About one extremal problem for open sets and partially non-overlapping domains
Sharp estimates of a product of the inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of any finite number of free poles on the rays.
Збережено в:
| Дата: | 2019 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Russian |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2019
|
| Назва видання: | Український математичний вісник |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/169442 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | About one extremal problem for open sets and partially non-overlapping domains / A.L. Targonskii, I.I. Targonskaya, K. Vashenko // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 228-238. — Бібліогр.: 22 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-169442 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1694422020-06-14T01:26:24Z About one extremal problem for open sets and partially non-overlapping domains Targonskii, A.L. Targonskaya, I.I. Vashenko, K. Sharp estimates of a product of the inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of any finite number of free poles on the rays. 2019 Article About one extremal problem for open sets and partially non-overlapping domains / A.L. Targonskii, I.I. Targonskaya, K. Vashenko // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 228-238. — Бібліогр.: 22 назв. — англ. 1810-3200 2010 MSC. Primary 30C70, 30C75. http://dspace.nbuv.gov.ua/handle/123456789/169442 ru Український математичний вісник Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
Russian |
| description |
Sharp estimates of a product of the inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of any finite number of free poles on the rays. |
| format |
Article |
| author |
Targonskii, A.L. Targonskaya, I.I. Vashenko, K. |
| spellingShingle |
Targonskii, A.L. Targonskaya, I.I. Vashenko, K. About one extremal problem for open sets and partially non-overlapping domains Український математичний вісник |
| author_facet |
Targonskii, A.L. Targonskaya, I.I. Vashenko, K. |
| author_sort |
Targonskii, A.L. |
| title |
About one extremal problem for open sets and partially non-overlapping domains |
| title_short |
About one extremal problem for open sets and partially non-overlapping domains |
| title_full |
About one extremal problem for open sets and partially non-overlapping domains |
| title_fullStr |
About one extremal problem for open sets and partially non-overlapping domains |
| title_full_unstemmed |
About one extremal problem for open sets and partially non-overlapping domains |
| title_sort |
about one extremal problem for open sets and partially non-overlapping domains |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2019 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169442 |
| citation_txt |
About one extremal problem for open sets and partially non-overlapping domains / A.L. Targonskii, I.I. Targonskaya, K. Vashenko // Український математичний вісник. — 2019. — Т. 16, № 2. — С. 228-238. — Бібліогр.: 22 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT targonskiial aboutoneextremalproblemforopensetsandpartiallynonoverlappingdomains AT targonskayaii aboutoneextremalproblemforopensetsandpartiallynonoverlappingdomains AT vashenkok aboutoneextremalproblemforopensetsandpartiallynonoverlappingdomains |
| first_indexed |
2025-07-15T04:15:47Z |
| last_indexed |
2025-07-15T04:15:47Z |
| _version_ |
1837684960434061312 |
| fulltext |
Український математичний вiсник
Том 16 (2019), № 2, 228 – 238
About one extremal problem for open sets and
partially non-overlapping domains
Andrey L. Targonskii, Irina I. Targonskaya, K. Vashenko
(Presented by O. Dovgoshey)
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.
2010 MSC. Primary 30C70, 30C75.
Key words and phrases. Inner radius of domain, quadratic differ-
ential, piecewise-separating transformation, the Green function, radial
systems of points, logarithmic capacity, variational formula.
1. Introduction
This paper belongs to the theory of extremal problems on classes of
disjoint domains, which is a separate direction in the geometric theory of
functions of a complex variable. The start of these investigations is asso-
ciated with the paper by M. A. Lavrent’ev [1]. He found the maximum
of some functional with respect to two simply connected domains with
two fixed points. We note that he applied this result to some aerodynam-
ics 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, includ-
ing Y.E. Alenitsin, M. A. Lebedev, J. Jenkins, P.M. Tamrazov, P. P. Ku-
farev, and others. Using the idea of P.M. Tamrazov, G. P. Bakhtina
solved, for the first time, the problem with the so-called “free poles” on a
unit circle (see, e.g., [3]).
