An extremal problem for the non-overlapping domains
Sharp estimates of product of inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of arbitrary finite number of free poles on the system points on the rays.
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irk-123456789-1693162020-06-11T01:26:26Z An extremal problem for the non-overlapping domains Targonskii, A. Sharp estimates of product of inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of arbitrary finite number of free poles on the system points on the rays. 2017 Article An extremal problem for the non-overlapping domains / A. Targonskii // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 126-134. — Бібліогр.: 18 назв. — англ. 1810-3200 2001 MSC. 30C70, 30C75 http://dspace.nbuv.gov.ua/handle/123456789/169316 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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Sharp estimates of product of inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of arbitrary finite number of free poles on the system points on the rays. |
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Targonskii, A. |
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Targonskii, A. An extremal problem for the non-overlapping domains Український математичний вісник |
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Targonskii, A. |
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Targonskii, A. |
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An extremal problem for the non-overlapping domains |
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An extremal problem for the non-overlapping domains |
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An extremal problem for the non-overlapping domains |
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An extremal problem for the non-overlapping domains |
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An extremal problem for the non-overlapping domains |
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extremal problem for the non-overlapping domains |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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An extremal problem for the non-overlapping domains / A. Targonskii // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 126-134. — Бібліогр.: 18 назв. — англ. |
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Український математичний вісник |
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AT targonskiia anextremalproblemforthenonoverlappingdomains AT targonskiia extremalproblemforthenonoverlappingdomains |
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2025-07-15T04:03:27Z |
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1837684183657349120 |
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Український математичний вiсник
Том 14 (2017), № 1, 126 – 134
An extremal problem
for the non-overlapping domains
Andrey Targonskii
(Presented by V. Ya. Gutlyanskii)
Abstract. Sharp estimates of product of inner radii for pairwise dis-
joint domains are obtained. In particular, we solve an extremal problem
in the case of arbitrary finite number of free poles on the system points
on the rays.
2001 MSC. 30C70, 30C75.
Key words and phrases. Inner radius of domain, quadratic differen-
tial, piecewise-separating transformation, Green function, radial systems
of points, logarithmic capacity, variational formula.
Introduction
This paper belongs to the theory of extremal problems on classes of
non-overlapping domain, which is a separate direction in geometric theory
of functions of a complex variable. The begin of these investigations
associated with the paper of M. A. Lavrent’ev [1] in 1934. He found
the maximum of some functional with respect to two simply connected
domains with two fixed points. We note that this result was needed him
for applying to some aerodynamics problems. In 1947, G. M. Goluzin
solved a similar problem for three fixed points on the complex plane
[2]. Then the topic began to evolve rapidly. In this connection we may
recall papers of many authors, including Y.E. Alenitsina, M. A. Lebedev,
J. Jenkins, P.M. Tamrazov, P. P. Kufareva and others. Using the idea
of P.M. Tamrazov, in 1975 G. P. Bakhtin solved first the problem with
so-called “free poles”, on the unit circle, see, e.g., [3].
Received 30.11.2016
The author is grateful to Prof. A. Bakhtin for suggesting problems and useful dis-
cussions. This research is partially supported by Grant of Ministry of Education and
Science of Ukraine (Project No. 0115U003027)
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
A. Targonskii 127
An important step for the development of this topic was papers of
V.N. Dubinin. He developed a new method of research that is method
of piecewise-separating transformation. He also first solved numerous
of extremal problems for an arbitrary but fixed multi connected non-
overlapping domains (see, e.g., [4–6]). Now this type of extremal problems
is used for investigations in holomorphic dynamics.
In the last decade actively used Bakhtin’s method of “managing func-
tional”. He managed to solve a series of extremal problems for so-called
“radial systems of points” (see, e.g., [4, 7–12]). In the present paper we
use the mentioned about Bakhtin’s method.
1. Theory
Let N, R — the sets natural and real numbers conformity, C — the
plain complex numbers, C = C
∪
{∞} — the Riemannian sphere.
