diameters of the sets c0k does not exceed the value
r1n(r1)
(n(r1)+c)L(z0)
< r1
L(z0)
. Hence, there exists a set c̃0=
{
z0 + tb : |t|= r
L(z0)
}
,
where r1 min{1,c}
2(n(r1)+c)
= η < r < r1, such that, for all z ∈ c̃0∣∣∣∣∂bF (z)F (z)
∣∣∣∣ ≤ PL(z) ≤ Pλb(r)L(z
0) ≤ Pλb (r1)L(z
0).
For any points z1 = z0 + t1b and z2 = z0 + t2b with c̃0 one has
ln
∣∣∣∣F (z0 + t2b)
F (z0 + t1b)
∣∣∣∣ ≤ ∫ t2
t1
∣∣∣∣∂bF (z0 + tb)
F (z0 + tb)
∣∣∣∣|dt|
≤ Pλ2 (r1)L(z
0)
πr
L(z0)
≤ πr1 P (r2)λb (r1) .
Therefore,
max
{
|F (z0 + tb)| : |t| = r
L(z0)
}
≤P2min
{
|F (z0 + tb)| : |t| = r
L(z0)
}
,
(4.10)
where P2 = exp {πr1 P (r2)λb (r1)} . If F (z0 + tb) ≡ 0, then inequality
(4.10) is obvious. By Theorem 3.3 the function F (z) has bounded L-
index in the direction b. Theorem 4.2 is proved.
5. Analog of Hayman’s Theorem
Below we formulate and prove criterion which is analog of Hayman’s
Theorem [18].
Theorem 5.1. Let L ∈ Qnb. A function F ∈ H̃n
b is of bounded L-index
in the direction b if and only if there exist p ∈ Z+ and C > 0 such that
for every z ∈ Cn one has
|∂p+1
b F (z)|
Lp+1(z)
≤ Cmax
{
|∂kbF (z)|
Lk(z)
: 0 ≤ k ≤ p
}
. (5.1)
170 Some criteria of boundedness of L-index...
Proof. The proof use ideas from the proof for entire functions of bounded
L-index in direction [6, 8]. Also there are known analogs of Hayman’s
theorem for other classes of analytic functions [4, 9].
Necessity. If Nb(F,L) < +∞, then by definition of boundedness
of L-index in direction we obtain (5.1) with p = Nb(F,L) and C =
(Nb(F,L) + 1)!
Sufficiency. Let inequality (5.1) be fulfilled, z0 ∈ Cn and K ={
t ∈ C : |t|≤1/L(z0)
}
. Since L ∈ Qnb, for every t ∈ K from (5.1) it follows
|∂p+1
b F (z0 + tb)|
Lp+1(z0)
≤
(
L(z0 + tb)
L(z0)
)p+1 |∂p+1
b F (z0 + tb)|
Lp+1(z0 + tb)
≤(λb(1))
p+1 |∂
p+1
b F (z0 + tb)|
Lp+1(z0 + tb)
≤C(λb(1))p+1 max
0≤k≤p
{
|∂kbF (z0 + tb)|
Lk(z0 + tb)
}
≤ C(λb(1))
p+1max
{(
L(z0)
L(z0 + tb)
)k |∂kbF (z0 + tb)|
Lk(z0)
: 0 ≤ k ≤ p
}
≤ C(λb(1))
p+1max
{
|∂kbF (z0 + tb)|
Lk(z0)
(λb(1))
k : 0 ≤ k ≤ p
}
≤ Bgz0(t),
(5.2)
where B = C(λb(1))
2p+1 and gz0(t)=max
{
|∂kbF (z0+tb)|
Lk(z0)
: 0 ≤ k ≤ p
}
.
Let us denote γ1=
{
t ∈ C : |t| = 1
4L(z0)
}
, γ2 =
{
t ∈ C : |t| = 2
L(z0)
}
.
Choose arbitrarily two points t1 ∈ γ1, t2 ∈ γ2 and connect them by a
piecewise analytic curve γ = (t = t(s), 0 ≤ s ≤ T ) such that gz0(t) ̸= 0
for t ∈ γ. We construct the curve γ such that its length |γ| does not
exceed
9
2L(z0)
.
