Separating transformation in a problem on extremal decomposition of the complex plane
The paper is devoted to investigation of the problems of geometric function theory of a complex variable. A general problem of the description of extremal configurations maximizing the product of the inner radii of mutually non-overlapping domains is studied.
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Інститут математики НАН України
2017
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Репозитарії
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552626410749952 |
|---|---|
| author | Denega, Iryna Targonskii, Andrey Denega, Iryna Targonskii, Andrey |
| author_facet | Denega, Iryna Targonskii, Andrey Denega, Iryna Targonskii, Andrey |
| author_institution_txt_mv | [
{
"author": "Iryna Denega",
"institution": "Institute of Mathematics of the National Academy of Sciences of Ukraine"
},
{
"author": "Andrey Targonskii",
"institution": "Zhytomyr Ivan Franko State University"
}
] |
| author_sort | Denega, Iryna |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-13T13:58:16Z |
| description | The paper is devoted to investigation of the problems of geometric function theory of a complex variable. A general problem of the description of extremal configurations maximizing the product of the inner radii of mutually non-overlapping domains is studied. |
| first_indexed | 2026-08-04T01:02:25Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2017, том 14, № 1, 147–155
УДК 517.54
I. V. Denega 1, A. L. Targonskii 2
1(Institute of Mathematics of the National Academy of Sciences of
Ukraine, Kyiv)
2(Zhytomyr Ivan Franko State University, Zhytomyr)
1 iradenega@yandex.ru, 2 targonsk@zu.edu.ua
Separating transformation in a
problem on extremal decomposition of
the complex plane
Dedicated to Prof. Yu. B. Zelinskii on the occasion of his 70th birthday
The paper is devoted to investigation of the problems of geometric functi-
on theory of a complex variable. A general problem of the description
of extremal configurations maximizing the product of the inner radii of
mutually non-overlapping domains is studied.
Робота присвячена дослiдженню вiдкритих проблем геометричної тео-
рiї функцiй комплексної змiнної. Зокрема, вивчається загальна пробле-
ма з описання екстремальних конфiгурацiй, що мiнiмiзують добуток
внутрiшнiх радiусiв попарно неперетинних областей.
Let N and R be the sets of natural and real numbers, respectively; C be
the Complex plane and let C = C∪{∞} be its one-point compactification,
R+ = (0,∞). Let r(B,a) be the inner radius of the domain B ⊂ C with
respect to a point a ∈ B [1, 2]. An inner radius is a generalization of a
conformal radius for multiply connected domains.
Definition 1. A finite collection of arbitrary fixed domains {Bk}nk=1,
n ∈ N, n > 2, such as Bk ⊂ C, Bk ∩ Bm = ∅, k 6= m, k,m = 1,n is called
a system of non-overlapping domains.
Definition 2. Let n ∈ N, n > 2. A set of points An :=
:=
{
ak ∈ C : k = 1,n
}
is called n-radial system if: |ak| ∈ R+, k = 1,n,
0 = arg a1 < arg a2 < . . . < arg an < 2π.
c© I. V. Denega, A. L. Targonskii, 2017
148 I. V. Denega, A. L. Targonskii
Denote Pk(An) := {w : arg ak < argw < arg ak+1},
θk := arg ak, an+1 := a1, θn+1 := 2π, αk := 1
π arg ak+1
ak
, αn+1 := α1,
k = 1, n. Let χ(t) = 1
2 (t+ t−1). For an arbitrary n-radial system of points
An = {ak}, k = 1,n, we assume
L(An) :=
n∏
k=1
χ
(∣∣∣ ak
ak+1
∣∣∣ 1
2αk
)
|ak|.
The class of n-radial systems of points for which L(An) = 1 automati-
cally includes all systems of n distinct points that are located on the unit
circle.
The goal of the present work is the construction of sharp upper bounds
for a functional of the form
Jn(γ) = [r (B0,0) r (B∞,∞)]
γ
n∏
k=1
r (Bk,ak) ,
where γ ∈ R+, An = {ak}nk=1 is n-radial system of points, B0, B∞,
{Bk}nk=1 is system of pairwise disjoint domains, ak ∈ Bk ⊂ C, k = 0, n,
∞ ∈ B∞ ⊂ C.
Note that to describe the extremal configurations of the domains we
use notion of quadratic differential (see, for example, [1, P. 63–70]).
