Extremal decomposition of multidimensional complex space for five domains
The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains contain...
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Zabolotni, i Y. Denega, I. 2020-06-12T17:53:31Z 2020-06-12T17:53:31Z 2018 Extremal decomposition of multidimensional complex space for five domains / Y. Zabolotnii, I. Denega // Український математичний вісник. — 2018. — Т. 15, № 3. — С. 431-441. — Бібліогр.: 21 назв. — англ. 1810-3200 010 MSC. 30C75, 32A30 https://nasplib.isofts.kiev.ua/handle/123456789/169414 The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1, 2.57] and generalized this result to the case of multidimensional complex space. en Інститут прикладної математики і механіки НАН України Український математичний вісник Extremal decomposition of multidimensional complex space for five domains Article published earlier |
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Extremal decomposition of multidimensional complex space for five domains Zabolotni, i Y. Denega, I. |
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Extremal decomposition of multidimensional complex space for five domains |
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extremal decomposition of multidimensional complex space for five domains |
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Zabolotni, i Y. Denega, I. |
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Zabolotni, i Y. Denega, I. |
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Український математичний вісник |
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The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1, 2.57] and generalized this result to the case of multidimensional complex space.
|
| issn |
1810-3200 |
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https://nasplib.isofts.kiev.ua/handle/123456789/169414 |
| citation_txt |
Extremal decomposition of multidimensional complex space for five domains / Y. Zabolotnii, I. Denega // Український математичний вісник. — 2018. — Т. 15, № 3. — С. 431-441. — Бібліогр.: 21 назв. — англ. |
| work_keys_str_mv |
AT zabolotniiy extremaldecompositionofmultidimensionalcomplexspaceforfivedomains AT denegai extremaldecompositionofmultidimensionalcomplexspaceforfivedomains |
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2025-11-26T00:08:44Z |
| last_indexed |
2025-11-26T00:08:44Z |
| _version_ |
1850593288916041728 |
| fulltext |
Український математичний вiсник
Том 15 (2018), № 3, 431 – 441
Extremal decomposition of multidimensional
complex space for five domains
Yaroslav Zabolotnii, Iryna Denega
(Presented by O. A. Dovgoshey)
Abstract. The paper is devoted to one open extremal problem in ge-
ometric function theory of complex variables associated with estimates
of functional defined on the systems of non-overlapping domains. We
consider problem of the maximum of product of inner radii of n non-
overlapping domains containing points of the unit circle and the power
γ of the inner radius of domain containing the origin. The problem was
formulated in 1994 in the Dubinin paper in the journal “Russian Math-
ematical Surveys” in the list of unsolved problems and then repeated in
his monograph in 2014. Currently it is not solved in general. In this
paper we obtained a solution of the problem for five simply connected
domains and power γ ∈ (1; 2, 57] and generalized this result to the case
of multidimensional complex space.
2010 MSC. 30C75, 32A30.
Key words and phrases. Inner radius of the domain, non-overlapping
domains, polycylindrical domain, radial system of points, separating
transformation, quadratic differential, Green’s function.
Let N, R be the sets of natural and real numbers, respectively, C be
the complex plane, C = C
∪
{∞} be its one point compactification and
R+ = (0,∞). Let n ∈ N. A set of points An :=
{
ak ∈ C : k = 1, n
}
is
called n-radial system if |ak| ∈ R+, k = 1, n, and 0 = arg a1 < arg a2 <
. . . < arg an < 2π. Let B be the domain in C. Let
gB(B, a) = hB,a(z) + log
1
|z − a|
be generalized Green function of the domain B relatively to a point a ∈ B.
If a→ ∞ then
gB(B,∞) = hB,∞(z) + log
1
|z|
.
Received 03.04.2018
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
432 Extremal decomposition of multidimensional complex...
By r(B, a) := exp(hB,a(z)) we denote the inner radius of the domain
B ⊂ C with respect to a point a ∈ B (see, for example, [1–5]). Denote
αk :=
1
π
arg
ak+1
ak
, αn+1 := α1, k = 1, n,
∑n
k=1
αk = 2, Γ := {w ∈ C := |w| = 1}.
A finite set of arbitrary domains {Bk}nk=1, n ∈ N, n ≥ 2 such as
Bk ⊂ C, Bk ∩ Bm = ∅, k ̸= m, k,m = 1, n, is called a system of non-
overlapping domains.
One of the main concepts in this paper is a notion of quadratic differ-
ential. Quadratic differential is a convenient tool for describing extremals
on the plane. A great collection of definitions and results about quadratic
differentials one can find in monographs [6, 7].