The function gz0(t(s)) is continuous on [0, T ]. Without loss of gen-
erality we may assume that the function t = t(s) is analytic on [0, T ].
Otherwise, one can consider each interval of analyticity of this function
separately and repeat the corresponding considerations, which are given
below on [0, T ]. First, we show that the function gz0(t(s)) is continu-
ously differentiable on [0, T ] except possibly a finite set of points. For
arbitrary k1, k2, 0 ≤ k1 ≤ k2 ≤ p, either
|∂k1b F (z0+t(s)b)|
Lk1 (z0)
≡ |∂k2b F (z0+t(s)b)|
Lk2(z0)
ot the equality
|∂k1b F (z0+t(s)b)|
Lk1 (z0)
=
|∂k2b F (z0+t(s)b)|
Lk2 (z0)
is true for a finite set of
points sk ∈ [0, T ]. Then we can split the segment [0, T ] onto a finite num-
ber of segments such that on each of them gz0(t(s)) ≡
|∂kbF (z0+t(z)b)|
Lk(z0)
for
some k, 0 ≤ k ≤ p. It means that the function gz0(t(s)) is continuously
A. Bandura, O. Skaskiv 171
differentiable with exception, perhaps, of a finite sets of points. Taking
into account (5.2), we obtain
dgz0(t(s))
ds
≤ max
{
d
ds
(
|∂kbF (z0 + t(s)b)|
Lk(z0)
)
: 0 ≤ k ≤ p
}
≤ max
{
|∂k+1
b F (z0 + t(s)b)||t′(s)|/Lk(z0) : 0 ≤ k ≤ p
}
= L(z0)|t′(s)|max
{
|∂k+1
b F (z0 + t(s)b)|/Lk+1(z0) : 0 ≤ k ≤ p
}
≤ Bgz0(t(s))|t′(s)|L(z0).
Hence, we have∣∣∣∣ln gz0(t2)gz0(t1)
∣∣∣∣= ∣∣∣∣∫ T
0
dgz0(t(s))
gz0(t(s))
∣∣∣∣ ≤BL(z0)∫ T
0
|t′(s)|ds=BL(z0)|γ|≤4.5B.
If we choose a point t2 ∈ γ2, such that
|F (z0 + t2b)| = max
{
|F (z0 + tb)| : |t| = 2/L(z0)
}
,
then we obtain
max
{
|F (z0+tb)| : |t| = 2
L(z0)
}
≤gz0(t2)≤gz0(t1) exp{9B/2}. (5.3)
Applying Cauchy’s inequality and using that t1 ∈ γ1 we obtain for
all j ∈ {1, . . . , p}
|∂jbF (z
0+t1b)|≤j!(4L(z0))j max
{
|F (z0+tb) : |t−t1| =
1
4L(z0)
}
≤j!(4L(z0))jmax
{
|F (z0 + tb) : |t− t0| =
1
2L(z0)
}
,
gz0(t1) ≤ p!(4)pmax
{
|F (z0 + tb) : |t− t0| =
1
2L(z0)
}
.
Therefore, from (5.3) it follows that
|F (z0 + t2b)| = max
{
|F (z0 + tb)| : |t| = β/L(z0)
}
≤gz0(t2)≤gz0(t1) exp
{
9B/2
}
≤p!(4)p exp
{
9B/2
}
×max
{
|F (z0 + tb)| : |t| = 1/(2L(z0))
}
.
By Proposition 3.1 we conclude that the function F has bounded L-index
in the direction b ∈ Cn. Theorem 5.1 is proved.
172 Some criteria of boundedness of L-index...
Using Theorem 5.1 we prove the following
Theorem 5.2. Let L ∈ Qnb. A function F ∈ H̃n
b has bounded L-index
in the direction b if and only if there exist numbers C ∈ (0,+∞) and
N ∈ N such that for all z ∈ Cn
N∑
k=0
|∂kbF (z)|
k!Lk(z)
≥ C
∞∑
k=N+1
|∂kbF (z)|
k!Lk(z)
. (5.4)
Proof. Proof of this theorem is similar to proof of its analogs for entire
functions of bounded L-index in direction [3] and for entire functions of
bounded l-index [25].