The following statement holds
Theorem 1. Let n ∈ N, n > 7, 0 < γ 6 γn, γn = 0,08n2. Then
for any n-radial system of points An = {ak}nk=1 such that L(An) = 1
and any set of mutually non-overlapping domains Bk, a0 = 0 ∈ B0 ⊂ C,
∞ ∈ B∞ ⊂ C, ak ∈ Bk ⊂ C, k = 1,n, the following inequality holds
[r (B0,0) r (B∞,∞)]
γ
n∏
k=1
r (Bk,ak) 6 [r (Λ0,0) r (Λ∞,∞)]
γ
n∏
k=1
r (Λk, λk) ,
where domains Λ0, Λ∞, Λk, and points 0,∞, λk, k = 1,n, are, respectively,
circular domains and poles of the quadratic differential
Q(w)dw2 = −γw
2n + (n2 − 2γ)wn + γ
w2(wn − 1)2
dw2. (1)
Separating transformation in a problem on . . . 149
Proof. Consider the system of functions
ζ = πk(w) = −i
(
e−iθkw
) 1
αk , k = 1,n.
The family of functions {πk(w)}nk=1 is called admissible for separating
transformation of domains B0, B∞, Bk, k = 1,n, with respect to the
angles {Pk}nk=1. Let Ω
(1)
k , k = 1,n, denote a domain of the plane Cζ
obtained as a result of the union of the connected component of the set
πk(Bk
⋂
P k) containing the point πk(ak) with the own symmetric reflecti-
on relative to the imaginary axis. In turn, by Ω
(2)
k , k = 1,n, we denote the
domain of the plane Cζ obtained as a result of the union of the connected
component of the set πk(Bk+1
⋂
P k) containing the point πk(ak+1) with
the own symmetric reflection relative to the imaginary axis, Bn+1 := B1,
πn(an+1) := πn(a1). In addition, by Ω
(0)
k we denote the domain of the
plane Cζ obtained as a result of the union of the connected component of
the set πk(B0
⋂
P k) containing the point ζ = 0 with the own symmetric
reflection relative to the imaginary axis. Accordingly, by Ω
(∞)
k we denote
the domain of the plane Cζ obtained as a result of the union of the
connected component of the set πk(B∞
⋂
P k) containing the point ζ =∞
with the own symmetric reflection relative to the imaginary axis. Denote
πk(ak) := ω
(1)
k , πk(ak+1) := ω
(2)
k , k = 1,n. It follows from the definition of
the function πk(w) that
|πk(w)− ω(1)
k | ∼
1
αk
|ak|
1
αk
−1 · |w − ak|, w → ak, w ∈ Pk,
|πk(w)− ω(2)
k | ∼
1
αk
|ak+1|
1
αk
−1 · |w − ak+1|, w → ak+1, w ∈ Pk,
|πk(w)| ∼ |w|
1
αk , w → 0, w ∈ Pk,
|πk(w)| ∼ |w|
1
αk , w →∞, w ∈ Pk.
Taking into account corresponding results of papers [2, 3], we have the
inequalities
r (Bk,ak) 6
r
(
Ω
(1)
k ,ω
(1)
k
)
· r
(
Ω
(2)
k ,ω
(2)
k
)
1
αk
|ak|
1
αk
−1 · 1
αk−1
|ak|
1
αk−1
−1
1
2
, k = 1,n, (2)
150 I. V. Denega, A. L. Targonskii
r (B0,0) 6
[
n∏
k=1
rα
2
k
(
Ω
(0)
k ,0
)] 1
2
,
r (B∞,∞) 6
[
n∏
k=1
rα
2
k
(
Ω
(∞)
k ,∞
)] 1
2
. (3)
An equality in (2)–(3) is fully investigated in [2, Theorem 1.9]. By using
this inequalities we have the expansion
Jn(γ) 6
n∏
k=1
(
r
(
Ω
(0)
k ,0
)
r
(
Ω
(∞)
k ,∞
)) γα2
k
2 ×
×
r
(
Ω
(1)
k ,ω
(1)
k
)
· r
(
Ω
(2)
k ,ω
(2)
k
)
1
αk
|ak|
1
αk
−1 · 1
αk−1
|ak|
1
αk−1
−1
1
2
.