An important step for the development of this topic was the papers
by V. N. Dubinin. He proposed a new method of piecewise separating
Received 01.05.2019
The author is grateful to Prof. Bakhtin for suggesting problems and useful discus-
sions.
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
A. L. Targonskii, I. I. Targonskaya, K. Vashenko 229
transformation and first solved a number of extremal problems for any
multiconnected disjoint domains (see, e.g., [4–6]). Now, this type of ex-
tremal problems is used in investigations in the holomorphic dynamics.
In the last decade, Bakhtin’s method of “managing functional” is ac-
tively used. A. K. Bakhtin solved a number of extremal problems for
the so-called “radial systems of points” (see, e.g., [4, 7–20]). Namely this
method will be applied in what follows.
Let N and R be the sets of natural and real numbers, respectively,
let C be the plane of complex numbers, and let C = C
∪
{∞} be the
Riemannian sphere, R+ = (0;∞).
For a fixed number n ∈ N of points
An = {ak}nk=1 ,
the following relations are valid:
0 = arg a1 < arg a2 < ... < arg an < 2π. (1.1)
For such systems of points, we consider the following sizes:
σk =
1
π
(arg ak+1 − arg ak) , k = 1, 2, ..., n, an+1 := a1.
Consider the system of angular domains
Mk := {w : arg ak < argw < arg ak+1} , k = 1, n, an+1 := a1,
and the following “managing functional” for an arbitrary system of points
An:
T(An) =
n∏
k=1
χ
(∣∣∣∣ akak+1
∣∣∣∣ 1
2σk
)
|ak|,
where χ(t) = 1
2(t+
1
t ).
Let D, D ⊂ C, be any open set, and let the point w = a ∈ D. By
D(a), we denote a connected component of the set D, which contains the
point a. For an arbitrary system of points An = {ak ∈ C : k = 1, n},
satisfying condition (1.1) and for the open set D, An ⊂ D, we denote, by
Dk (as) , the connected component of the set D (as)
∩
Mk containing the
point as, k = 1, n, s = k, k + 1, an+1 := a1.
We say that the open set D, An ⊂ D, satisfies the condition of dis-
jointness relative to the system of points An, if the relation
Dk(ak)
∩
Dk(ak+1) = ∅, (1.2)
230 About one extremal problem for open sets...
k = 1, n, on all angles Mk, holds.
System domains {Bk}nk=1, k = 1, n, the define system partially non-
overlapping domains, if
D :=
n∪
k=1
Bk, (1.3)
is open set, if she meets a condition (1.2).
Let
gB (z, a) = hB,a(z) + log
1
|z − a|
be the generalized Green function of a domain B relative to the point
a ∈ B. If a = ∞, then
gB (z,∞) = hB,∞(z) + log
1
|z|
.
The quantity
r(B, a) := exp (hB,a(z))
stands for the internal radius of the domain B ⊂ C relative to the point
a ∈ B (see [4–6,16,17,21]).
We use the concept of a quadratic differential. We note that the main
results on the theory of quadratic differentials can be found in work [22].
In what follows, we consider the following problems.
Problem 1. Let n ∈ N, n ≥ 2, α ≥ 0. To find a maximum of the
functional
n∏
k=1
(|ak+1 − ak|α · r (D, ak)) ,
where An = {ak}nk=1 is any ray system of points that satisfies condition
(1.1), D is an open set that satisfies condition (1.2), ak ∈ D ⊂ C, and to
describe all extremals (k = 1, n).
Problem 2. Let n ∈ N, n ≥ 2, α ≥ 0. To find a maximum of the
functional
n∏
k=1
(|ak+1 − ak|α · r (Bk, ak)) ,
where An = {ak}nk=1 is any ray system of points that satisfies condition
(1.1), {Bk}nk=1 is any collection of partially non-overlapping domains that
satisfies condition (1.3), ak ∈ Bk ⊂ C, and to describe all extremals
(k = 1, n).