The emergence in the theory of univalent functions of extremal prob-
lems of non-overlapping domains associated with the paper of M. Lavren-
tyev, who in 1934 solved the problem of product of conformal radii of
two mutually non-overlapping simply connected domains. According to
Lavrentyev’s result, when a1 and a2 be finite points then for any pair of
non-overlapping simply connected domainsD1 andD2 such that ak ∈ Dk,
k = 1, 2, the inequality holds
r(D1, a1)r(D2, a2) ≤ |a1 − a2|2, (1)
where equality is attained for the half-planesDk and points ak, symmetric
with respect to their common boundary. The value
T2 :=
{
r(D1, a1)r(D2, a2)|a1 − a2|−2, a1 + a2 ̸= ∞
r(D1, a1)r(D2, a2), a1 + a2 = ∞,
(2)
ak ∈ Dk ⊂ C, k = 1, 2, is an invariant relative to the fractional-linear
mapping of the plane C on itself. That is for any function of the type
f(z) = (az + b)/(cz + d) where ad− bc ̸= 0 the inequality holds
T2(a1, a2, D1, D2) = T2(f(a1), f(a2), f(D1), f(D2)).
Thus, the inequality (1) can be considered as upper estimation of the
mebius invariant
T2 ≤ 1.
Y. Zabolotnii, I. Denega 433
In the case of three or more points many authors [3–5] considered esti-
mations of a more general mebius invariant of the form
Tn :=
n∏
k=1
r(Dk, ak){
′∏
16k<l6n
|ak − al|
} 2
n−1
, (3)
(here the stroke in the product means that for an infinitely distant point
under the corresponding factor we mean the unit). In order to get Tn
it is necessary for n pairwise non-overlapping domains Dk and points
ak ∈ Dk, k = 1, n, n ≥ 2, multiply all possible pairwise products of type
(2) and take the root of degree 2(n− 1).
In 1951 G.Goluzin summarized the Lavrentyev problem in the case of
a finite number of n, n > 3 mutually non-overlapping simply connected
domains Bk, ak ∈ Bk ⊂ C ({ak}nk=1 are arbitrary fixed finite and various
points of the complex plane), and at n = 3 obtained an exact estimate
for T3 [2, p. 165]
T3 ≤
64
81
√
3
,
where equality is achieved, with the precision of fractional-linear trans-
formations, only for domains that are equal angles and points lying on
the bisectors of these angles at identical distances from their common
vertex.
The case n > 3 is more complicated and significantly different from
the situation n 6 3, since by the fractional-linear mapping any three
predefined points can be converted into three points that are convenient
to prove.
For n = 4 G. Kuz’mina [3, p. 25] showed that the problem of esti-
mating T4 reduces to the smallest capacity problems in a certain family
of continuums and obtained the exact inequality
T4 ≤ 32 · 4−8/3.
Equality is achieved only in the case when with precision to fractional-
linear transformation a1 = −a2 = 1, a3 = −a4 = i(2 −
√
3), and the
boundary of the set
4∪
k=1
Dk consists with segments [
√
3− 2, 2−
√
3], rays
{z|z = it, t ∈ [1, +∞)} and {z|z = it, t ∈ [−∞, −1)}, part of the
circular arc ρ, passing through points −1, i, 2 −
√
3, which lies in the
434 Extremal decomposition of multidimensional complex...
first quadrant, and circular arcs that we get from ρ by transformations
w → −w, w → w, w → −w.
For n ≥ 5 full solution (3) is not obtained at this time. Since, the
evaluation of the product of conformal radii of mutually non-overlapping
domains if n ≥ 5 without any restriction on the domains Bk and points
ak, k = 1, ..., 5 is quite difficult and interesting problem.
In 1975 N. Lebedev in the monograph [8, p. 32–33] considered the
more general extremal problem on product of conformal radii.
Problem 1. There are n various fixed points ak, k = 1, n, n > 3,
on a plane w. Functions w = fk(z), k = 1, n, are regular in the circle
|z| < 1 and univalent map circle |z| < 1 onto non-overlapping domains
Bk, which contain the corresponding points ak, k = 1, n, and in such a
way, that fk(0) = ak, k = 1, n. What about maximum of product
n∏
k=1
|f ′k(0)|γk −→ max, γk > 0, n > 3,
relatively to any functions fk(z), k = 1, n?
However, this problem is generally not solved so far. Further this
problem was generalized to more general classes of multiply connected
domains replacing conformal radius to the inner radius.
Consider an extremal problem which was formulated in 1994 in the
works of V. Dubinin [4, 9] in the lists of unsolved problems.
