Some extremal problems in the theory of nonoverlapping domains with free poles on rays
We obtain new results on the maximization of the product of powers of the interior radii of pairwise disjoint domains with respect to certain systems of points in the extended complex plane.
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| Date: | 2006 |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| author | Bakhtin, A. K. Targonskii, A. L. Бахтин, А. К. Таргонский, А. Л. Бахтин, А. К. Таргонский, А. Л. |
| author_facet | Bakhtin, A. K. Targonskii, A. L. Бахтин, А. К. Таргонский, А. Л. Бахтин, А. К. Таргонский, А. Л. |
| author_sort | Bakhtin, A. K. |
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| datestamp_date | 2020-03-18T19:57:46Z |
| description | We obtain new results on the maximization of the product of powers of the interior radii of pairwise disjoint domains with respect to certain systems of points in the extended complex plane. |
| first_indexed | 2026-03-24T02:44:56Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 517.54
A. K. Baxtyn, A. L. Tarhonskyj
(Yn-t matematyky NAN Ukrayn¥, Kyev)
NEKOTORÁE ∏KSTREMAL|NÁE ZADAÇY
TEORYY NENALEHAGWYX OBLASTEJ
SO SVOBODNÁMY POLGSAMY NA LUÇAX
∗∗∗∗
We obtain new results on the maximization of a product of inner radius powers for domains that are
mutually disjoint. The maximization is performed with respect to some system of points in the extended
complex plane.
Otrymano novi rezul\taty pro maksymizacig dobutku stepeniv vnutrißnix radiusiv oblastej, qki
poparno ne peretynagt\sq, vidnosno deqkyx system toçok u rozßyrenij kompleksnij plowyni.
Vvedenye. V heometryçeskoj teoryy funkcyj kompleksnoj peremennoj πks-
tremal\n¥e zadaçy o nenalehagwyx oblastqx sostavlqgt aktyvno razvyvagwe-
esq napravlenye, voznyknovenye kotoroho svqzano s rabotoj M. A. Lavrent\eva
[1]. V πtoj rabote b¥la vperv¥e postavlena y reßena zadaça o maksymume pro-
yzvedenyq konformn¥x radyusov dvux vzaymno neperesekagwyxsq odnosvqzn¥x
oblastej v rasßyrennoj kompleksnoj ploskosty C . V dal\nejßem zadaçy ta-
koho typa rassmatryvalys\ mnohymy avtoramy (sm., naprymer, rabot¥ [2 – 5] y
pryvedennug v nyx byblyohrafyg). V nastoqwej rabote reßen¥ nov¥e πkstre-
mal\n¥e zadaçy o nenalehagwyx oblastqx — s tak naz¥vaem¥my svobodn¥my
polgsamy na luçax.
Perejdem k formulyrovke rezul\tatov. Pust\ { }Bk k
n
=1 — systema poparno
neperesekagwyxsq oblastej v C . Pry kaΩdom k n= 1, tol\ko koneçnoe
çyslo komponent svqznosty mnoΩestva C \ Bk moΩet soderΩat\ vnutry sebq
kakug-to yz oblastej Bj , j n= 1, , j ≠ k ; takye komponent¥ budem naz¥vat\
suwestvenn¥my. Oblast\, poluçennug ysklgçenyem yz C vsex suwestvenn¥x
komponent svqznosty mnoΩestva C \ Bk , budem oboznaçat\ B̃k . Qsno, çto
B Bk k⊂ ˜ , k n= 1, , y { }B̃k k
n
=1 — systema koneçnosvqzn¥x vzaymno neperesekag-
wyxsq oblastej bez yzolyrovann¥x hranyçn¥x toçek. ∏tu systemu oblastej bu-
dem naz¥vat\ zapolnenyem system¥ poparno neperesekagwyxsq oblastej
{ }Bk k
n
=1.
Dlq oblasty B ⊂ C y toçky a B∈ oboznaçym çerez r ( B, a ) vnutrennyj
radyus oblasty B otnosytel\no toçky a (vse opredelenyq, yspol\zuem¥e v
∗∗∗∗
V¥polnena pry çastyçnoj fynansovoj podderΩke Hosudarstvennoj prohramm¥ Ukrayn¥
#;0102Y000917.