Let 0 < θ < 1. If the function F is of bounded L-index in the direction
b, then by Lemma 2.1 F is also of bounded L∗-index in the direction b,
where L∗(z) = θL(z). Denote N∗ = Nb(F,L∗) and N = Nb(F,L). Thus,
max
{
|∂kbF (z)|
k!Lk(z)
: 0 ≤ k ≤ N∗
}
= max
{
|∂kbF (z)|
k!Lk∗(z)
θk : 0 ≤ k ≤ N∗
}
≥ θN
∗
max
{
|∂kbF (z)|
k!Lk∗(z)
: 0 ≤ k ≤ N∗
}
≥ θN
∗ |∂jbF (z)|
j!Lj∗(z)
= θN
∗−j |∂
j
bF (z)|
j!Lj(z)
for all j ≥ 0 and
∞∑
j=N∗+1
|∂jbF (z)|
j!Lj(z)
≤ max
{
|∂kbF (z)|
k!Lk(z)
: 0 ≤ k ≤ N∗
} ∞∑
j=N∗+1
θj−N
∗
=
θ
1− θ
max
{
|∂kbF (z)|
k!Lk(z)
: 0 ≤ k ≤ N∗
}
≤ θ
1− θ
N∗∑
k=0
|∂kbF (z)|
k!Lk(z)
,
i.e. we obtain (5.4) with N = N∗ and C = 1−θ
θ .
Now we prove the sufficiency. From (5.4) we obtain
|∂N+1
b F (z)|
(N + 1)!LN+1(z)
≤
∞∑
k=N+1
|∂kbF (z)|
k!Lk(z)
≤ 1
C
N∑
k=0
|∂kbF (z)|
k!Lk(z)
≤ N + 1
C
max
{
|∂kbF (z)|
k!Lk(z)
: 0 ≤ k ≤ N
}
.
Applying Theorem 5.1, we obtain the desired conclusion.
Using Theorems 2.2 and 5.1 we obtain this corollary.
A. Bandura, O. Skaskiv 173
Corollary 5.1. Let L ∈ Qnb, F ∈ Hn
b, (∀p ∈ N) ∂pbF ∈ Hn
b, G be a
bounded domain in Cn such that ∀z ∈ G F (z + tb) ̸≡ 0. The function F
has bounded L-index in the direction b if and only if there exist p ∈ Z+
and C > 0 such that for all z ∈ Cn \G the inequality (5.1) holds.
6. Functions having bounded value L-distribution
in direction
An entire function f(z) (z ∈ C) is said ot be of bounded value dis-
tribution [16, 18, 24], if there exist p ≥ 0, R > 0 such that the equation
f(z) = w has at least p roots in any disc of radius R.
One of the remarkable properties generating big interest to functions
of bounded index is the following fact proved by W. Hayman [18]: an
entire function has bounded value distribution if and only if it deriva-
tive has bounded index. Leter, there was introduced a concept of entire
function of bounded value l-distribution [19], and this property was gen-
eralized for entire functions of bounded l-index [27]. For entire bivariate
functions of bounded index in joint variables similar results are partially
obtained in [23].
Definition 6.1. Function F ∈ H̃n
b is said to be of bounded value L-
distribution in a direction b if for all p ∈ N ∀w ∈ C ∀z0 ∈ Cn such
that F (z0 + tb) ̸≡ w, the inequality holds n
(
1
L(z0)
, z0, 1
F−w
)
≤ p, i.e.
the equation F (z0 + tb) = w has at most p solutions in the disc {t :
|t| ≤ 1
L(z0)
}. In other words, the function F (z0 + tb) is p-valent in
{t : |t| ≤ 1
L(z0)
}.
The corresponding Sheremeta result [27] we will generalize for the
functions from the class H̃n
b, which have bounded value L-distribution in
direction b.
Proposition 6.1. Let L ∈ Qnb. An entire function F ∈ H̃n
b is a function
of bounded value L-distribution in the direction b if and only if the direc-
tional derivative ∂bF ∈ H̃n
b has bounded L-index in the same direction
b.
Proof. This is similar to the proof of the corresponding proposition for
analytic functions in the unit ball [2].