For fix number n ∈ N system points
An = {ak}nk=1
the relations are executed:
0 = arg a1 < arg a2 < ... < arg an < 2π. (1.1)
For such systems of points we will consider the following sizes:
σk =
1
π
(arg ak+1 − arg ak) , k = 1, 2, ..., n, an+1 := a1.
Let’s consider system of angular domains:
Mk := {w : arg ak < argw < arg ak+1} , k = 1, n, an+1 := a1.
Let’s consider the following “operating”, functionalities for arbitrary
An-system points
T(An) =
n∏
k=1
χ
(∣∣∣∣ akak+1
∣∣∣∣ 1
2σk
)
|ak|,
where χ(t) = 1
2(t+
1
t ).
Let {Bk}nk=1 — arbitrary non-overlapping domains such that
ak ∈ Bk, Bk ⊂ C, k = 1, n. (1.2)
Let
gB (z, a) = hB,a(z) + log
1
|z − a|
128 An extremal problem for...
generalized Green’s function of domains B with respect to a point a ∈ B.
If a = ∞, then
gB (z,∞) = hB,∞(z) + log
1
|z|
.
The value of
r(B, a) := exp (hB,a(z))
the define of inner radius domain B ⊂ C with respect to a point a ∈ B
(see [4–6,13–15]).
We use the concept of a quadratic differential. Recall that a quadratic
differential on a Riemann surface S is a map
φ : TS → C
satisfying
φ(λυ) = λ2φ(υ)
for all υ ∈ TS and all λ ∈ C, TS — tangent space. If z ∈ U → C, is a
chart defined on some open set U ⊂ S then φ is equal on U to
φU (z)dz
2
for some function φU defined on z(U).
Suppose that two charts z : U → C and w : V → C on S overlap, and
let
h := w ◦ z−1
be the transition function. If φ is represented both as φU (z)dz2 and
φV (w)dw
2 on U ∩ V , then we have
φV (h(z))
(
h′(z)
)2
= φU (z).
One way to say this is that quadratic differentials transform under pull-
backs by the square of the derivative. As the main results associated with
it can be found in [16].
2. Results
Subject of studying of our work are the following problems.
Problem. Let n ∈ N, n ≥ 2, α ≥ 0. Maximum functional be found
n∏
k=1
(|ak+1 − ak|α · r (Bk, ak)) ,
where An = {ak}nk=1 — arbitrary system points on the rays, the satisfied
condition (1.1), {Bk}nk=1 — arbitrary set non-overlapping domains, the
satisfied condition (1.2), ak ∈ Bk ⊂ C, and all extremal the describe
(k = 1, n).
A. Targonskii 129
Lemma 1. The function
P (τ) = ln sin
πτ
2
is convex for τ ∈ (0, 2).
Proof. Find the second-order derivative
P ′′(τ) =
π
2
·
(
ctg
πτ
2
)′
= −
(π
2
)2
· 1
sin2 πτ2
.
Consequently,
P ′′(τ) < 0, for 0 < τ < 2.
Theorem 1. Let n ∈ N, n ≥ 2, α ≥ 0. Then for all system points
An = {ak}nk=1, the satisfied condition (1.1) and
(|ak| − |ak+1|)2 = 4 sin2
πσk
2
(1− |ak||ak+1|) , k = 1, n,
and arbitrary set non-overlapping domains {Bk}nk=1, the satisfied condi-
tion (1.2), be satisfied inequality
n∏
k=1
(|ak+1 − ak|α · r (Bk, ak)) ≤
(
2α+2
n
· sinα π
n
)n
· T(An) .
The equality obtain in this inequality, when points ak and domains Bk
are, conformity, the poles and the circular domains of the quadratic dif-
ferential
Q(w)dw2 = − wn−2
(wn − 1)2
dw2. (2.3)
Proof. The theorem of the proof leans on a method of the piece-dividing
transformation developed by Dubinin (see [4–6]).
Function
ζk (w) = −i
(
e−i arg akw
) 1
σk , k = 1, 2, . . . , n (2.4)
realizes univalent and conformal transformations of domain Mk to the
right half-plane Reζ > 0, for all k = 1, n.