© A. K. BAXTYN, A. L. TARHONSKYJ, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12 1715
1716 A. K. BAXTYN, A. L. TARHONSKYJ
nastoqwej rabote, pryveden¥, naprymer, v [5]; otmetym lyß\, çto v otlyçye ot
[6] m¥ ponymaem vnutrennyj radyus otnosytel\no beskoneçno udalennoj toçky
v sm¥sle opredelenyq, predloΩennoho v [2]). Dlq borelevskoho mnoΩestva
E ⊂ C çerez cap E m¥ oboznaçaem eho loharyfmyçeskug emkost\. Kak ob¥ç-
no, i — mnymaq edynyca.
Vsgdu nyΩe n — celoe neotrycatel\noe (natural\noe) çyslo, n ≥ 3,
χ( ) : ( ) /t t t= + −1 2 , t > 0. Dlq nabora toçek { } { }\ak k
n
= ⊂1 0C takoho, çto 0 =
= arg a1 < arg a2 < … < arg an < 2π, poloΩym
σ πk k ka a: (arg arg ) /= −+1 , k n= −1 1, , σ π πn na: ( arg ) /= −2 ,
a an+ =1 1: , µ χ σ( ) /{ } :
( )
a a a ak k
n
k
n
k k k
k
=
=
+=
∏
−
1
1
1
2 1
.
V prynqt¥x v¥ße oboznaçenyqx spravedlyv¥ sledugwye utverΩdenyq.
Teorema�1. Kakov¥ b¥ ny b¥ly poloΩytel\n¥e dejstvytel\n¥e çysla α y
µ0 , toçky a an1 0, , { }\… ∈C y poparno neperesekagwyesq oblasty B B0 1, ,…
… ⊂+, ,B Bn n 1 C takye, çto arg ( ) /a k nk = −2 1π y a Bk k∈ , k n= 1, ,
µ( ){ }ak k
n
=1 ≤ µ0 , 0 0∈B , ∞ ∈ +Bn 1, ymeet mesto neravenstvo
[ ]( , ) ( , ) ( , )r B r B r B an
k
n
k k0 1
1
0 +
=
∞ ∏α ≤ [ ]( , ) ( , ) ( , )r B r B r B an
k
n
k k0
0
1
0
1
0 00 +
=
∞ ∏α , (1)
hde toçky a an1
0 0, ,… y oblasty B B B Bn n0
0
1
0 0
1
0, , , ,… + qvlqgtsq sootvetstven-
no polgsamy y kruhov¥my oblastqmy kvadratyçnoho dyfferencyala
Q w dw( ) 2 = –
α µ α αµ
µ
w n w
w w
dw
n n
n
n
2
0
2 2
2
0
2
22+ − +
−
( )
( )
(2)
( a Bk k
0 0∈ pry k n= 1, , 0 0
0∈B , ∞ ∈ +Bn 1
0
) . Znak ravenstva v (1) dostyhaetsq
tohda y tol\ko tohda, kohda pry vsex k n= +0 1, v¥polnen¥ ravenstva
B̃ Bk k= 0
y cap B Bk k
0 0\ = .
Sledstvye�1. Kakov¥ b¥ ny b¥ly poloΩytel\noe dejstvytel\noe çyslo
µ0
, toçky a an1 0, , { }\… ∈C y poparno neperesekagwyesq oblasty B1, …
… , Bn ⊂ C takye, çto arg ( ) /a k nk = −2 1π y a Bk k∈ , k n= 1, , µ( ){ }ak k
n
=1 ≤
≤ µ0 , ymeet mesto neravenstvo
k
n
k kr B a
=
∏
1
( , ) ≤
k
n
k kr B a
=
∏
1
0 0( , ) , (3)
hde toçky a an1
0 0, ,… y oblasty B Bn1
0 0, ,… , a Bk k
0 0∈ , k n= 1, , qvlqgtsq so-
otvetstvenno polgsamy y kruhov¥my oblastqmy kvadratyçnoho dyfferencya-
la
Q w dw( ) 2 = – w
w
dw
n
n
−
−
2
0
2
2
( )µ
.
Sledstvye;1 poluçaetsq yz teorem¥;1 predel\n¥m perexodom pry α → 0 .
Teorema�2. Kakov¥ b¥ ny b¥ly dejstvytel\n¥e çysla α ∈( ; , ]0 0 18 y µ0 >
> 0
, toçky a an1 0, , { }\… ∈C y poparno neperesekagwyesq oblasty B0,
B B Bn n1 1, , ,… ⊂+ C takye, çto 0 = arg a1 < … < arg an < 2π , a Bk k∈ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
NEKOTORÁE ∏KSTREMAL|NÁE ZADAÇY TEORYY … 1717
k n= 1, , µ( ){ }ak k
n
=1 ≤ µ0 , 0 0∈B , ∞ ∈ +Bn 1, v¥polnqetsq neravenstvo (1), hde
toçky a an1
0 0, ,… y oblasty B B B Bn n0
0
1
0 0
1
0, , , ,… + — te Ωe, çto y v teoreme;1.
