Estimates of the inner radii of non-overlapping domains
The paper is devoted to extremal problems of the geometric function theory of complex variable related with estimates of functionals defined on systems of non-overlapping domains. Till now, many such problems have not been solved, though some partial solutions are available. In the paper improved me...
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irk-123456789-1694312020-06-14T01:26:23Z Estimates of the inner radii of non-overlapping domains Denega, I. The paper is devoted to extremal problems of the geometric function theory of complex variable related with estimates of functionals defined on systems of non-overlapping domains. Till now, many such problems have not been solved, though some partial solutions are available. In the paper improved method is proposed for solving problems on extremal decomposition of the complex plane. The main results of the paper generalize and strengthening some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane. 2019 Article Estimates of the inner radii of non-overlapping domains / I. Denega // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 46-56. — Бібліогр.: 13 назв. — англ. 1810-3200 2010 MSC. 30C75 http://dspace.nbuv.gov.ua/handle/123456789/169431 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The paper is devoted to extremal problems of the geometric function theory of complex variable related with estimates of functionals defined on systems of non-overlapping domains. Till now, many such problems have not been solved, though some partial solutions are available. In the paper improved method is proposed for solving problems on extremal decomposition of the complex plane. The main results of the paper generalize and strengthening some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane. |
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Denega, I. |
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Denega, I. Estimates of the inner radii of non-overlapping domains Український математичний вісник |
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Denega, I. |
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Denega, I. |
| title |
Estimates of the inner radii of non-overlapping domains |
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Estimates of the inner radii of non-overlapping domains |
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Estimates of the inner radii of non-overlapping domains |
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Estimates of the inner radii of non-overlapping domains |
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Estimates of the inner radii of non-overlapping domains |
| title_sort |
estimates of the inner radii of non-overlapping domains |
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Інститут прикладної математики і механіки НАН України |
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http://dspace.nbuv.gov.ua/handle/123456789/169431 |
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Estimates of the inner radii of non-overlapping domains / I. Denega // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 46-56. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT denegai estimatesoftheinnerradiiofnonoverlappingdomains |
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2025-07-15T04:15:04Z |
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2025-07-15T04:15:04Z |
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1837684917898575872 |
| fulltext |
Український математичний вiсник
Том 16 (2019), № 1, 46 – 56
Estimates of the inner radii
of non-overlapping domains
Iryna Denega
(Presented by O. A. Dovgoshey)
Dedicated to the memory of Professor Bogdan Bojarski
Abstract. The paper is devoted to extremal problems of the geometric
function theory of complex variable related with estimates of functionals
defined on systems of non-overlapping domains. Till now, many such
problems have not been solved, though some partial solutions are avail-
able. In the paper improved method is proposed for solving problems
on extremal decomposition of the complex plane. The main results of
the paper generalize and strengthening some known results in the theory
of non-overlapping domains with free poles to the case of an arbitrary
arrangement of systems of points on the complex plane.
2010 MSC. 30C75.
Key words and phrases. Inner radius of domain, non-overlapping
domains, Green’s function, transfinite diameter, theorem on minimizing
of the area, the Cauchy inequality.
1. Inequalities for the inner radii of symmetric
non-overlapping domains on the unit circle
Let N, R be the sets of natural and real numbers, respectively, C
be the complex plane, C = C
⋃{∞} be a one point compactification and
R+ = (0,∞). Let r(B, a) be an inner radius of the domain B ⊂ C relative
to a point a ∈ B. The inner radius of the domain B is connected with
Green’s generalized function gB(z, a) of the domain B by the relations
gB(z, a) = − ln |z − a|+ ln r(B, a) + o(1), z → a,
gB(z,∞) = ln |z|+ ln r(B,∞) + o(1), z → ∞.
Received 24.12.2018
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
I. Denega 47
The system of points An := { ak ∈ C, k = 1, n}, n ∈ N, n > 2 is called
n-radial, if |ak| ∈ R+ for k = 1, n and
0 = arg a1 < arg a2 < ... < arg an < 2π.
Denote an+1 := a1, αk := 1
π arg
ak+1
ak
, αn+1 := α1, k = 1, n,
n∑
k=1
αk = 2.
Consider the following extremal problem.
Problem. For any fixed values of γ ∈ (0, n] to find the maximum of
the functional
In(γ) = rγ (B0, 0)
n∏
k=1
r (Bk, ak) ,
where B0, B1, B2,..., Bn, n > 2, are mutually non-overlapping domains
in C and B1, . . . , Bn are symmetric about the unit circle, a0 = 0, |ak| = 1,
k = 1, n, ak ∈ Bk ⊂ C, k = 0, n.