Elementary calculations show that
Jn(γ) 6
(
n∏
k=1
αk
)
n∏
k=1
|ak|
1
αk + |ak+1|
1
αk
(|ak||ak+1|)
1
2αk
· |ak|×
×
n∏
k=1
(
r
(
Ω
(0)
k ,0
)
r
(
Ω
(∞)
k ,∞
))γα2
k
·
r
(
Ω
(1)
k ,ω
(1)
k
)
· r
(
Ω
(2)
k ,ω
(2)
k
)
(
|ak|
1
αk + |ak+1|
1
αk
)2
1
2
.
Hence
Jn(γ) 6
(
n∏
k=1
αk
)
n∏
k=1
(∣∣∣∣ akak+1
∣∣∣∣ 1
2αk
+
∣∣∣∣ak+1
ak
∣∣∣∣ 1
2αk
)
|ak|×
×
n∏
k=1
(
r
(
Ω
(0)
k ,0
)
r
(
Ω
(∞)
k ,∞
))γα2
k
·
r
(
Ω
(1)
k ,ω
(1)
k
)
· r
(
Ω
(2)
k ,ω
(2)
k
)
(
|ak|
1
αk + |ak+1|
1
αk
)2
1
2
.
Separating transformation in a problem on . . . 151
Thus
n∏
k=1
(∣∣∣∣ akak+1
∣∣∣∣ 1
2αk
+
∣∣∣∣ak+1
ak
∣∣∣∣ 1
2αk
)
|ak| =
n∏
k=1
χ
(∣∣∣ ak
ak+1
∣∣∣ 1
2αk
)
|ak| = L(An),
and we obtain
Jn(γ) 6 2n ·
(
n∏
k=1
αk
)
· L(An)×
×
n∏
k=1
r
(
Ω
(1)
k ,ω
(1)
k
)
· r
(
Ω
(2)
k ,ω
(2)
k
)
(
|ak|
1
αk + |ak+1|
1
αk
)2 (
r
(
Ω
(0)
k ,0
)
r
(
Ω
(∞)
k ,∞
))γα2
k
1
2
.
For convenience, right side of the last inequality we multiply and divide
by the value √γ and get the inequality
Jn(γ) 6
(
2
√
γ
)n
·
(
n∏
k=1
αk
√
γ
)
· L(An)×
×
n∏
k=1
r
(
Ω
(1)
k ,ω
(1)
k
)
· r
(
Ω
(2)
k ,ω
(2)
k
)
(
|ak|
1
αk + |ak+1|
1
αk
)2 (
r
(
Ω
(0)
k ,0
)
r
(
Ω
(∞)
k ,∞
))γα2
k
1
2
,
where |ak|
1
αk + |ak+1|
1
αk = |ω(2)
k − ω
(1)
k |, k = 1,n. Equality in the last
inequality is attained if and only if the equality is attained in inequalities
(2) – (3) for all k = 1,n. Each expression contained in braces in the last
inequality is value of the functional
Kτ = [r (B0,0) r (B∞,∞)]
τ2
· r (B1,a1) r (B2,a2)
|a1 − a2|2
(4)
on the system of non-overlapping domains {Ω(0)
k ,Ω
(1)
k ,Ω
(2)
k ,Ω
(∞)
k } and
corresponding points system {0, ω(1)
k , ω
(2)
k ,∞}, k = 1,n. Functional
evaluation (4) in the case of fixed poles was first obtained by V.N. Dubi-
nin, then by G.V. Kuzmina, E. Emelyanov, A. L. Tarhonskii. Note that
in our case the points a1 and a2 of the functional (4) are not fixed and not
symmetrical. So, fractional-linear transformations can not reduce them to
−1 and 1. V.N. Dubinin and G.V. Kuzmina considered the case when
152 I. V. Denega, A. L. Targonskii
the points a1 and a2 are symmetrical and equal to 1 and −1, respecti-
vely. Therefore decisive role is played corollary of Theorem 4.1.1 in [1,
p. 169]. From here, based on invariance of the functional (4) as in the
proof of Theorem 4.1.1 in [1], we have that Kτ 6 Φ(τ), τ > 0, where
Φ(τ) = τ2τ
2 · |1− τ |−(1−τ)2 · (1 + τ)−(1+τ)
2
. Then
Jn(γ) 6
(
2
√
γ
)n( n∏
k=1
αk
√
γ
)[
n∏
k=1
Φ(τk)
]1/2
=
=
(
2
√
γ
)n [ n∏
k=1
(
τ
2τ2
k+2
k · |1− τk|−(1−τk)
2
· (1 + τk)−(1+τk)
2
)] 1
2
,
where τk =
√
γ · αk, k = 1,n. Let
S(x) = x2x
2+2 · |1− x|−(1−x)
2
· (1 + x)
−(1+x)2 and Ψ(x) = ln(S(x)).