A. L. Targonskii, I. I. Targonskaya, K. Vashenko 231
2. The case of open set
Lemma 2.1. The function
P (τ) = ln sin
πτ
2
is convex for τ ∈ (0, 2).
The proof of the lemma can be found in the papers [20].
Theorem 2.1. Let n ∈ N, n ≥ 2, α ≥ 0. Then for all system points
An = {ak}nk=1, the satisfied condition (1.1) and condition
(|ak| − |ak+1|)2 = 4 sin2
πσk
2
(1− |ak||ak+1|) , k = 1, n, (2.1)
arbitrary open set D, the satisfied condition (1.2), ak ∈ D ⊂ C, k = 1, n,
be satisfied inequality
n∏
k=1
(|ak+1 − ak|α · r (D, ak)) ≤
(
2α+2
n
· sinα π
n
)n
· T (An) .
The equality obtain in this inequality, when D =
n∪
k=1
Bk, where points ak
and domains Bk are, conformity, the poles and the circular domains of
the quadratic differential
Q(w)dw2 = − wn−2
(wn − 1)2
dw2. (2.2)
Proof. At once we will note that from the condition of unapplying fol-
lows that capC\D > 0 and set D possesses Green’s generalized function
gD(z, a), where
gD(z, a) =
gD(a)(z, a), z ∈ D(a),
0, z ∈ C\D(a),
lim
ζ→z
gD(a)(ζ, a), ζ ∈ D(a), z ∈ ∂D(a)
– Green’s generalized function open set D concerning a point a ∈ D, and
gD(a)(z, a) – Green’s function domain D(a) concerning a point a ∈ D(a).
Further, we will use methods of works [4, 6, 7]. Sets we will consider
E0 = C\D; E(ak, t) = {w ∈ C : |w − ak| 6 t}, k = 1, n, n > 2, n ∈ N,
t ∈ R+. The condenser we will enter into consideration for rather small
t > 0
C (t, D, An) = {E0, E1} ,
232 About one extremal problem for open sets...
where
E1 =
n∪
k=1
E(ak, t).
Capacity of the condenser C (t, D, A2n,2m−1) is called as ( [4], [6])
cap C (t, D, An) = inf
∫ ∫ [
(G′
x)
2 + (G′
y)
2
]
dxdy,
where an infimum undertakes on all continuous and to the lipschicevym
in C functions G = G(z), such that G
∣∣∣
E0
= 0, G
∣∣∣
E1
= 1.
Let is named the module of condenser C, reverse the capacity of
condenser
|C| = [capC]−1
From a theorem 1 [16] get
|C (t,D,A2n,2m−1) | =
1
2π
· 1
n
· log 1
t
+M(D,An) + o(1), t→ 0, (2.3)
where
M(D,An) =
1
2π
· 1
n2
·
[ n∑
k=1
log r(D, ak) + 2
∑
k ̸=q
gD(ak, aq)
]
. (2.4)
Function
ζk (w) = −i
(
e−i arg akw
) 1
σk , k = 1, 2, . . . , n (2.5)
realizes univalent and conformal transformations of domain Mk to the
right half-plane Reζ > 0, for all k = 1, n.
From a formula (2.5) we receive the following asymptotic expressions
|ζk (w)− ζk (am)| ∼
1
σk
|am|
1
σk
−1 |w − am| ,
w → am, k = 1, 2, ..., n, m = k, k + 1. (2.6)
It’s obvious that
ζk (ak) = −i|ak|
1
σk , ζk (ak+1) = i|ak+1|
1
σk , k = 1, 2, ..., n, an+1 := a1.
(2.7)
For any domain ∆ ∈ C the define (∆)∗ :=
{
w ∈ C : w ∈ ∆
}
.