This problem belongs to the class of extremal problems with free
poles. Problems of this type have been studied in many papers (see, for
example, [1–13]). In 1968 P. M. Tamrazov in the paper [10] first attracted
the attention of experts to the study of the extremal problems associated
with quadratic differentials with nonfixed poles possessing a definite free-
dom. And he solved a significant extremal problem of the geometric
theory of functions of a complex variable with five free simple poles. In
the works [1, 9] a very efficient method of separating transformation was
developed with the help of which it was possible to solve some difficult
problems with free poles on a circle.
Problem posed above in the case γ = 1 was formulated as an open
problem in the paper [1]. For γ = 1 and n > 2 this problem was solved
by L. V. Kovalev [3, 4]. Namely, it was shown that under its conditions
the inequality is true
r(B0, 0)
n∏
k=1
r(Bk, ak) 6 r (D0, 0)
n∏
k=1
r (Dk, dk) ,
where dk, Dk, k = 0, n, are, respectively, poles and circular domains of
the quadratic differential
Q(w)dw2 = −w
2n + 2(n2 − 1)wn + 1
w2(wn − 1)2
dw2.
However, for the values of γ 6= 1, this problem has not been solved for
a long time. Only in 2018 in the paper [5] it was solved for n > 2 and
γ ∈ (0, 1). In the paper we propose a method which allows to obtained
estimate of the maximum of the functional In(γ) for this problem for all
n > 2 and γ ∈ (1, n]. We obtain the following result.
48 Estimates of the inner radii of non-overlapping...
Theorem 1. Let n ∈ N, n > 2, γ ∈ (1, n]. Then, for any system of dif-
ferent points {ak}nk=1 of the unit circle and any mutually non-overlapping
domains Bk, ak ∈ Bk ⊂ C, k = 0, n, a0 = 0, and Bk, k = 1, n, are sym-
metric about the unit circle |ak| = 1, the following inequality holds
rγ (B0, 0)
n∏
k=1
r (Bk, ak) 6 n−
γ
2
(
n∏
k=1
r (Bk, ak)
)1− γ
n
. (1.1)
Proof. From the definition of the Green function we have
r (B0, 0) = r
(
B+
0 ,∞
)
, B+ =
{
z;
1
z
∈ B
}
.
Let E ⊂ C be a compact infinite set of the complex plane. Let d(E)
be the transfinite diameter of a compact set E ⊂ C. Then the following
relation holds
r (B0, 0) = r
(
B+
0 ,∞
)
=
1
d(C \B+
0 )
6
1
d(
n⋃
k=1
B
+
k )
. (1.2)
By virtue of the well-known Polya theorem [6, p. 28], [7, p. 34], the
inequality
µE 6 πd2(E),
where µE denotes the Lebesgue measure of a compact set E, is valid.
From whence, we get
d(E) >
(
1
π
µE
) 1
2
.
Then relation (1.2) yields
r (B0, 0) 6
1
d(
n⋃
k=1
B
+
k )
6
1√
1
πµ(
n⋃
k=1
B
+
k )
=
[
1
π
n∑
k=1
µB
+
k
]− 1
2
. (1.3)
For a bounded domain B, a ∈ B we consider the class of all regular
functions ϕ(z), ϕ(a) = 0, ϕ′(a) = 1, given in the domain B, and the area
of an image of the domain B at the mapping by an arbitrary function
ϕ(z). It follows from the theorem of minimization of areas [7, p. 34] that
∫∫
B
|ϕ′(z)|2dxdy > πr2 (B, a) , (1.4)
I. Denega 49
where r (B, a) is the inner radius of the domain B with respect to the
point a. Let us set ϕ1(z) = (z − a). Then relation (1.4) yields
S(B) = µ(B) > πr2 (B, a) .
Inequality (1.3) implies directly that
r (B0, 0) 6
[
1
π
n∑
k=1
µB
+
k
]− 1
2
6
[
1
π
n∑
k=1
µB+
k
]− 1
2
6
[
n∑
k=1
r2
(
B+
k , a
+
k
)
]− 1
2
.
From whence, we get the inequality
r (B0, 0) 6
1
[
n∑
k=1
r2
(
B+
k , a
+
k
)] 1
2
.