Problem 2. [4, p. 68]. Show that for pairwise disjoint domains B0,
B1, B2, . . . , Bn, n ≥ 2, in C, system of points An = {ak}nk=1, a0 = 0,
|ak| = 1, k = 1, n, and number γ ≤ n, maximum of the product
In(γ) = rγ (B0, 0)
n∏
k=1
r (Bk, ak) ,
is attained for configuration of domains Bk and points ak possessing ro-
tational n-symmetry.
Currently it is not solved in general only partial results are known. In
1988 in [10] this problem was solved for γ = 1 and n ≥ 2, and from the
method of this work it implies that the result is true and for 0 < γ < 1. In
1996 Kovalev [11] got the solution to this problem with some restrictions
on the geometry location of sets of points on the unit circle and, namely,
for n ≥ 5 and subclass points systems satisfying condition
0 < αk ≤ 2/
√
γ, k = 1, n.
Y. Zabolotnii, I. Denega 435
It is clear that these conditions are sufficiently stringent conditions, sig-
nificantly narrowing the set of feasible configurations. It should be noted
that the Kovalev result is interesting not only by itself, but the method of
study is important too. In 2003 in the paper of G. Kuz’mina [12] in the
case of simply connected domains this problem has also been studied for
γ ∈ (0, 1] by another method. In 2008 Bakhtin [5, p. 255] complemented
the ideas and methods of previous works and thus he showed that the
result of V. Dubinin holds for an arbitrary γ ∈ (0,∞) but since some
number n0(γ). It should be noted in the case γ > 1 method developed
in the paper of V. Dubinin [10] can not be applied. In papers [13–18]
particular cases of the Dubinin problem were considered. Note that the
cases n = 2, 3, 4 with γ > 1 are the most difficult cases in this issue.
It is well known (see, for example, [19–21]) that
Cl = C× C× C× ...× C︸ ︷︷ ︸
l-times
, l ∈ N.
Let Cl
= C× C× C× ...× C︸ ︷︷ ︸
l-times
be compactification of the space Cl where
the set of infinitely distant points has complex dimension l − 1 (see, for
example, [19–21]). And let Ω(ω1, ω2, ..., ωl) be a point of the space Cl
with coordinates ωk, k = 1, l. By [D]l (cartesian degree of the domain
D ∈ C) we denote the cartesian product D ×D × . . .×D︸ ︷︷ ︸
l-times
. Besides by
[d]l (cartesian degree of a point d ∈ C) we denote the point in Cl which
has coordinates (d, ..., d)︸ ︷︷ ︸
l−times
.
A domain B = B1×B2×B3× ...×Bl is called polycylindrical domain
in Cl (see [19]), where the domains Bk ⊂ C, k = 1, l, and Bk are called
coordinate domains.
A system {Bk}lk=1
(
Bk = B
(k)
1 × . . .×B
(k)
l , k = 1, l
)
is called a sys-
tem of polycylindrical non-overlapping domains if for each fixed p0, p0 =
1, l, the system of domains
{
B
(k)
p0
}
, k = 1, l, is the system of non-
overlapping domains in C.
A value
R(B,Ω) =
[
l∏
k=1
r(Bk, ωk)
] 1
l
is called generalized inner radius of the polycylindrical domain B in point
Ω (Ω ∈ B). If l = 1 then the value R(B,Ω) is usual inner radius of the
domain B ⊂ C relatively to a point Ω.
436 Extremal decomposition of multidimensional complex...
In the paper we consider the Problem 2 for five pairwise disjoint sim-
ply connected domains in the complex plane. Currently the theorem 1
of the paper [15] is the best result for n = 4 without any additional
conditions on the location of the points ak, k = 1, 4, where the Dubinin
problem was solved for γ ∈ (1; 2, 09]. In the paper the following result is
obtained.
Theorem 1. Let Bk = B
(k)
1 × B
(k)
2 × B
(k)
3 × ... × B
(k)
l , k = 0, 4, is a
system of mutually non-overlapping polycylindrical domains in Cl and
points Ωk = (ω
(k)
1 , ω
(k)
2 , ..., ω
(k)
l ), k = 0, 4, satisfy the following conditions
Ω0 = (0, 0, ..., 0), Ωk ∈ Bk and for any m = 1, l, B0
m, B1
m, B2
m, B3
m, B4
m
are pairwise disjoint simply connected domains in C, |ωk
m| = 1, k = 1, 4,
and γ ∈ (1; 2, 57]. Then the following inequality holds
Rγ(B0,Ω0)
4∏
k=1
R(Bk,Ωk) ≤
(γ4 )
γ
4(
1− γ
16
)4+ γ
4
(
1−
√
γ
4
1 +
√
γ
4
)2
√
γ
.
