Estimation of the product of inner radii of partially nonoverlapping domains
We present new results on the maximization of products of positive powers of inner radii of some special domain systems in the extended complex plane $\overline{{\mathbb C}}$ with respect to points of finite sets such that any two distinct points $z_1, z_2 \in {\mathbb C}\setminus \{0\}$ of such set...
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| Дата: | 2008 |
|---|---|
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| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509269477031936 |
|---|---|
| author | Podvysotskii, R. V. Подвысоцкий, Р. В. Подвысоцкий, Р. В. |
| author_facet | Podvysotskii, R. V. Подвысоцкий, Р. В. Подвысоцкий, Р. В. |
| author_sort | Podvysotskii, R. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:23Z |
| description | We present new results on the maximization of products of positive powers of inner radii of some special domain systems in the extended complex plane $\overline{{\mathbb C}}$ with respect to points of finite sets such that any two distinct points $z_1, z_2 \in {\mathbb C}\setminus \{0\}$ of such set belong to different rays emerging from the origin. |
| first_indexed | 2026-03-24T02:38:25Z |
| format | Article |
| fulltext |
UDK 517.54
R. V. Podv¥sockyj (Yn-t matematyky NAN Ukrayn¥, Kyev)
OCENKA PROYZVEDENYQ VNUTRENNYX RADYUSOV
ÇASTYÇNO NENALEHAGWYX OBLASTEJ
We present new results on the maximization of products of positive powers of inner radii of some special
domain systems in the extended complex plane C with respect to points of finite sets such that any two
distinct points z1 , z2 ∈C \ 0{ } of such set belong to different rays emerging from the origin.
Navedeno novi rezul\taty pro maksymizacig dodatnyx stepeniv vnutrißnix radiusiv deqkyx spe-
cial\nyx system oblastej u rozßyrenij kompleksnij plowyni C vidnosno toçok skinçennyx
mnoΩyn takyx, wo dvi bud\-qki rizni toçky z1 , z2 ∈C \ 0{ } mnoΩyny leΩat\ z poçatku koor-
dynat.
V teoryy odnolystn¥x funkcyj πkstremal\n¥e zadaçy o proyzvedenyy vnutren-
nyx radyusov oblastej sostavlqgt aktyvno razvyvagweesq napravlenye. Voz-
nyknovenye πtoho napravlenyq svqzano s klassyçeskoj rabotoj M. A. Lavren-
t\eva [1], v kotoroj b¥la vperv¥e postavlena y reßena zadaça o proyzvedenyy
komfortn¥x radyusov dvux vzaymno neperesekagwyxsq odnosvqzn¥x oblastej.
Vposledstvyy πta tematyka razvyvalas\ v rabotax mnohyx avtorov (sm., nap-
rymer, [2 – 7]).
V dannoj rabote rassmatryvagtsq zadaçy podobnoho typa dlq çastyçno nena-
lehagwyx oblastej.
1. Oboznaçenyq y opredelenyq. Pust\ N, R — mnoΩestva natural\n¥x y
vewestvenn¥x çysel, R
+ = ( ; )0 ∞ , C — ploskost\ kompleksn¥x çysel, C =
= C ∪ ∞{ } — ee odnotoçeçnaq kompaktyfykacyq, r B a( , ) — vnutrennyj radyus
oblasty B � C otnosytel\no toçky a B∈ [2, 5]. V dannoj rabote yspol\zu-
etsq teoryq kvadratyçn¥x dyfferencyalov, s kotoroj moΩno oznakomyt\sq v
monohrafyy [2]. Systemu toçek An : ak k
n{ } =1, ak ∈ C\ 0{ }, k = 1, n , takug, çto
0 = arga1 < arga2 < … < argan < 2π, budem naz¥vat\ luçevoj systemoj toçek.
Oboznaçym
P Ak n( ) : = w a a ak w k: arg arg arg< <{ }+1 ,
σk : = 1
1π
(argak + – arg )ak , k = 1, n ,
Qsno, çto σkk
n
=∑ 1
= 2. Rassmotrym proyzvol\nug luçevug systemu toçek An :
ak k
n{ } =1. Pust\ Bk k
n{ } = 0 — lgboj nabor oblastej takyx, çto 0 ∈ B � C , ak ∈
∈ Bk � C , k = 1, n .Tohda oboznaçym çerez Bk
( )1
svqznug komponentu mnoΩestva
Bk ∩ P Ak n( ) , soderΩawug toçku ak , a çerez Bk
( )2
svqznug komponentu mno-
Ωestva Bk ∩ P Ak n– ( )1 , soderΩawug toçku ak , k = 1, n , P An0( ) : P An n( ) . Bk
( )0
oboznaçaet svqznug komponentu mnoΩestva B0 ∩ P Ak n( ) , k = 1, n , soderΩawug
toçku w = 0. Pust\ D — proyzvol\noe otkr¥toe mnoΩestvo, D = An ∪ 0{ }.