With regard for the relation
r
(
B+
k , a
+
k
)
=
r (Bk, ak)
|ak|2
we arrive at the inequality
r (B0, 0) 6
1
n∑
k=1
r2(Bk,ak)
|ak|4
1
2
.
This result and the assumption of Theorem 1 yield the relation
rγ (B0, 0)
n∏
k=1
r (Bk, ak) 6
n∏
k=1
r (Bk, ak)
[
n∑
k=1
r2(Bk ,ak)
|ak|4
] γ
2
. (1.5)
The Cauchy inequality yields automatically the relation
1
n
n∑
k=1
r2 (Bk, ak)
|ak|4
>
[
n∏
k=1
r2 (Bk, ak)
|ak|4
] 1
n
.
And using the equality
n∏
k=1
|ak| = 1,
50 Estimates of the inner radii of non-overlapping...
we get easily
[
n∑
k=1
r2 (Bk, ak)
|ak|4
] γ
2
>
n
[
n∏
k=1
r2 (Bk, ak)
|ak|4
] 1
n
γ
2
> n
γ
2
[
n∏
k=1
r (Bk, ak)
] γ
n
.
Eventually, from (1.5) we obtain
rγ (B0, 0)
n∏
k=1
r (Bk, ak) 6 n−
γ
2
[
n∏
k=1
r (Bk, ak)
]1− γ
n
.
Thus Theorem 1 is proved.
Remark 1. If γ = n, then from Theorem 1 the following inequality
holds
rn (B0, 0)
n∏
k=1
r (Bk, ak) 6 n−
n
2 .
Using Theorem 5.1.1 [8] from Theorem 1 we have the following state-
ment.
Corollary 1. Let n ∈ N, n > 2, γ ∈ (1, n]. Then, for any system of dif-
ferent points {ak}nk=1 of the unit circle and any mutually non-overlapping
domains Bk, ak ∈ Bk ⊂ C, k = 0, n, a0 = 0, and Bk, k = 1, n, are sym-
metric about the unit circle |ak| = 1, the following inequality holds
rγ (B0, 0)
n∏
k=1
r (Bk, ak) 6 2(n−γ) · n− γ
2 ·
(
n∏
k=1
αk
)1− γ
n
.
From Theorem 6.11 [9, p.172] we obtain the following inequality.
Corollary 2. Let n ∈ N, n > 2, γ ∈ (1, n]. Then, under the conditions
of the Corollary 1 the following inequality holds
rγ (B0, 0)
n∏
k=1
r (Bk, ak) 6 n−
γ
2
(
4
n
)n−γ
.
Let
I0n(γ) = rγ(D0, d0) ·
n∏
k=1
r(Dk, dk),
where Dk and dk, k = 0, n are the circular domains and, respectively, the
poles of the quadratic differential
Q(w)dw2 = −γw
2n + 2(n2 − γ)wn + γ
w2(wn − 1)2
dw2.
I. Denega 51
From [1–5] we have
I0n(γ) =
(
4
n
)n
(
2γ
n2
) γ
n
∣∣∣1− 2γ
n2
∣∣∣
n
2
+ γ
n
∣∣∣∣
n−√
2γ
n+
√
2γ
∣∣∣∣
√
2γ
.
We made a comparative analysis of estimate of the maximum of functional
In(γ) obtained in Theorem 1 and the estimate I0n(γ) when γ = n for
n = 2, 10 (see table below).
n I0n(n) n−
n
2 n−
n
2 − I0n(n)
n−n
2 −I0n(n)
I0n(n)
2 0,2500000000 0,5000000000 0,2500000000 1,0000000000
3 0,0897092419 0,1924500897 0,1027408478 1,1452649211
4 0,0273370712 0,0625000000 0,0351629288 1,2862727153
5 0,0070194764 0,0178885438 0,0108690674 1,5484156742
6 0,0015467153 0,0046296296 0,0030829143 1,9932008548
7 0,0002977029 0,0011019372 0,0008042344 2,7014669766
8 0,0000508051 0,0002441406 0,0001933355 3,8054339019
9 0,0000077826 0,0000508053 0,0000430227 5,5280820279
10 0,0000010811 0,0000100000 0,0000089189 8,2502515613
2. Inequalities for the inner radii of non-overlapping
domains on the complex plane
From method of proof of the Theorem 1 we can obtain estimates of
the next functional
Jn(γ) = [r (B0, 0) r (B∞,∞)]γ
n∏
k=1
r (Bk, ak) ,
considered, for example, in the papers [1, 8, 9, 13], in which for Jn(γ)
in particular cases for some values of γ, the following inequality was
established
Jn(γ) 6
(
4
n
)n
(
4γ
n2
) 2γ
n
∣∣∣1− 4γ
n2
∣∣∣
2γ
n
+n
2
∣∣∣∣
n− 2
√
γ
n+ 2
√
γ
∣∣∣∣
2
√
γ
.