One of the extremal system is the system
{Bk}4k=0 =
{[
B
(0)
0
]l
,
[
B
(0)
1
]l
,
[
B
(0)
2
]l
,
[
B
(0)
3
]l
,
[
B
(0)
4
]l}
,
{Ωk}4k=0 =
{
[0]l ,
[
ω
(0)
1
]l
,
[
ω
(0)
2
]l
,
[
ω
(0)
3
]l
,
[
ω
(0)
4
]l}
,
where domains B
(0)
k and points ω
(0)
k , k = 0, 4, are, respectively, circular
domains and poles of the quadratic differential
G(w)dw2 = −(16− γ)w4 + γ
w2(w4 − 1)2
dw2. (4)
Proof. Perform the following transformations
Rγ(B0,Ω0) ·
4∏
k=1
R(Bk,Ωk) =
[
l∏
m=1
r(B
(0)
m ,Ω
(0)
m )
] γ
l
·
4∏
k=1
[
l∏
m=1
r(B
(k)
m ,Ω
(k)
m )
] 1
l
=
[
l∏
m=1
[
(r(B
(0)
m ,Ω
(0)
m ))γ
4∏
k=1
r(B
(k)
m ,Ω
(k)
m )
]] 1
l
.
Then for ∀m = 1, l the domains B
(k)
m , k = 0, n, make up the system of
pairwise disjoint simply connected domains in the complex plane C. 2
Thus, prove the following lemma.
Y. Zabolotnii, I. Denega 437
Lemma 1. Let n = 4, γ ∈ (1; 2, 57]. Then the following sharp estimate
holds
rγ(B0, 0)
4∏
k=1
r(Bk, ak) ≤
(γ4 )
γ
4(
1− γ
16
)4+ γ
4
(
1−
√
γ
4
1 +
√
γ
4
)2
√
γ
, (5)
where B0, B1, B2, B3, B4 are pairwise disjoint simply connected domains
in C, a0 = 0, |ak| = 1, k = 1, 4, r(Bj , aj) is the inner radius of the
domain Bj with respect to the point aj (aj ∈ Bj), j = 0, 4. Equality
is attained if dk and Dk, k = 0, n, are, respectively, poles and circular
domains of the quadratic differential (4).
Proof. Find the value
I0n(γ) = rγ (D0, 0)
n∏
k=1
r (Dk, dk) ,
where 0 ∪ {dk}nk=1 and {Dk}nk=0 are, respectively, poles and circular do-
mains of the quadratic differential (4). From results of the papers [4,5,11]
and properties of the separating transformation we have
I0n(γ) =
(
2
n
)n
2
4γ
n2+6
(
2
√
γ
n
) 4γ
n2
(
2− 2
√
γ
n
) 1
2
(
2− 2
√
γ
n
)2 (
2 +
2
√
γ
n
) 1
2
(
2+
2
√
γ
n
)2
n
2
.
Using elementary calculations we obtain that the following estimate holds
I0n(γ) =
(
4
n
)n
(
4γ
n2
) γ
n(
1− γ
n2
)n+ γ
n
(
1−
√
γ
n
1 +
√
γ
n
)2
√
γ
.
Thus, for n = 4 we have
I04 (γ) =
(γ4 )
γ
4
(1− γ
16)
4+ γ
4
(
1−
√
γ
4
1 +
√
γ
4
)2
√
γ
.
The following statement holds.
Lemma 2. [18] Let n ∈ N, n ≥ 2, γ > 0. And let {B0, B1, B2, . . . , Bn}
be the system of pairwise disjoint simply connected domains such that
0 ∈ B0, ak ∈ Bk ∩ Γ, k = 1, n, and rγ (B0, 0)
n∏
k=1
r (Bk, ak) > I0n(γ).
Then we have inequality
r (B0, 0) ≤ n
− n
2(n−γ) I0n(γ)
− 1
n−γ .
438 Extremal decomposition of multidimensional complex...
Taking lemma 2 and result of the paper [13] into account we consider
the case
r (B0, 0) ≤ 4
− 2
(4−γ) I04 (γ)
− 1
4−γ , α0 >
2
√
γ
, α0 = max
k
αk.
We will prove that under the given conditions the following inequality
holds
rγ(B0, 0)
4∏
k=1
r(Bk, ak)
( γ
4 )
γ
4
(1− γ
16)
4+
γ
4
(
1−
√
γ
4
1+
√
γ
4
)2
√
γ
< 1.
From Lemma 2
rγ(B0, 0) < 4
− 2γ
(4−γ) I04 (γ)
− γ
4−γ .