From a formula (2.4) we receive the following asymptotic expressions
|ζk (w)− ζk (am)| ∼
1
σk
|am|
1
σk
−1 |w − am| ,
130 An extremal problem for...
w → am, k = 1, 2, ..., n, m = k, k + 1. (2.5)
It’s obvious that
ζk (ak) = −i|ak|
1
σk , ζk (ak+1) = i|ak+1|
1
σk , k = 1, 2, ..., n. (2.6)
Family of functions {ζk(w)}nk=1, set by equality (2.4), it is possible for
by piece-dividing transformation (see [4–6]) domains
{
Bk : k = 1, n
}
in
relation to the system of corners {Mk}nk=1. For any domain ∆ ∈ C the de-
fine (∆)∗ :=
{
w ∈ C : w ∈ ∆
}
. Let G
(1)
k the define connected component
ζk
(
Bk
∩
Mk
)∪ (
ζk
(
Bk
∩
Mk
))∗
, containing a point (−i), G(2)
k−1 – the
define connected component ζk−1
(
Bk
∩
Mk−1
)∪ (
ζk−1
(
Bk
∩
Mk−1
))∗
,
containing a point i, k = 1, n, M0 := Mn, ζ0 := ζn, G
(2)
0 := G
(2)
n . It
is clear, that, G
(s)
k generally speaking, domains are multiconnected do-
mains, k = 1, n, s = 1, 2. Pair of domains G
(2)
k−1 and G
(1)
k grows out of
piece-dividing transformation domainsBk concerning families {Mk−1,Mk},
{ζk−1, ζk} in point ak, k = 1, n.
From the Theorem 1.9 [13] (see also [5,6]) and the formulae (2.5), we
have the inequalities
r (Bk, ak) ≤
[
|ak|
1− 1
σk · σk · r
(
G
(1)
k , ζk (ak)
)
· σk−1
× |ak|
1− 1
σk−1 · r
(
G
(2)
k−1, ζk−1 (ak)
)] 1
2
, k = 1, 2, ..., n. (2.7)
From the condition that the points ak, k = 1, 2, .., n, we get that
|ak+1 − ak| = 2 sin
πσk
2
, k = 1, 2, ..., n. (2.8)
Using formulas (2.7), (2.8) it is received the following ratio:
n∏
k=1
(|ak+1 − ak|α · r (Bk, ak)) ≤ 2nα ·
n∏
k=1
σk |ak|
|ak|
1
2σk · |ak|
1
2σk−1
×
n∏
k=1
sinα
πσk
2
·
n∏
k=1
(
r
(
G
(1)
k , ζk (ak)
)
· r
(
G
(2)
k , ζk (ak+1)
)) 1
2
. (2.9)
Inequalities Lavrent’ev using [1] and (2.6), we get:
r
(
G
(1)
k , ζk (ak)
)
· r
(
G
(2)
k , ζk (ak+1)
)
≤
(
|ak|
1
σk + |ak+1|
1
σk
)2
, k = 1, 2, ..., n.
A. Targonskii 131
Taking into account the last inequality, the expression (2.9) can be
written as follows:
n∏
k=1
(|ak+1 − ak| · r (Bk, ak)) ≤ 2nα ·
n∏
k=1
σk sin
α πσk
2
×
n∏
k=1
|ak|
1
σk + |ak+1|
1
σk
|ak|
1
2σk · |ak|
1
2σk−1
|ak| .
It’s obvious that
n∏
k=1
|ak|
1
σk + |ak+1|
1
σk
|ak|
1
2σk · |ak|
1
2σk−1
|ak| = 2n · T(An) .
Also,
n∏
k=1
σk ≤
(
2
n
)n
.
The equality obtain in this inequality, if and only if
σ1 = σ2 = ... = σn =
2
n
.