Equality in this inequality is achieved when 0, ∞, ak and B0, B∞, Bk,
k = 1, n, are, respectively, poles and circular domains of the quadratic
differential
Q(w)dw2 = −γw
2n + (n2 − 2γ)wn + γ
w2(wn − 1)2
dw2.
52 Estimates of the inner radii of non-overlapping...
Namely, the following result holds.
Theorem 2. Let n ∈ N, n > 2, γ ∈ (0, n+2
2 ]. Then, for any fixed
system of different points An = {ak}nk=1 ∈ C/{0,∞} and any mutually
non-overlapping domains B0, B∞, Bk, a0 = 0 ∈ B0 ⊂ C, ∞ ∈ B∞ ⊂ C,
ak ∈ Bk ⊂ C, k = 1, n, the following inequality holds
Jn(γ) 6 (n+ 1)−
n+1
n+2
γ
(
n∏
k=1
r (Bk, ak)
)1− 2γ
n+2
(
n∏
k=1
|ak|
) 2γ
n+2
. (2.1)
Proof. Using inequalities (1.2), (1.3) and (1.4), we have
r (B0, 0) 6
1
[
r2
(
B+
∞, 0
)
+
n∑
k=1
r2
(
B+
k , a
+
k
)] 1
2
=
1
[
r2 (B∞,∞) +
n∑
k=1
r2(Bk,ak)
|ak |4
] 1
2
,
r (B∞,∞) 6
1
[
r2 (B0, 0) +
n∑
k=1
r2 (Bk, ak)
] 1
2
.
Taking into account the Cauchy inequality
1
n+ 1
(
r2 (B0, 0) +
n∑
k=1
r2 (Bk, ak)
)
>
[
r2 (B0, 0)
n∏
k=1
r2 (Bk, ak)
] 1
n+1
.
Then
(
r2 (B0, 0) +
n∑
k=1
r2 (Bk, ak)
) 1
2
> (n + 1)
1
2
[
r (B0, 0)
n∏
k=1
r (Bk, ak)
] 1
n+1
.
Analogically,
(
r2 (B∞,∞) +
n∑
k=1
r2 (Bk, ak)
|ak|4
) 1
2
>(n+1)
1
2
[
r (B∞,∞)
n∏
k=1
r (Bk, ak)
|ak|2
] 1
n+1
.
Thus,
r (B∞,∞) 6
1
(n+ 1)
1
2 (r (B0, 0))
1
n+1
(
n∏
k=1
r (Bk, ak)
) 1
n+1
,
I. Denega 53
r (B0, 0) 6
(
n∏
k=1
|ak|
) 2
n+1
(n + 1)
1
2 (r (B∞,∞))
1
n+1
(
n∏
k=1
r (Bk, ak)
) 1
n+1
.
Further, we obtain the relations
r (B0, 0) r (B∞,∞)
6
(
n∏
k=1
|ak|
) 2
n+1
(n+ 1) (r (B0, 0) r (B∞,∞))
1
n+1
(
n∏
k=1
r (Bk, ak)
) 2
n+1
,
(r (B0, 0) r (B∞,∞))1+
1
n+1 6
(
n∏
k=1
|ak|
) 2
n+1
(n+ 1)
(
n∏
k=1
r (Bk, ak)
) 2
n+1
,
r (B0, 0) r (B∞,∞) 6
(
n∏
k=1
|ak|
) 2
n+2
(n+ 1)
n+1
n+2
(
n∏
k=1
r (Bk, ak)
) 2
n+2
,
from which inequality (2.1) of the Theorem 2 follows
Jn(γ) 6
n∏
k=1
r (Bk, ak)
(
n∏
k=1
|ak|
) 2
n+2
(n+ 1)
n+1
n+2
γ
(
n∏
k=1
r (Bk, ak)
) 2γ
n+2
= (n+ 1)−
n+1
n+2
γ
(
n∏
k=1
r (Bk, ak)
)1− 2γ
n+2
(
n∏
k=1
|ak|
) 2γ
n+2
.