Suppose that F is of bounded value L-distribution in direction b, i.e.
for all z0 ∈ Cn such that F (z0 + tb) ̸≡ const the function F (z0 + tb) is
p-valent in every disc {t : |t| ≤ 1
L(z0)
}.
To prove the proposition we need the following proposition from [26,
p. 48, Theorem 2.8].
174 Some criteria of boundedness of L-index...
Theorem 6.1 ( [26]). Let D0 = {t : |t − t0| < R}, 0 < R < ∞. If
analytic in D0 function f is p-valent in D0, then for j > p
|f (j)(t0)|
j!
Rj ≤ (Aj)2pmax
{
|f (k)(t0)|
k!
Rk : 1 ≤ k ≤ p
}
, (6.1)
where A = 2p
√
p+2
2
√
8eπ2 .
By Theorem 6.1 inequality (6.1) holds with R = 1
L(z0)
for the function
F (z0+ tb), as a function of single variable t ∈ C for every fixed z0 ∈ Cn.
Then it is easy to deduce that for every m ∈ N the following equality
g
(p)
z0
(t) = ∂pbF (z
0 + tb) holds. Take j = p+1 and t0 = 0 in Theorem 6.1.
From (6.1) it follows∣∣∣∂p+1
b F (z0)
∣∣∣
(p+ 1)!Lp+1(z0)
≤(A(p+ 1))2pmax
{∣∣∂kbF (z0)∣∣
k!Lk(z0)
: 1 ≤ k ≤ p
}
⇒∣∣∣∂p+1
b F (z0)
∣∣∣
Lp+1(z0)
≤ (p+ 1)!(A(p+ 1))2pmax
{∣∣∂kbF (z0)∣∣
Lk(z0)
: 1 ≤ k ≤ p
}
×max
{
1
k!
: 1 ≤ k ≤ p
}
⇒
∣∣∂pb∂bF (z0)∣∣
Lp(z0)
≤ L(z0) · (p+ 1)!A2p(p+ 1)2pmax
∣∣∣∂k−1
b ∂bF (z
0)
∣∣∣
Lk(z0)
:
0 ≤ k − 1 ≤ p− 1} ⇒
∣∣∂pb∂bF (z0)∣∣
Lp(z0)
≤ (p+ 1)!A2p(p+ 1)2pmax
∣∣∣∂k−1
b ∂bF (z
0)
∣∣∣
Lk−1(z0)
: 0 ≤ k − 1 ≤ p− 1
.
Now we need analog of Hayman’s Theorem proved above. Thus, for ∂bF
inequality (5.1) holds with p− 1 instead p and with C = (p+1)!A2p(p+
1)2p. In Theorem 6.1 the constant A ≥ max
j>p
p+2
2 (8eπ
2
)p(1− 1
j )
j does not
depend from z0, because the parameter p is independent of z0. Hence,
the quantity C = (p+ 1)!A2p(p+ 1)2p does not depend of z0. Therefore
by Theorem 5.1 the function ∂bF has bounded L-index in the direction
b.
Conversely, let ∂bF be a function of bounded L-index in the direction
b. By Theorem 5.1 there exist p ∈ Z+ and C ≥ 1 such that for every
A. Bandura, O. Skaskiv 175
z ∈ Cn the following inequality holds
|∂p+1
b F (z)|
Lp+1(z)
≤ Cmax
{
|∂kbF (z)|
Lk(z)
: 1 ≤ k ≤ p
}
. (6.2)
Let us consider a disc K0 =
{
t ∈ C : |t| ≤ 1
L(z0)
}
, z0 ∈ Cn.
One should observe that if L ∈ Qnb, z
0 ∈ Cn then for all r > 0 the
inequality |t| ≤ r
L(z0)
and definition of class Qnb yield
L(z0)/λb(r) ≤ L(z0 + tb) ≤ λb(r)L(z
0). (6.3)
Now from (6.2) and (6.3) for z = z0 + tb, t ∈ K one has
|∂p+1
b F (z0 + tb)|
(p+ 1)!(Cλb(1)L(z0))p+1
≤ max
{
|∂kbF (z0 + tb)|
k!
1
(Cλb(1)L(z0))k
×
(
L(z0 + tb)
Cλb(1)L(z0)
)p+1−k
: 1 ≤ k ≤ p
}
≤ C
p+ 1
× max
1≤k≤p
{
|∂kbF (z0 + tb)|
k!