Then, we have:
n∏
k=1
(|ak+1 − ak| · r (Bk, ak)) ≤
(
2α+2
n
)n
· T(An) ·
n∏
k=1
sinα
πσk
2
. (2.10)
The equality obtain in this inequality, when points ak and domains Bk
are, conformity, the poles and the circular domains of the quadratic dif-
ferential
Q(ζ)dζ2 =
dζ2
(ζ2 + 1)2
. (2.11)
Using the Lemma that the function α ln sin πσk
2 , is convex for σk ∈
(0; 2) , α ≥ 0. Hence, when σk ∈ (0; 2), then
α
n
·
n∑
k=1
ln sin
πσk
2
≤ α ln sin
(
π
2
· 1
n
n∑
k=1
σk
)
.
Given that
n∑
k=1
σk = 2,
132 An extremal problem for...
we obtain
n∏
k=1
sinα
πσk
2
≤ sinnα
π
n
. (2.12)
The equality obtain in this inequality, if and only if
σ1 = σ2 = ... = σn =
2
n
.
Then from (2.10) using formulas (2.12) it is received the following
ratio
n∏
k=1
(|ak+1 − ak| · r (Bk, ak)) ≤
(
2α+2
n
)n
· T(An) · sinnα
π
n
.
The equality obtain in this inequality, when points ak and domains
Bk are, conformity, the poles and the circular domains of the quadratic
differential (2.3). It is derived from the square of the quadratic differen-
tial (2.11) conversion using
ζ = −iw
n
2 .
Provided that, |ak| = 1, k = 1, 2, ..., n we obtain the well known
result.
Corollary 1. [4–6]. Let n ∈ N, n ≥ 2, α > 0. Then for all system points
An = {ak}nk=1, the satisfied condition (1.1) and |ak| = 1, k = 1, 2, ..., n,
and arbitrary set non-overlapping domains {Bk}nk=1, the satisfied condi-
tion (1.2), be satisfied inequality
n∏
k=1
(|ak+1 − ak|α · r (Bk, ak)) ≤
(
2α+2
n
· sinα π
n
)n
.
The equality obtain in this inequality, when points ak and domains Bk
are, conformity, the poles and the circular domains of the quadratic dif-
ferential (2.3).
As a consequence, at α = 0, |ak| = 1, k = 1, 2, ..., n we obtain the
well known result.
Corollary 2. [4–6]. Let n ∈ N, n ≥ 2. Then for all system points An =
{ak}nk=1, the satisfied condition (1.1) and |ak| = 1, k = 1, 2, ..., n, and
A. Targonskii 133
arbitrary set non-overlapping domains {Bk}nk=1, the satisfied condition
(1.2), be satisfied inequality
n∏
k=1
r (Bk, ak) ≤
(
4
n
)n
.
The equality obtain in this inequality, when points ak and domains Bk
are, conformity, the poles and the circular domains of the quadratic dif-
ferential (2.3).
References
[1] M. A. Lavrent’ev, On the theory of conformal mappings // Tr. Fiz.-Mat. Inst.
Akad. Nauk SSSR, 5 (1934), 159–245.
[2] G. M. Goluzin, Geometric theory of functions of a complex variable, Nauka,
Moscow, 1966.
[3] G. P. Bakhtina, Variational methods and quadratic differentials in problems for
disjoint domains, PhD thesis, Kiev, 1975.
[4] A. K. Bakhtin, G. P. Bakhtina, Yu. B. Zelinskii, Topological-algebraic structures
and geometric methods in complex analysis, Inst. Math. NAS Ukraine, Kiev, 2008.
[5] V. N. Dubinin, Separating transformation of domains and problems of extremal
division // Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Ros. Akad. Nauk,
168 (1988), 48–66.
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a complex variable // Usp. Mat. Nauk, 49 (1994), No. 1, 3–76.
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open sets // Ukr. Math. J., 61 (2009), No. 5, 716–733.
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lapping domains with free poles on rays // Ukr. Math. J., 58 (2006), No. 12,
1950–1954.
Contact information
Andrey Targonskii Zhitomir State University,
Department of Mathematics,
Zhitomir, Ukraine
E-Mail: targonsk@mail.ru,
targonsk@zu.edu.ua
|