Otmetym, çto v teoremax;1,;2 y v sledstvyy;1 πkstremal\n¥j nabor polgsov
y oblastej udovletvorqet uslovyqm, naklad¥vaem¥m na nabor¥ toçek ak y ob-
lastej Bk
. Dokazatel\stva vsex sformulyrovann¥x utverΩdenyj osnovan¥ na
prymenenyy tak naz¥vaemoho kusoçno-razdelqgweho preobrazovanyq, predlo-
Ωennoho v rabotax V. N. Dubynyna (sm. [2]).
1. Dokazatel\stvo teorem¥�2. Pust\ α ∈( ; , ]0 0 18 , µ 0 > 0
, toçky
a an1, ,… ∈C y poparno neperesekagwyesq oblasty B B B Bn n0 1 1, , , ,… ⊂+ C
udovletvorqgt uslovyqm teorem¥;2. Oboznaçym a an+ =1 1: , B Bn+ =1 1: ,
σ σ0 := n , ∆k k kw a w a: { : arg arg arg }= ∈ < < +C 1 pry k n= −1 1, ,
∆n nw a w: { : arg arg }= ∈ < <C 2π , y pust\ ζk w( ) — odnoznaçnaq vetv\ funk-
cyy – i w i ak
kexp{ arg } /−( )1 σ , otobraΩagwaq oblast\ ∆ k na poluploskost\
Im ζ > 0, k = 1, … , n . Tohda pry vsex k = 1, … , n v¥polnqgtsq ravenstva
ζ ζ
σ
σ
k k k
k
k kw a a w a ok( ) ( ) ( ( ))/− = − +−1 1 11 1 , w → ak ,
ζ ζ
σ
σ
k k k
k
k kw a a w a ok( ) ( ) ( ( ))
/− = − ++ +
−
+1 1
1 1
1
1 1 1 , w → ak + 1 ,
ζ σ
k w w k( ) /= 1 , w k∈∆ .
Pust\ k n∈ …{ , , }1 . Rassmotrym obraz ζk kB( )0 ∩ ∆ otkr¥toho mnoΩestva
B k0 ∩ ∆ pry otobraΩenyy ζk , obæedynym eho s mnoΩestvom, symmetryçn¥m
ζk kB( )0 ∩ ∆ otnosytel\no mnymoj osy, y voz\mem soderΩawug naçalo koordy-
nat O svqznug komponentu vnutrennosty zam¥kanyq poluçennoho takym obra-
zom mnoΩestva. V rezul\tate poluçym oblast\, soderΩawug toçku O, koto-
rug budem oboznaçat\ çerez Gk
0. Posledovatel\no zamenqq v pred¥duwej
konstrukcyy oblast\ B0 na B∞ , Bk , B k + 1 , a toçku O sootvetstvenno na
beskoneçno udalennug toçku y toçky w i ak k
k− = − 1/σ
y w i ak k
k+
+= 1
1/σ
,
poluçaem oblasty Gk
∞ , Gk
−
y Gk+
+
1, soderΩawye sootvetstvenno beskoneçno
udalennug toçku y toçky wk
−
y wk
+
, G Gn+
+ +=1 1: . Sohlasno teoreme;1.9 [2], yz
pryvedenn¥x v¥ße asymptotyçeskyx ravenstv v¥tekagt neravenstva
r B ak k( , ) ≤
r G w
a
r G w
a
k k
k
k
k k
k
k
k k
( , ) ( , )
/ /
/
− −
−
−
+
−
+
−
−−
1 11 1
1 1
1
1 1
1 2
1
σ σ
σ σ
, k = 1, … , n,
r B( , )0 0 <
k
n
kr G k
=
∏ [ ]
1
0 2
0
2
( , )
/σ
, r B( , )∞ ∞ ≤
k
n
kr G k
=
∞∏ ∞[ ]
1
22
( , )
/σ
,
otkuda, v¥polnqq oçevydn¥e preobrazovanyq, poluçaem cepoçku sootnoßenyj
r B r B r B a
k
n
k k( , ) ( , ) ( , )0
1
0 ∞
=
∞[ ] ∏α ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1718 A. K. BAXTYN, A. L. TARHONSKYJ
≤
k
n
k k k
k k k k
k k
r G r G
r G w r G w
a a
k
k
=
∞
− − + +
+
−∏ ∞[ ] ( )
1
0 2
1
1 1
1 2
0
2
σ
ασ
σ( , ) ( , )
( , ) ( , )/
/
/
=
= 2
1 1
1 2
n
k
n
k
k
k
k
a
a
a
k
= +
∏
σ χ
σ/( )
×
× r G r G
r G w r G w
a a
k k
k k k k
k k
k
k k
( , ) ( , )
( , ) ( , )
/ /
/
0
1
1
1 2
1 2
0
2
∞
− − + +
+
∞[ ]
+( )
ασ
σ σ
=
= 2 00
1 1
0
2
1 2
2
n
k
n
k
k
n
k k
k k k k
k k
r G r G
r G w r G w
w w
kµ σ
ασ
= =
∞
− − + +
+ −∏ ∏
∞[ ]
−
( , ) ( , )
( , ) ( , )
/
.