1
(Cλb(1)L(z0))k
1
Cp+1−k
}
≤ max
{
|∂kbF (z0 + tb)|
k!
1
(Cλb(1)L(z0))k
: 1 ≤ k ≤ p
}
. (6.4)
To prove the proposition we need such a statement from [26, p. 44, The-
orem 2.7].
Theorem 6.2 ( [26, p.44, Theorem 2.7]). Let D0 = {t ∈ C : |t − t0| <
R}, 0 < R < +∞, and f(t) is an analytic function in D0. If for all
t ∈ D0(
R
2
)p+1 |f (p+1)(t)|
(p+ 1)!
≤ max
{(
R
2
)k |f (k)(t)|
k!
: 1 ≤ k ≤ p
}
, (6.5)
then f(t) is p-valent in {t ∈ C : |t − t0| ≤ R
25
√
p+1
}, i.e., f(t) attains
every value at more p times.
From inequality (6.4) it follows inequality (6.5) with R = 2
Cλb(1)L(z0)
and t0 = 0. By Theorem 6.2 the function F (z0 + tb) is p-valent in the
disc {t ∈ C : |t| ≤ ρ
L(z0)
}, ρ = 2
25Cλb(1)
√
p+1
.
Let tj be an arbitrary point in K0 and K∗
j = {t ∈ C : |t − ti| ≤
ρ
L(z0+tjb)
}. Since by definition of class Qnb L(z0 + tjb) ≤ λb(1)L(z
0),
one has Kj =
{
t ∈ C : |t − tj | ≤ ρ
λb(1)L(z0)
}
⊂ K∗
j . We will repeat
176 Some criteria of boundedness of L-index...
the similar considerations for the set
{
t ∈ C : |t− tj | ≤ 1
L(z0+tjb)
}
. As a
consequence, we deduce that F (z0+ tb) is p-valent in K∗
j . But Kj ⊂ K∗
j ,
then F (z0 + tb) is p-valent in Kj .
Finally, we note that every closed disc of radius R∗ can be covered
by a finite number m∗ of closed discs of radius ρ∗ < R∗ with the centers
in the disc. Moreover, m∗ < B∗(R∗/ρ∗)
2, where B∗ > 0 is an absolute
constant. Hence, K0 can be covered finite number m of discs Kj , where
m ≤ 625B∗(p + 1)C2(λb(1))
2/4. Since the function F (z0 + tb) in Kj is
p-valent, it is mp-valent in K0.
In view of arbitrariness of z0, the statement is proved.
7. Existence theorem for functions of bounded L-index in
direction
For the one-dimensional case, some time ago mathematicians were
interested in the following two problems: the problem of the existence of
an entire function of bounded l-index for a given l, and the problem of
the existence of a function l for a given entire function f such that f is
of bounded l-index [12,13,17,29]. It is clear that similar problems can be
posed for the multidimensional case [5, 7].
We note that the solution of the first problem for the one-dimensional
case is given by a canonical product. The solution of the first problem in
the multidimensional case also exists in the class of canonical products
with “planar” zeros.
In particular, the following proposition is true.
We consider the function F (z0 + tb) where z0 ∈ Cn is fixed. If
F (z0 + tb) ̸≡ 0, then we denote by pb(z0 + a0kb) the multiplicity of the
zero a0k of the function F (z0 + tb). If F (z0 + tb) ≡ 0 for some z0 ∈ Cn,
then we put pb(z0 + tb) = −1.
Theorem 7.1. In order that for a function F ∈ H̃n
b there exist a positive
continuous function L(z) such that F (z) is of bounded L-index in the
direction b it is necessary and sufficient that ∃p ∈ Z+ ∀z0 ∈ Cn ∀k
pb(z
0 + a0kb) ≤ p.
Proof. Necessity. To simplify the notation we denote everywhere in the
proof p0k ≡ pb(z
0 + ak0b). One can prove the necessity using the defini-
tion of bounded L-index in direction. Indeed, assume on the contrary
that ∀p ∈ Z+ ∃z0 ∃k p0k > p. It means that ∂
p0k
b F (z0 + a0kb) ̸= 0 and
∂jbF (z
0 + a0kb) = 0 for all j ∈ {1, . . . , p0k − 1}. Therefore, the L-index in
the direction b at the point z0 + a0kb is not less than p0k > p
Nb(F,L, z
0 + a0kb) > p.