S(x) is logarithmically convex function on the interval [0, x0],
x0 ≈ 0,88441. Further similarly [3, 4] we consider the following extremal
problem
n∏
k=1
S(xk) −→ max,
n∑
k=1
xk = 2
√
γ, xk = αk
√
γ.
Let X(0) =
{
x
(0)
k
}n
k=1
be any extremal set of points in above menti-
oned problem. In a similar way as in [4], we obtain the following result:
if 0 < x
(0)
k < x
(0)
j , then
Ψ′(x
(0)
k ) = Ψ′(x
(0)
j ), (5)
where k,j = 1,n, k 6= j,
Ψ′(x) = 4x ln(x)− 2(x− 1) ln |x− 1| − 2(x+ 1) ln(x+ 1) +
2
x
(see Fig. 1).
Separating transformation in a problem on . . . 153
Fig. 1: Graph of the function y = Ψ′(x)
By using similar arguments as in the paper [4] and the relation (5) we
will prove that x(0)1 = x
(0)
2 = · · · = x
(0)
n . Taking into account properties
of the function Ψ′(x) and conditions of Theorem we obtain the following
relation: (x1 − 0,56)n + (x2 − x1) > 0 for n > 7. Therefore, we have
nx1 + (x2 − x1) > 0,56n. Finally we get
(n− 1)x1 + x2 > 2
√
γn, γn = 0,08n2, n > 7.
Thus, in accordance with [4], we agree to say that the set of points{
x
(0)
k
}n
k=1
, n > 7, can not be extremal if x(0)n ∈ (x0, 2]. Consequently, for
an extremal set
{
x
(0)
k
}n
k=1
is only possible in the case when x(0)k ∈ (0, x0],
k = 1,n, and x
(0)
1 = x
(0)
2 = . . . = x
(0)
n . For γ < γn, n > 7, all previ-
154 I. V. Denega, A. L. Targonskii
ous arguments are valid. The equality case is straightforward to verify.
Theorem 1 is proved.
Corollary 1. Under conditions of Theorem 1 the following inequality holds
[r(B0,0)r(B∞,∞)]
γ
n∏
k=1
r(Bk, ak) 6
6
(
4
n
)n
(
4γ
n2
)( 4γ
n2 )∣∣∣ 2√γn − 1
∣∣∣( 2
√
γ
n −1
)2 (
2
√
γ
n + 1
)( 2
√
γ
n +1
)2
n
2
.
Equality is attained if 0,∞, ak, and B0, B∞, Bk, k = 1,n, are, respectively,
poles and circular domains of the quadratic differential (1).
Corollary 2. Let n ∈ N, n > 7, 0 < γ 6 γn, γn = 0,08n2. Then for
any different points on the circle |ak| = 1, k = 1,n, and any pairwise non-
overlapping domains Bk, a0 = 0 ∈ B0 ⊂ C, ∞ ∈ B∞ ⊂ C, ak ∈ Bk ⊂ C,
k = 1,n, the following inequality holds
[r (B0,0) r (B∞,∞)]
γ
n∏
k=1
r (Bk,ak) 6 [r (Λ0,0) r (Λ∞,∞)]
γ
n∏
k=1
r (Λk, λk) ,
where domains Λ0, Λ∞, Λk, and points 0,∞, λk, k = 1,n, are, respectively,
circular domains and poles of the quadratic differential (1).
References
[1] Bakhtin A.K., Bakhtina G.P., Zelinskii Yu. B. Topological-algebraic
structures and geometric methods in complex analysis // Proceedings of
the Institute of Mathematics of NAS of Ukraine, 2008. — 308 p. (in Russi-
an).
[2] Dubinin V.N. Symmetrization method in geometric function theory of
complex variables // Successes Mat. Science. — 1994. — 49, No. 1 (295). —
P. 3 – 76 (in Russian); Engl. transl. in: Russian Math. Surveys. — 1994. —
49, № 1. — P. 1 – 79.