Remark 2. If γ = n+2
2 and
n∏
k=1
|ak| 6 1 then from Theorem 2 the
following inequality holds
Jn(γ) 6 (n+ 1)−
n+1
2 .
54 Estimates of the inner radii of non-overlapping...
From Theorem 2 we have the following statements.
Corollary 3. Let n ∈ N, n > 2, γ ∈ (0, n+2
2 ]. Then, for any sys-
tem of different points {ak}nk=1 of the unit circle and any mutually non-
overlapping domains B0, B∞, Bk, a0 = 0 ∈ B0 ⊂ C, ∞ ∈ B∞ ⊂ C,
ak ∈ Bk ⊂ C, k = 1, n, the following inequality holds
Jn(γ) 6 (n + 1)−
n+1
n+2
γ
(
2n
n∏
k=1
αk
)1− 2γ
n+2
.
Corollary 4. Let n ∈ N, n > 2, γ ∈ (0, n+2
2 ]. Then, under the condi-
tions of the Corollary 3 the following inequality holds
Jn(γ) 6 (n+ 1)−
n+1
n+2
γ
(
4
n
)n− 2nγ
n+2
.
From Theorem 2 on condition B0 ⊂ U , we obtain a result giving some
estimate in the problem stated in the paper of G. P. Bakhtina [2].
Theorem 3. Let n ∈ N, n > 2, γ ∈ (0, n+2
2 ] and B0 ⊂ U . Then, for
any system of different points {ak}nk=1 of the unit circle and any mutually
non-overlapping domains Bk, ak ∈ Bk ⊂ C, k = 0, n, a0 = 0, and Bk,
k = 1, n, are symmetric about the unit circle |ak| = 1, the following
inequality holds
r2γ (B0, 0)
n∏
k=1
r (Bk, ak)
6 (n+ 1)−
γ(n+1)
n+2
(
n∏
k=1
r (Bk, ak)
)1− 2γ
n+2
(
n∏
k=1
|ak|
) 2γ
n+2
.
Let p, q ∈ N. A set of points Ap,q :=
{
ak,s ∈ C : k = 1, p, s = 1, q
}
is called (p, q)-radial system, if for all k = 1, p and s = 1, q the relations
hold
0 < |ak,1| < . . . < |ak,q| <∞;
arg ak,1 = arg ak,2 = . . . = arg ak,q =: θk =: θk(Ap,q);
0 = θ1 < θ2 < . . . < θp < θp+1 := 2π.
Then, from method of proofs of the above presented theorems we obtain
the following corollaries for Ap,q-radial systems points.
Corollary 5. Let p, q ∈ N, p > 2, γ ∈ (0, pq]. Then, for any fixed
(p, q)-radial system of points Ap,q = {ak,s}, k = 1, p, s = 1, q, and any
I. Denega 55
mutually non-overlapping domains B0, Bk,s, ak,s ∈ Bk,s ⊂ C, k = 1, p,
s = 1, q, a0 = 0 ∈ B0 ⊂ C, the inequality holds
rγ (B0, 0)
p∏
k=1
q∏
s=1
r (Bk,s, ak,s)
6 (pq)−
γ
2
(
p∏
k=1
q∏
s=1
r (Bk,s, ak,s)
)1− γ
pq
(
p∏
k=1
q∏
s=1
|ak,s|
) 2γ
pq
.
Corollary 6. Let p, q ∈ N, p > 2, γ ∈ (0, pq+2
2 ]. Then, for any fixed
(p, q)-radial system of points Ap,q = {ak,s}, k = 1, p, s = 1, q, and
any mutually non-overlapping domains B0, B∞, Bk,s, ak,s ∈ Bk,s ⊂ C,
k = 1, p, s = 1, q, a0 = 0 ∈ B0 ⊂ C, ∞ ∈ B∞ ⊂ C, the inequality holds
[r (B0, 0) r (B∞,∞)]γ
p∏
k=1
q∏
s=1
r (Bk,s, ak,s)
6 (pq + 1)
− pq+1
pq+2
γ
(
p∏
k=1
q∏
s=1
r (Bk,s, ak,s)
)1− 2γ
pq+2
(
p∏
k=1
q∏
s=1
|ak,s|
) 2γ
pq+2
.
Acknowlegements
The author thanks the Prof. Alexander Bakhtin for posing problems,
careful analysis of this work and remarks.
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Contact information
Iryna Denega Institute of mathematics of National
Academy of Sciences of Ukraine,
Kyiv, Ukraine
E-Mail: iradenega@gmail.com
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