A. Bandura, O. Skaskiv 177
If p→ +∞, then we obtain that Nb(F,L, z
0+a0kb) → +∞. But this con-
tradicts the boundedness of L-index in the direction b of the function F.
Sufficiency. If F (z0 + tb) ≡ 0 for some z0 ∈ Cn, then inequality
(1.1) is obvious.
Let p be the least integer such that ∀z0 ∈ Cn F (z0+ tb) ̸≡ 0, and ∀k
pk(z
0) ≤ p. For any point z ∈ Cn we choose z0 ∈ Cn and t0 ∈ C so that
z = z0 + t0b and the point z0 lies on the hyperplane ⟨z,m⟩ = 1, where
⟨b,m⟩ = 1 (actually it is sufficient that ⟨b,m⟩ ̸= 0, i.e. the hyperplane
is not parallel to b). Therefore, t0 = ⟨z,m⟩ − 1, z0 = z − (⟨z,m⟩ − 1)b.
We put KR = {t ∈ C : max{0, R− 1} ≤ |t| ≤ R+ 1} for all R ≥ 0 and
m1(z
0, R) = min
a0k∈KR
max
0≤s≤p
{
|∂sbF (z0 + a0kb)|
s!
}
.
Since F is a slice entire function, there exists ε = ε(z0, R) > 0 such
that ∣∣∂s0b F (z0 + tb)
∣∣
s0!
≥ m1(z
0, R)
2
for some s0 = s(a0k) ∈ {0, . . . , p} and for all t ∈ KR ∩ {t ∈ C : |t− a0k| <
ε(R, z0)} and for all k. We denote G0
ε =
∪
a0k∈KR
{t ∈ C : |t − a0k| < ε},
m2(z
0, R) = min{|F (z0 + tb)| : |t| ≤ R+ 1, t /∈ G0
ε},
Q(R, z0) = min
{
m1(R, z
0)
2
,m2(R, z
0), 1
}
.
We take R = |t0|. Then at least one of the numbers |F (z0 + t0b)|,∣∣∂bF (z0 + t0b)
∣∣ , . . . , |∂pbF (z0+t0b)|
p! is not less than Q(R, z0) (respectively,
for t0 ∈ G0
ε
|∂s0b F (z0+t0)b)|
p0k!
and for t /∈ Gε |F (z0 + t0b)|).
Hence,
max
{
|∂jbF (z
0 + t0b)|
j!
: 0 ≤ j ≤ p
}
≥ Q(R, z0). (7.1)
On the other hand, for |t0| = R and j ≥ p + 1 Cauchy’s inequality is
valid
|∂jbF (z
0 + t0b)|
j!
≤ 1
2π
∫
|τ−t0|=1
|F (z0 + τb)|
|τ − t0|j+1
|dτ |
≤ max{|F (z0 + τb)| : |τ | ≤ R+ 1}. (7.2)
178 Some criteria of boundedness of L-index...
We choose a positive continuous function L(z) such that
L(z0 + t0b) ≥ max
{
max{|F (z0 + tb)| : |τ | ≤ R+ 1}
Q(R, z0)
, 1
}
.
From (7.1) and (7.2) with |t0| = R and j ≥ p+ 1 we obtain
|∂jbF (z0+t0b)|
j!Lj(z0+t0b)
max
{
|∂kbF (z0+t0b)|
k!Lk(z0+t0b)
: 0 ≤ k ≤ p
} ≤ L−j(z0 + tb)
Q(R, z0)L−p(z0 + tb)
×max{|F (z0 + tb)| : |τ | ≤ R+ 1} ≤ Lp+1−j(z0 + tb) ≤ 1.
Since z = z0 + t0b, we have∣∣∣∂jbF (z)∣∣∣
j!Lj(z)
≤ max
{∣∣∂kbF (z)∣∣
k!Lk(z)
: 0 ≤ k ≤ p
}
.
In view of arbitrariness of z the function F has bounded L-index in the
direction b.
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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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