Esly svqzn¥e komponent¥ D Pk∩[ ], k = 1, n , otnosytel\no toçek 0, ak ,
ak +1 vzaymno ne peresekagtsq pry k = 1, n , to mnoΩestvo D udovletvorqet
uslovyg nenalehanyq otnosytel\no luçevoj system¥ toçek An . Systema
Bk k
n{ } = 0 udovletvorqet uslovyg nenalehanyq, esly otkr¥toe mnoΩestvo D =
© R. V. PODVÁSOCKYJ, 2008
1004 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
OCENKA PROYZVEDENYQ VNUTRENNYX RADYUSOV ÇASTYÇNO … 1005
=
Bkk
n
= 0∪ udovletvorqet uslovyg nenalehanyq otnosytel\no luçevoj system¥
toçek An ∪ 0{ }.
Dlq proyzvol\noj luçevoj system¥ toçek An = ak k
n{ } =1 y α ∈ +
R poloΩym
µα( )An : = χ
σ
ασ
α σ σa
a
ak
kk
n
k
k
n
k
k
k k
+=
+ +
=
∏ ∏
1
1
2
1
1
2
1 1
4
1
2
1
–
( )– ,
hde an +1 : = a1, σ0 : = σn , χ( )t = 1
2
t( + t–1). Cel\g dannoj rabot¥ qvlqetsq
poluçenye toçn¥x ocenok sverxu dlq funkcyonalov vyda Jα = r Bα( 0 ,
0) r B ak kk
n
( , )=∏ 1
, hde α ∈ +
R , An = ak k
n{ } =1 — luçevaq systema toçek, Bk k
n{ } = 0
— nabor oblastej, udovletvorqgwyx uslovyg nenalehanyq otnosytel\no luçe-
voj system¥ toçek An . Podobn¥e zadaçy rassmatryvagtsq v rabotax [2 – 4].
2. Osnovn¥e rezul\tat¥.
Teorema 1. Pust\ dan¥ çysla n ∈N , n ≥ 2 y α , 0 < α ≤ 0,2. Tohda dlq
proyzvol\noj luçevoj system¥ toçek An = ak k
n{ } =1 takoj , çto µα( )An = 1,
y lgboho nabora oblastej Bk k
n{ } = 0 , udovletvorqgweho uslovyg nenalehanyq
otnosytel\no luçevoj system¥ toçek An , v¥polnqetsq neravenstvo
r B r B ak k
k
n
α( , ) ( , )0
1
0
=
∏ t ≤ σ α
α
α
α
α
α
α
k
k
n n n n
n
n
n
n
n
n= +
∏
( ) +
1
2
2
2
2 4( )
–
–
.
Znak ravenstva v πtom neravenstve dostyhaetsq tohda, kohda ak , k = 1, n , y
Bs , s = 0, n , qvlqgtsq sootvetstvenno polgsamy y kruhov¥my oblastqmy
kvadratyçnoho dyfferencyala
Q w dw( ) 2 = –
–
–
n w
w w
dw
n
n
2
2 2
2
1
α α( ) +
( )
. (1)
Teorema 2. Pust\ n ∈N , n ≥ 3. Tohda dlq lgboho α ∈ +
R , 0 < α ≤ 7
16
2n ,
lgboj luçevoj system¥ toçek An = ak k
n{ } =1 takoj , çto µα( )An = 1, σk ≤
≤ 1 75.
α
, k = 1, n , y lgboho nabora oblastej Bk k
n{ } = 0 , udovletvorqgweho us-
lovyg nenalehanyq otnosytel\no luçevoj system¥ toçek An , v¥polnqetsq
neravenstvo
r B r B ak k
k
n
α( , ) ( , )0
1
0
=
∏ ≤ r D r D ak k
k
n
α( , ) ,0
0
1
0 ( )
=
∏ ,
hde ak
0
, k = 1, n , y Ds , s = 0, n , — sootvetstvenno polgs¥ y kruhov¥e oblas-
ty kvadratyçnoho dyfferencyala (1).
Dokazatel\stvo teorem¥ 1 bazyruetsq na metode razdelqgweho preobra-
zovanyq (sm. [5, 6]). Rassmotrym otkr¥toe mnoΩestvo D = Bkk
n
=1∪ ∪ B0 . Dlq
dostatoçno mal¥x t ∈ +
R obrazuem mnoΩestva
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
1006 R. V. PODVÁSOCKYJ
E0 = C \D ; Ut = w w t∈ ≤{ }C: ,
E tk ( ) = w w a tk∈ ≤{ }C: – , k = 1, n .