Further, using Theorem 1 of the paper [3] the following inequality holds
4∏
k=1
r(Bk, ak)
≤ 9
4
8
3
(|a1 − a2| · |a1 − a3| · |a2 − a3| · |a1 − a4| · |a2 − a4| · |a3 − a4|)
2
3 .
Let α0 = α4 then we obtain
4∏
k=1
r(Bk, ak) ≤
144
4
8
3
(
sin
πα1
2
sin
πα2
2
sin
πα3
2
) 2
3
(6)
×
(
sin
π(α1 + α2)
2
sin
π(α2 + α3)
2
sin
π(α1 + α2 + α3)
2
) 2
3
.
If α0 >
2√
γ then α1 + α2 + α3 < 2 − 2√
γ . Consider the right side of the
expression (6) under these conditions. The following inequality holds
4∏
k=1
r(Bk, ak)
≤ 144
4
8
3
sin2
(
2− 2
√
γ
)(π
6
)
sin
4
3
(
2− 2
√
γ
)(π
3
)
sin
2
3
(
2− 2
√
γ
)(π
2
)
.
Thus,
rγ(B0, 0)
4∏
k=1
r(Bk, ak)
( γ
4
)
γ
4
(1− γ
16
)4+
γ
4
(
1−
√
γ
4
1+
√
γ
4
)2
√
γ
Y. Zabolotnii, I. Denega 439
≤
144 sin2
(
2− 2√
γ
) (
π
6
)
sin
4
3
(
2− 2√
γ
) (
π
3
)
sin
2
3
(
2− 2√
γ
) (
π
2
)
4
8
3
+ 2γ
(4−γ)
(
I04 (γ)
) 4
4−γ
:= Q(γ).
Substituting in the right side of the last inequality the value γ = 2, 57,
we obtain
rγ(B0, 0)
4∏
k=1
r(Bk, ak)
( γ
4
)
γ
4
(1− γ
16
)4+
γ
4
(
1−
√
γ
4
1+
√
γ
4
)2
√
γ
≤ 1.
Therefore for γ = 2, 57 the inequality (5) is valid. Let now γ < 2, 57.
Prove that the functionQ(γ) monotonically increases for 1 < γ ≤ 2, 57. It
is clear that the functions sin
(
2− 2√
γ
) (
π
6
)
, sin
(
2− 2√
γ
) (
π
3
)
,
sin
(
2− 2√
γ
) (
π
2
)
are increasing on this interval. And thus the numerator
of the expression for the function Q(γ) is increasing. Further, consider
the denominator of the expression Q(γ). So,
4
8
3
+ 2γ
(4−γ)
(
I04 (γ)
) 4
4−γ = 4
8
3
(
2γ
(
I04 (γ)
)) 4
4−γ .
Prove that the expression
(
2γ
(
I04 (γ)
)) 4
4−γ =
2γ
(γ4 )
γ
4
(1− γ
16)
4+ γ
4
(
1−
√
γ
4
1 +
√
γ
4
)2
√
γ
4
4−γ
monotonically decreases for 1 < γ ≤ 2, 57. Taking logarithmic derivative
of the last expression we have that for 1 < γ ≤ 2, 57 it is negative and it
means that denominator of the expression Q(γ) decreases. Summarizing
all of the above we obtain that the expression Q(γ) increases on the
interval 1 < γ ≤ 2, 57 and therefore for any 1 < γ < 2, 57 the following
inequality holds
Q(γ) < Q(2, 57) < 1.
Hence, for any γ such that 1 < γ < 2, 57 and for any configuration
of domains Bk and points ak, k = 0, 4, satisfying all conditions of the
Lemma 1, the inequality (5) holds. Lemma 1 is proved. 2
Therefore, we obtain the inequality of the Theorem 1
Rγ(B0,Ω0)
4∏
k=1
R(Bk,Ωk) ≤
l∏
m=1
(γ4 )
γ
4
(1− γ
16)
4+ γ
4
(
1−
√
γ
4
1 +
√
γ
4
)2
√
γ
1
l
440 Extremal decomposition of multidimensional complex...
=
(γ4 )
γ
4
(1− γ
16)
4+ γ
4
(
1−
√
γ
4
1 +
√
γ
4
)2
√
γ
.
Theorem 1 is proved. 2
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Contact information
Yaroslav Zabolotnii Institute of mathematics of National
Academy of Sciences of Ukraine,
Kyiv, Ukraine
E-Mail: yaroslavzabolotnii@gmail.com
Iryna Denega Institute of mathematics of National
Academy of Sciences of Ukraine,
Kyiv, Ukraine
E-Mail: iradenega@gmail.com
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