LetΩ
(1)
k define connected component ζk
(
D
∩
Mk
)∪ (
ζk
(
D
∩
Mk
))∗
,
containing a point ζk (ak), a Ω
(2)
k−1 – connected component
A. L. Targonskii, I. I. Targonskaya, K. Vashenko 233
ζk−1
(
D
∩
Mk−1
)∪ (
ζk−1
(
D
∩
Mk−1
))∗
, containing a point ζk−1 (ak),
k = 1, n, Ω
(2)
0 := Ω
(2)
n . It is clear, that Ω
(s)
k generally speaking, domains
are multiconnected domains, k = 1, n, s = 1, 2. Pair of domains Ω
(2)
k−1
and Ω
(1)
k grows out of piece-dividing transformation open set D concern-
ing families {Mk−1,Mk}, {ζk−1, ζk} in point ak, k = 1, n, M0 := Mn,
ζ0 := ζn.
Let’s consider condensers
Ck (t, D, An) =
(
E
(k)
0 , E
(k)
1
)
,
where
E(k)
s = ζk
(
Es
∩
P k
)∪[
ζk
(
Es
∩
P k
)]∗
,
k = 1, n, s = 0, 1, {Mk}nk=1 – the system of corners corresponding to
system of points An; operation [A]∗ compares to any the set A ⊂ C a
set, symmetric a set A is relative unit circle |w| = 1. From this it follows
that to the condenser C (t, D, An), at dividing transformation is relative
{Pk}nk=1 and {ζk}nk=1, there corresponds a set of condensers the system of
corners corresponding to system of points An; operation [A]∗ compares
to any the set A ⊂ C a set, symmetric a set A is relative unit circle |w| =
1. From this it follows that to the condenser C (t, D, An), at dividing
transformation is relative {Mk}nk=1 and {ζk}nk=1, there corresponds a set
of condensers {Ck (t, D, An)}nk=1, symmetric relatively {z : |z| = 1}.
According to works [4, 16], we will receive
capC (t,D,An) >
1
2
n∑
k=1
capCk (t,D,An) . (2.8)
From here follows
|C (t,D,An) | 6 2
(
n∑
k=1
|Ck (t,D,An) |−1
)−1
. (2.9)
The formula (2.3) gives a module asymptotics C (t, D, An) at t→ 0,
and M (D,An) is the given module of a set D relatively An. Using
formulas (2.6) and that fact that a setD meets the condition of unapplied
in relation to the system of points An, for condensers we will receive
similar asymptotic representations Ck (t,D,An), k = 1, n
|Ck (t,D,An) | =
1
4π
log
1
t
+Mk (D,An) + o(1), t→ 0, (2.10)
234 About one extremal problem for open sets...
where
Mk (D,An) =
1
8π
·
log r
(
Ω
(1)
k , ζk (ak)
)
1
σk
|ak|
1
σk
−1
+ log
r
(
Ω
(2)
k−1, ζk−1 (ak)
)
1
σk−1
|ak|
1
σk−1
−1
.
By means of (2.10), we receive
|Ck (t,D,An)|−1 =
4π
log 1
t
·
(
1 +
4π
log 1
t
Mk (D,An) + o
(
1
log 1
t
))−1
=
4π
log 1
t
−
(
4π
log 1
t
)2
Mk (D,An) + o
( 1
log 1
t
)2
, t→ 0. (2.11)
Further, from (2.11), follows that
n∑
k=1
|Ck (t,D,An)|−1
=
4πn
log 1
t
−
(
4π
log 1
t
)2
·
n∑
k=1
Mk (D,An)+o
( 1
log 1
t
)2
, t→ 0. (2.12)
In turn, allows (2.12) to receive the following asymptotic representa-
tion (
n∑
k=1
|Ck (t,D,An)|−1
)−1
=
log 1
t
4πn
·
(
1− 4π
n log 1
t
·
n∑
k=1
Mk (D,An) + o
(
1
log 1
t
))−1
=
log 1
t
4πn
+
1
n2
·
n∑
k=1
Mk (D,An) + o(1), t→ 0. (2.13)
Inequalities, (2.8) and (2.9) taking into (2.3) and (2.13) allow to no-
tice that
1
2π
· 1
n
· log 1
t
+M(D,An) + o(1) 6
log 1
t
2πn
+
2
n2
·
n∑
k=1
Mk (D,An) + o(1).