Rassmotrym kondensator C t D An( , , ) = E0{ , Ut , E t1( ) , … , E tn( )}. Vvedem klass
funkcyj Πα( )c vsex vewestvenn¥x, neprer¥vn¥x y lypßycev¥x v C funk-
cyj G = G z( ) takyx, çto G = 0 v okrestnosty mnoΩestva E
0
, GUt
= α ,
G E tk ( ) = 1, k = 1, n . Emkost\g kondensatora C t D An( , , ) naz¥vaetsq velyçyna
(sm. [5]) capC t D An( , , ) = inf ′( )[∫∫ Cx
2 + ′( ) ]C dxdyy
2
, hde nyΩnqq hran\ beretsq
po mnoΩestvu. Modul\ kondensatora C t D An( , , ) opredelqetsq v¥raΩenyem
C t D An( , , ) = capC t D An( , , ) –[ ] 1
. Opredelqem asymptotyku modulq kondensato-
ra C t D An( , , ) (sm. [6])
C t D An( , , ) = 1
2π α
1 1
n t+
log + M D An( , ) + o( )1 , t → 0, (2)
hde
M D An( , ) = 1
2π α
α α1 1 0 2 02
1( )
log log ( , ) ( , )
n t
r D g a
k
n
D k+
+
=
∑ +
+
k
n
k
k p
D p kr D a g a a
= ≠
∑ ∑+
1
log ( , ) ( , ) , (3)
g z aB( , ) — obobwennaq funkcyq Hryna otkr¥toho mnoΩestva B otnosytel\no
toçky a B∈ , B a( ) — svqznaq komponenta mnoΩestva B, soderΩawaq toçku a,
pry πtom g z aB( , ) = g z aB a( )( , ) , esly z B a∈ ( ), y g z aB( , ) = 0, esly z ∈C \ B,
g z aB a( )( , ) — funkcyq Hryna oblasty B a( ) otnosytel\no toçky a B a∈ ( ). Ras-
smotrym razdelqgwee preobrazovanye (sm. [5]) kondensatora C t D An( , , ) otno-
sytel\no semejstv uhlov P Ak n k
n( ){ } =1 y funkcyj z wk k
n( ){ } =1, hde z wk ( ) =
= (– )1 k i e wi ak
k
– arg( )
1
σ , k = 1, n .
Rassmotrym kondensator¥ C t D Ak n( , , ) = E k
0
( )( , Ut
k( ) , E k
1
( ) , E k
2
( )) , hde
E k
0
( ) = z E Pk k0 ∩( ) ∪ z E Pk k0 ∩( ){ }∗
,
Ut
k( ) = z U Pk t k∩( ) ∪ z U Pk t k∩( ){ }∗
,
E k
1
( ) = z E t Pk k k( ) ∩( ) ∪ z E t Pk k k( ) ∩( ){ }∗
, k = 1, n , E tn +1( ) = E t1( ) ,
A{ }∗ = w w A∈ ∈{ }∗
C : – .
KaΩdomu kondensatoru C t D Ak n( , , ) sopostavym klass Vk vsex vewestven-
n¥x, neprer¥vn¥x y lypßycev¥x v C funkcyj G = G z( ) takyx, çto G = 0 v
okrestnosty mnoΩestva E k
0
( )
, G Ut
k( ) = α , G Ep
k( ) = 1, k = 1, n , p = 1, 2. Pry
razdelqgwem preobrazovanyy kondensatoru C t D An( , , ) sootvetstvuet nabor
kondensatorov C t D Ak n k
n( , , ){ } =1, pryçem v sylu rezul\tatov rabot [5, 6] v¥pol-
nqetsq neravenstvo capC t D An( , , ) ≥
1
2
cap
=1k
n
nC t D A∑ ( , , ) . Otsgda neposred-
stvenno poluçaem sootnoßenye
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
OCENKA PROYZVEDENYQ VNUTRENNYX RADYUSOV ÇASTYÇNO … 1007
C t D An( , , ) ≤ 2 1
1
1
C t D Ak n
l
n
( , , ) –
–
=
∑
. (4)
Spravedlyv¥ sledugwye asymptotyçeskye predstavlenyq:
z w z ak k m( ) – ( ) ~ 1
1 1
σ
σ
k
m ma w ak
–
– , w → am , k = 1, 2, … , n,
(5)m = k, k + 1, z wk ( ) ~ w k
1
σ , w → 0.