Separating transformation in a problem on . . . 155
[3] Dubinin V.N. Separating transformation of domains and problems on
extremal decomposition // Notes scientific. sem. Leningr. Dep. of Math.
Inst. AN USSR. — 1988. — 168. — P. 48 – 66 (in Russian); Engl. transl. in:
J. Soviet Math. — 1991. — 53, No. 3. — P. 252 — 263.
[4] Kovalev L.V. On the problem of extremal decomposition with free poles
on a circle // Dal’nevostochnyi Mat. Sb. — 1996. — 2. — P. 96 – 98 (in
Russian).
[5] Lavrentev M.A. To the theory of conformal mappings // Trudy Fiz.-Mat.
Inst. AN SSSR. — 1934. — 5. — P. 159 – 245 (in Russian).
[6] Goluzin G.M. Geometric theory of functions of a complex variable. — Amer.
Math. Soc., Providence, R.I., 1969. — 676 p.
[7] Hayman V.K. Multivalent functions. — Cambridge University Press,
1958. — 151 p.
[8] Jenkins J.A. Univalent functions and conformal mapping. — M.: Publishing
House of Foreign Literature, 1962. — 256 p. (in Russian).
[9] Kolbina L. I. Conformal mapping of the unit circle onto non-overlapping
domains // Bulletin of Leningrad University. — 1955. — 5. — P. 37 – 43 (in
Russian).
[10] Kuzmina G.V. Problems on extremal decomposition of the riemann
sphere // Notes scientific. sem. Leningr. Dep. of Math. Inst. AN USSR. —
2001. — 276. — P. 253 – 275 (in Russian); Engl. transl. in: J. of Math. Sci —
2003. — 118, No. 1. — P. 4880 – 4894.
[11] Bakhtin A.K., Denega I. V. Addendum to a theorem on extremal
decomposition of the complex plane // Bulletin de la société des sciences et
des lettres de Lódź, Recherches sur les déformations. — 2012. — 62, № 2. —
P. 83 – 92.
[12] Bakhtin A.K., Bakhtina G.P., Vjun V.E. On some inequalities in the
theory of non-overllaping domains // Zb. Pr. Inst. Mat. NAN Ukr. — K.:
In-te of mathematics of NAS of Ukraine, 2014. — 11, № 1. — P. 141 – 152
(Ukrainian).
[13] Dvorak I. Ya. On the inner radii of symmetric nonoverlapping domains //
Ukrainskii Matematychnyi Visnyk. — 2016. — 13, No. 2. — P. 10 – 19 (in
Russian).
|
| id | oai:trim.imath.kiev.ua:article-95 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:02:25Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/28/e1cdd1edb521fe902227ac9af61af028.pdf |
| spelling | oai:trim.imath.kiev.ua:article-952018-02-13T13:58:16Z Separating transformation in a problem on extremal decomposition of the complex plane Denega, Iryna Targonskii, Andrey Denega, Iryna Targonskii, Andrey The paper is devoted to investigation of the problems of geometric function theory of a complex variable. A general problem of the description of extremal configurations maximizing the product of the inner radii of mutually non-overlapping domains is studied. Інститут математики НАН України 2017-04-25 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/95 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 14 No. 1 (2017): Vol. 14 No. 1 (2017): Analysis and Applications; 147-155 Сборник Трудов Института математики НАН Украины; Том 14 № 1 (2017): Том 14 № 1 (2017): Анализ и приложения; 147-155 Збірник Праць Інституту математики НАН України; Том 14 № 1 (2017): Аналіз та застосування; 147-155 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/95/87 Авторське право (c) 2017 Праці Інституту математики НАН України |
| spellingShingle | Denega, Iryna Targonskii, Andrey Denega, Iryna Targonskii, Andrey Separating transformation in a problem on extremal decomposition of the complex plane |
| title | Separating transformation in a problem on extremal decomposition of the complex plane |
| title_full | Separating transformation in a problem on extremal decomposition of the complex plane |
| title_fullStr | Separating transformation in a problem on extremal decomposition of the complex plane |
| title_full_unstemmed | Separating transformation in a problem on extremal decomposition of the complex plane |
| title_short | Separating transformation in a problem on extremal decomposition of the complex plane |
| title_sort | separating transformation in a problem on extremal decomposition of the complex plane |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/95 |
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