(2.14)
From (2.14) at t→ 0 we receive that
M(D,An) 6
2
n2
·
n∑
k=1
Mk (D,An) . (2.15)
A. L. Targonskii, I. I. Targonskaya, K. Vashenko 235
Formulas (2.4), (2.10) and (2.15) lead to the following expression
1
2π
· 1
n2
·
[ n∑
k=1
log r(D, ak) + 2
∑
k ̸=q
gD(ak, aq)
]
≤ 1
4πn2
·
n∑
k=1
log
r
(
Ω
(1)
k , ζk (ak)
)
1
σk
|ak|
1
σk
−1
+
n∑
k=1
log
r
(
Ω
(2)
k−1, ζk−1 (ak)
)
1
σk−1
|ak|
1
σk−1
−1
.
Thus, taking into (2.1), we receive
n∏
k=1
(|ak+1 − ak|α · r (D, ak)) ≤ 2nα ·
n∏
k=1
σk |ak|
|ak|
1
2σk · |ak|
1
2σk−1
×
n∏
k=1
sinα
πσk
2
·
n∏
k=1
(
r
(
Ω
(1)
k , ζk (ak)
)
· r
(
Ω
(2)
k , ζk (ak+1)
)) 1
2
. (2.16)
The equality obtain in this inequality, when points ak and domains
Bk are, conformity, the poles and the circular domains of the quadratic
differential
Q(ζ)dζ2 =
dζ2
(ζ2 + 1)2
. (2.17)
Using the Lemma 2.1 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,
we obtain
n∏
k=1
sinα
πσk
2
≤ sinnα
π
n
. (2.18)
The equality obtain in this inequality, if and only if
σ1 = σ2 = ... = σn =
2
n
.
Then from (2.16) using formulas (2.18) it is received the following
ratio
n∏
k=1
(|ak+1 − ak|α · r (Bk, ak)) ≤
(
2α+2
n
)n
· T (An) · sinnα
π
n
.
236 About one extremal problem for open sets...
The equality obtain in this inequality, when points ak and domains
Bk are, conformity, the poles and the circular domains of the quadratic
differential (2.2). It is derived from the square of the quadratic differen-
tial (2.17) conversion using
ζ = −iw
n
2 .
3. The case of partially non-overlapping domains
Theorem 3.1. Let n ∈ N, n ≥ 2, α ≥ 0. Then for all system points
An = {ak}nk=1, the satisfied condition (1.1) and condition (2.1), arbi-
trary system partially non-overlapping domains Bk, the satisfied condi-
tion (1.3), ak ∈ Bk ⊂ C, k = 1, n, 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, where points ak and domains Bk
are, conformity, the poles and the circular domains of the quadratic dif-
ferential (2.2).
Proof. In case of partially non-overlapping domains, the open set is en-
tered by a representation (1.3), which satisfies (1.2). From here, we have
Bk ⊂ D, k = 1, n. (3.1)
From (3.1), we receive, using results works [5, 6, 21]
r (Bk, ak) ≤ r (D, ak) , k = 1, n. (3.2)
Multiplying inequalities (3.2), we draw a conclusion
n∏
k=1
(|ak+1 − ak|α · r (Bk, ak)) ≤
n∏
k=1
(|ak+1 − ak|α · r (D, ak)) .
We receive an end result using the theorem 2.1.
References
[1] M. A. Lavrent’ev, On the theory of conformal mappings // Tr. Fiz.-Mat. Inst.