Yspol\zuq (2), (3) y (5), poluçaem asymptotyçeskye ravenstva
C t D Ak n( , , ) = 1
2
1
2
1
π ασ+ k t
log + M D Ak n( , ) + o( )1 , t → 0, (6)
hde
M D Ak n( , ) =
= 1
2
1
2
0
1 12
2 0
1 1 2 2
1 1
1
1 1π ασ
σ α
σ σ
σ σ
( )
log , log
, ,( )
( ) ( ) ( ) ( )
– –+
( ) +
( ) ( )
+
k
k k
k k k k
k
k
k
k
r
r w r w
a ak k
Ω
Ω Ω
, (7)
z ak k( ): = wk
( )1
, z ak k( )+1 : = wk
( )2
, k = 1, n , z an n( )+1 : = wn
( )2 , a Ωk
( )0 , Ωk
s( )
— obæe-
dynenye svqzn¥x komponent¥ mnoΩestv z D Pk k∩( ) , soderΩawyx toçky 0 y
wk
s( )
, k = 1, n , s = 1, 2, sootvetstvenno s yx symmetryçn¥m otobraΩenyem otno-
sytel\no mnymoj osy. S uçetom (6) pryxodym k sootnoßenyg
k
n
k nC t D A
=
∑
1
1
1
( , , ) –
–
=
= 1
4
1
π α( )
log
n t+
+N
1
4
22
2
1( – )
( ) ( , )
n
M D Ak k n
k
n
α
ασ+
=
∑ N+N o( )1 . (8)
Yz sootnoßenyj (2), (4) y (8) ymeem
M D Ak n( , ) ≤ 1
2
22
2
1( )
( ) ( , )
n
M D Ak k n
k
n
+
+
=
∑α
ασ . (9)
Yspol\zuq metod rabot¥ [4], poluçaem ocenku yssleduemoho funkcyonala
Jα = r B r B ak k
k
n
α( , ) ( , )0
1
0
=
∏ ≤
≤ σ µα
ασ
k
k
n
n k k k
k
n
A r G r G i r G ik
= =
∏ ∏
( ) ( ) ( ){ }
1
0 1 2
1
1
22
0( ) , , – , –( ) ( ) ( ) . (10)
V rezul\tate provedenn¥x v¥çyslenyj ysxodnaq zadaça ob ocenke velyçyn¥
Jα svedena k ocenke sverxu dlq funkcyonala I3( )β = r Bβ2
0 0( , ) r B( , )1 0 r B( 2,
– )i na klasse troek poparno neperesekagwyxsq oblastej B B B0 1 2, ,{ } takyx,
çto 0 0∈B , i B∈ 1, –i B∈ 2 , Bk � C , k = 0, 1, 2. Yzvestno [5, 4], çto I3( )β ≤
≤ ψ β( ) = 2
2 6β + ββ2
2( – β
β
)
– ( – )1
2
2 2
(2 + β
β
)
– ( )1
2
2 2+
, b ∈ 0 2,[ ]. Funkcyonal
log ( )ψ β v¥pukla vverx na otrezke ( , , )0 0 8 . Otsgda
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
1008 R. V. PODVÁSOCKYJ
r B r B ak k
k
n
α( , ) ( , )0
1
0
=
∏ ≤
≤ σ µ ψ α
αk
k
n
n
n
A
n=
∏
1
22
( ) , tk = α σk , 0 ≤ tk ≤ 0 8, . (11)
Tohda s uçetom σkk
n
=∑ 1
= 2 ubeΩdaemsq, çto neravenstva (11) v¥polnqgtsq
pry α ≤ 0 8
4
, = 0,2.
Yz rabot [4, 5] poluçaem, çto znak ravenstva v neravenstve (10) dostyhaetsq,
kohda toçky ak y oblasty Bk qvlqgtsq sootvetstvenno polgsamy y kruhov¥-
my oblastqmy kvadratyçnoho dyfferencyala (1).
TeoremaN1 dokazana.
Dokazatel\stvo teorem¥ 2. Netrudno pokazat\, çto funkcyq ln ( )β ψ β2
v¥pukla vverx na otrezke 0 1 75, ,( ). Tohda yz (10) s uçetom [4] poluçaem
neravenstvo
r B r B ak k
k
n
α( , ) ( , )0
1
0
=
∏ ≤ µ α ψα( )
–
A t tn
n
k
k
n
k
2 2
1
1
2
=
∏ ( )
≤
≤ α µ α ψ α
α
–
( )
n
n
n
A
n n
2
2 22 2
hde tk = α σk , 0 ≤ tk ≤ 1 75, .