(4)
Opredelym funkcyg Ψ ( τ ) , τ > 0, ravenstvom
Ψ( ) : ( , ) ( , )
( , ) ( , )
τ τ τ τ
τ τ
= ∞( ) −
∞
−r B r B
r B i r B ii i
0 0
4
2
,
hde B0
τ , Bi
τ , B i−
τ , B∞
τ
— kruhov¥e oblasty kvadratyçnoho dyfferencyala
Q w dw( ) 2 = – τ τ τ2 4 2 2 2
2 2 2
22 2
1
w w
w w
dw
+ − +
+
( )
( )
,
soderΩawye sootvetstvenno toçky 0, i, – i y ∞ . V [3] (dokazatel\stvo teore-
m¥;6) pokazano, çto pry vsex τ ∈( , )0 1 ymeet mesto ravenstvo Ψ ( τ ) =
= τ τ ττ τ τ2 1 12 2 2
1 1( ) ( )( ) ( )− +− − − + . Otsgda neposredstvenn¥m v¥çyslenyem poluça-
em, çto vtoraq proyzvodnaq funkcyy ln[ ( )]τ τ2Ψ ravna – 2
1
2
2
2τ τ
τ
− +
−
ln y
(vsledstvye ee vozrastanyq) ymeet edynstvenn¥j koren\ na ( , )0 1 , prynadleΩa-
wyj yntervalu ( , ; , )0 85 0 9 . Poπtomu funkcyq ln[ ( )]τ τ2Ψ v¥pukla vverx na
promeΩutke ( ; , ]0 0 85 . S druhoj storon¥, sohlasno teoreme;1 yz rabot¥ [4],
v¥polnqgtsq neravenstva
r G r G
r G w r G w
w w
k k
k k k k
k k
k( , ) ( , )
( , ) ( , )0
20
2
∞
− − + +
+ −
∞[ ]
−
ασ
≤ Ψ( )σ αk , k n= 1, , (5)
podstavlqq kotor¥e v (4) y yspol\zuq to, çto σ αk ≤ 2 0 18, < 0,85 pry α ≤
≤ 0,18 y neravenstvo Jensena dlq funkcyy ln[ ( )]τ τ2Ψ , poluçaem
r B r B r B a
k
n
k k( , ) ( , ) ( , )0
1
0 ∞
=
∞[ ] ∏α ≤ 2 0
1 1
1 2
n
k
n
k
k
n
kµ σ σ α
= =
∏ ∏
Ψ( )
/
≤
≤ 2 0
2
1
2
1 2
n n
k
n
k kµ α σ α σ α−
=
∏
/
/
( ) ( )Ψ ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
NEKOTORÁE ∏KSTREMAL|NÁE ZADAÇY TEORYY … 1719
≤ 2 20
2
1
1 2
1 2
n n
k
n
knµ α α σ α−
=
−∏
/
/
( ) ( )Ψ ≤ 4 20
1 2
n
n
n n
( )−µ αΨ( )
/
.
Pust\ B B0 0
0= , B Bn n+ +=1 1
0
y pry vsex k = 1, … , n spravedlyv¥ ravenstva
a ak k= 0
y B Bk k= 0 , hde toçky a an1
0 0, ,… y oblasty B B B Bn n0
0
1
0 0
1
0, , , ,… + opre-
delen¥ v formulyrovke teorem¥;1. Tohda vo vsex pryvedenn¥x v¥ße cepoçkax
sootnoßenyj neravenstva prevrawagtsq v ravenstva, çto y zaverßaet dokaza-
tel\stvo teorem¥;2.