Akad. Nauk SSSR, 5 (1934), 159–245.
A. L. Targonskii, I. I. Targonskaya, K. Vashenko 237
[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.
[6] V. N. Dubinin, Method of symmetrization in the geometric theory of functions of
a complex variable // Usp. Mat. Nauk, 49 (1994), No. 1, 3–76.
[7] A. K. Bakhtin, Inequalities for the inner radii of nonoverlapping domains and
open sets // Ukr. Math. J., 61 (2009), No. 5, 716–733.
[8] A. K. Bakhtin, A. L. Targonskii, Extremal problems and quadratic differential //
Nonlin. Oscillations, 8 (2005), No. 3, 296–301.
[9] A. L. Targonskii, Extremal problems of partially nonoverlapping domains on a
Riemann sphere // Dop. NAN Ukr., 9 (2008), 31–36.
[10] A. Targonskii, Extremal problems on the generalized (n; d)-equiangular system of
points // An. St. Univ. Ovidius Constanta, 22 (2014), No. 2, 239–251.
[11] A. Targonskii, Extremal problem (2n; 2m-1)-system points on the rays // An. St.
Univ. Ovidius Constanta, 24 (2016), No. 2, 283–299.
[12] A. L. Targonskii, Extremal problems for partially non-overlapping domains on
equiangular systems of points // Bull. Soc. Sci. Lett. Lodz, 63 (2013), No. 1,
57–63.
[13] A. Targonskii, I. Targonskaya, On the One Extremal Problem on the Riemann
Sphere // International Journal of Advanced Research in Mathematics, 4 (2016),
1–7.
[14] A. K. Bakhtin, A. L. Targonskii, Generalized (n, d)-ray systems of points and in-
equalities for nonoverlapping domains and open sets // Ukr. Math. J., 63 (2011),
No. 7, 999–1012.
[15] A. K. Bakhtin, A. L. Targonskii, Some extremal problems in the theory of nonover-
lapping domains with free poles on rays // Ukr. Math. J., 58 (2006), No. 12,
1950–1954.
[16] V. N. Dubinin, Asymptotic representation of the modulus of a degenerating con-
denser and some its applications // Zap. Nauchn. Sem. Peterburg. Otdel. Mat.
Inst., 237 (1997), 56–73.
[17] V. N. Dubinin, Capacities of condensers and symmetrization in geometric func-
tion theory of complex variables, Dal’nayka, Vladivostok, 2009.
238 About one extremal problem for open sets...
[18] A. Targonskii, An extremal problem for the nonoverlapping domains // Ukr.
Math. Bull., 14 (2017), No. 1, 126–134; transl. Journal of Mathematical Sciences,
227 (2017), No. 1, 98– 104.
[19] A. Targonskii, I. Targonskaya, Extreme problem for partially nonoverlapping do-
mains on a Riemann sphere // Ukr. Math. Bull., 15 (2018), No. 1, 94–102; transl.
Journal of Mathematical Sciences, 235 (2018), No. 1, 74–80.
[20] A. L. Targonskii, About one extremal problem for projections of the points on unit
circle // Ukrainian Mathematical Bulletin, 15 (2018), No. 3, 418–430.
[21] W. K. Hayman, Multivalent functions, Cambridge University, Cambridge, 1958.
[22] J.A. Jenkins, Univalent functions and conformal mapping, Springer, Berlin, 1958.
Contact information
Andrei Leonidovich
Targonskii
I. Franko Zhitomir State University,
Zhitomir, Ukraine
E-Mail: targonsk@zu.edu.ua
Irina Igorevna
Targonskaya
I. Franko Zhitomir State University,
Zhitomir, Ukraine
E-Mail: targonsk@zu.edu.ua
Katya Vaschenko I. Franko Zhitomir State University,
Zhitomir, Ukraine
E-Mail: katyavaschenko@ukr.net
|