Qsno takΩe, çto
r Dα( , )0 0 r D ak kk
n
, 0
1 ( )=∏ = α α– n
n
2
22
ψ α2 2
n
n
.
Otsgda sleduet utverΩdenye teorem¥.
Sleduet otmetyt\, çto teoremaN1 obobwaet sootvetstvugwug teoremu yz ra-
bot¥ [4] na sluçaj çastyçno nenalehagwyx oblastej, a takΩe usylyvaet y
obobwaet teoremuN4 yz rabot¥ [5], a teoremaN2 obobwaet sootvetstvugwug teo-
remu yz rabot¥ [4].
1. Lavrent\ev M. A. K teoryy komformn¥x otobraΩenyj // Tr. Fyz.-mat. yn-ta AN SSSR. –
1934. – 5 – S. 159 – 245.
2. DΩenkyns DΩ. A. Odnolystn¥e funkcyy y konformn¥e otobraΩenyq. – M.: Yzd-vo ynostr.
lyt., 1962. – 256 s.
3. Baxtyna H. P. NVaryacyonn¥e metod¥ y kvadratyçn¥e dyfferencyal¥ v zadaçax o nenale-
hagwyx oblastqx: Avtoref. dys. … kand. fyz.-mat. nauk. – Kyev, 1975. – 11 s.
4. Baxtyna H. P., Baxtyn A. K. Razdelqgwee preobrazovanye y zadaçy o nenalehagwyx oblas-
tqx // Zb. prac\ In-tu matematyky NAN Ukra]ny. – 2006. – 3, # 4. – S. 273 – 281.
5. Dubynyn V. N. Razdelqgwee preobrazovanye oblastej y zadaçy ob πkstremal\nom razbyenyy
// Zap. nauç. sem. Lenynh. otd-nyq Mat. yn-ta AN SSSR. – 1988. – 168. – S. 48 – 66.
6. Dubynyn V. N. Asymptotyka modulq v¥roΩdagwehosq kondensatora y nekotor¥e ee pryme-
nenyq // Zap. nauç. sem. LOMY. – 1997. – 237. – S. 56 – 73.
7. Baxtyn A. K. Neravenstva dlq vnutrennyx radyusov nenalehagwyx oblastej y otkr¥t¥x
mnoΩestv // Dop. NAN Ukra]ny – 2006. – # 10. – S. 7 – 13.
Poluçeno 03.11.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
|
| id | umjimathkievua-article-3218 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:25Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c3/f827dbbef5aec0cf9b6dfe2f40d638c3.pdf |
| spelling | umjimathkievua-article-32182020-03-18T19:48:23Z Estimation of the product of inner radii of partially nonoverlapping domains Оценка произведения внутренних радиусов частично неналегающих областей Podvysotskii, R. V. Подвысоцкий, Р. В. Подвысоцкий, Р. В. We present new results on the maximization of products of positive powers of inner radii of some special domain systems in the extended complex plane $\overline{{\mathbb C}}$ with respect to points of finite sets such that any two distinct points $z_1, z_2 \in {\mathbb C}\setminus \{0\}$ of such set belong to different rays emerging from the origin. Наведено нові результати про максимізацію додатних степенів внутрішніх радіусів деяких спеціальних систем областей у розширеній комплексній площині $\overline{{\mathbb C}}$ відносно точок скінченних множин таких, що дві будь-які різні точки $z_1, z_2 \in {\mathbb C}\setminus \{0\}$ множини лежать з початку координат. Institute of Mathematics, NAS of Ukraine 2008-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3218 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 7 (2008); 1004–1008 Український математичний журнал; Том 60 № 7 (2008); 1004–1008 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3218/3182 https://umj.imath.kiev.ua/index.php/umj/article/view/3218/3183 Copyright (c) 2008 Podvysotskii R. V. |
| spellingShingle | Podvysotskii, R. V. Подвысоцкий, Р. В. Подвысоцкий, Р. В. Estimation of the product of inner radii of partially nonoverlapping domains |
| title | Estimation of the product of inner radii of partially nonoverlapping domains |
| title_alt | Оценка произведения внутренних радиусов частично неналегающих областей |
| title_full | Estimation of the product of inner radii of partially nonoverlapping domains |
| title_fullStr | Estimation of the product of inner radii of partially nonoverlapping domains |
| title_full_unstemmed | Estimation of the product of inner radii of partially nonoverlapping domains |
| title_short | Estimation of the product of inner radii of partially nonoverlapping domains |
| title_sort | estimation of the product of inner radii of partially nonoverlapping domains |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3218 |
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