2. Dokazatel\stvo teorem¥�1. Dokazatel\stvo πtoj teorem¥ v znaçytel\-
noj stepeny povtorqet dokazatel\stvo teorem¥;2. Ymenno, pry v¥polnenyy us-
lovyj teorem¥;1 σk n= 2 / dlq vsex k = 1, … , n , a v neravenstvax (5)
Ψ Ψ( ) ( )σ α αk n= −2 1 , k n= 1, .
Podstavlqq πty ravenstva v (4), ymeem
r B r B r B a
k
n
k k( , ) ( , ) ( , )0
1
0 ∞
=
∞[ ] ∏α ≤ 4 20
1 2
n
n
n n
( )−µ αΨ( )
/
.
UtverΩdenye o znake ravenstva v teoreme;2 proverqetsq tak Ωe, kak y v [5] (do-
kazatel\stvo teorem¥;2).
Teorema dokazana.
1. Lavrent\ev M. A. K teoryy konformn¥x otobraΩenyj // Tr. Fyz.-mat. yn-ta AN SSSR. –
1934. – 5. – S. 159 – 245.
2. Dubynyn V. N. Metod symmetryzacyy v heometryçeskoj teoryy funkcyj kompleksnoho pe-
remennoho // Uspexy mat. nauk. – 1994. – 49, # 1;(295). – S. 3 – 76.
3. Dubynyn V. N. Razdelqgwee preobrazovanye oblastej y zadaçy ob πkstremal\nom razbyenyy
// Zap. nauç. sem. Lenynhr. otd-nyq Mat. yn-ta AN SSSR. – 1988. – 168. – S. 48 – 66.
4. Baxtyn A. K. ∏kstremal\n¥e zadaçy o nenalehagwyx oblastqx so svobodn¥my polgsamy
na okruΩnosty // Dopov. NAN Ukra]ny. – 2004. – # 8. – S. 7 – 15.
5. Baxtyn A. K. ∏kstremal\n¥e zadaçy o nenalehagwyx oblastqx so svobodn¥my polgsamy
na okruΩnosty // Ukr. mat. Ωurn. – 2006. – 58, # 7. – S.;867 – 886.
6. Xejman V. K. Mnoholystn¥e funkcyy. – M.: Yzd-vo ynostr. lyt., 1960. – 180 s.
Poluçeno 09.12.2003,
posle dorabotky — 06.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
|
| id | umjimathkievua-article-3567 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:44:56Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f5/a5d6432853d4a332a8cbbc71b138daf5.pdf |
| spelling | umjimathkievua-article-35672020-03-18T19:57:46Z Some extremal problems in the theory of nonoverlapping domains with free poles on rays Некоторые экстремальные задачи теории неналегающих областей со свободными полюсами на лучах Bakhtin, A. K. Targonskii, A. L. Бахтин, А. К. Таргонский, А. Л. Бахтин, А. К. Таргонский, А. Л. We obtain new results on the maximization of the product of powers of the interior radii of pairwise disjoint domains with respect to certain systems of points in the extended complex plane. Отримано нові результати про максимізацію добутку степенів внутрішніх радіусів областей, які попарно не перетинаються, відносно деяких систем точок у розширеній комплексній площині. Institute of Mathematics, NAS of Ukraine 2006-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3567 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 12 (2006); 1715–1719 Український математичний журнал; Том 58 № 12 (2006); 1715–1719 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3567/3868 https://umj.imath.kiev.ua/index.php/umj/article/view/3567/3869 Copyright (c) 2006 Bakhtin A. K.; Targonskii A. L. |
| spellingShingle | Bakhtin, A. K. Targonskii, A. L. Бахтин, А. К. Таргонский, А. Л. Бахтин, А. К. Таргонский, А. Л. Some extremal problems in the theory of nonoverlapping domains with free poles on rays |
| title | Some extremal problems in the theory of nonoverlapping domains with free poles on rays |
| title_alt | Некоторые экстремальные задачи теории неналегающих областей со свободными полюсами на лучах |
| title_full | Some extremal problems in the theory of nonoverlapping domains with free poles on rays |
| title_fullStr | Some extremal problems in the theory of nonoverlapping domains with free poles on rays |
| title_full_unstemmed | Some extremal problems in the theory of nonoverlapping domains with free poles on rays |
| title_short | Some extremal problems in the theory of nonoverlapping domains with free poles on rays |
| title_sort | some extremal problems in the theory of nonoverlapping domains with free poles on rays |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3567 |
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