Application of a separating transformation to estimates of inner radii of open sets
We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets.
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| Date: | 2007 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2007
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3390 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509471005999104 |
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| author | Bakhtin, A. K. V'yun, V. E. Бахтин, А. К. Вьюн, В. Е. Бахтин, А. К. Вьюн, В. Е. |
| author_facet | Bakhtin, A. K. V'yun, V. E. Бахтин, А. К. Вьюн, В. Е. Бахтин, А. К. Вьюн, В. Е. |
| author_sort | Bakhtin, A. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:53:10Z |
| description | We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets. |
| first_indexed | 2026-03-24T02:41:37Z |
| format | Article |
| fulltext |
UDK 517.54
A. K. Baxtyn, V. E. V\gn (Yn-t matematyky NAN Ukrayn¥, Kyev)
PRYMENENYE RAZDELQGWEHO PREOBRAZOVANYQ
K OCENKAM VNUTRENNYX RADYUSOV
OTKRÁTÁX MNOÛESTV
∗∗∗∗
We obtain solutions of new extremal problems of the geometric theory of functions of complex variable
associated with estimates of inner radii of nonoverlapping domains. Some already known results are
generalized to the case of open sets.
OderΩano rozv’qzky novyx ekstremal\nyx zadaç heometryçno] teori] funkcij kompleksno]
zminno], pov’qzanyx z ocinkamy vnutrißnix radiusiv neperetynnyx oblastej. Uzahal\neno deqki
raniß vidomi rezul\taty na vypadok vidkrytyx mnoΩyn.
∏kstremal\n¥e zadaçy heometryçeskoj teoryy funkcyj kompleksnoj peremen-
noj, svqzann¥e s ocenkamy vnutrennyx radyusov nenalehagwyx oblastej y ot-
kr¥t¥x mnoΩestv, voznykly v svqzy s rabotoj M.5A.5Lavrent\eva [1], hde b¥la
reßena zadaça o proyzvedenyy konformn¥x radyusov dvux vzaymno neperese-
kagwyxsq oblastej. ∏ta rabota v¥zvala potok yssledovanyj mnohyx avtorov
(sm. [2 – 16]).
Pust\ n, m ∈ N . Systemu toçek An m, : = { }, : , , ,a k n p mk p ∈ = =C 1 1 na-
zovem ( , )n m -luçevoj, esly pry vsex k = 1, n y p = 1, m v¥polnqgtsq
sootnoßenyq
0 < ak,1 < ak,2 < … < ak m, < ∞ ,
arg ,ak 1 = arg ,ak 2 = … = arg ,ak m = : θk = : θk n mA( ), ,
0 = θ1 < θ2 < … < θn < θn+1 : = 2 π , an p+1, : = a p1, .
Opredelym umnoΩenye ( , )n m -luçevoj system¥ An m, = { },ak p na çyslo t > 0
sledugwym obrazom: t An m, : = { },tak p , k = 1, n , p = 1, m . V sluçae m = 1 na-
zovem ( , )n 1 -luçevug systemu toçek n-luçevoj y rassmotrym bolee prost¥e
oboznaçenyq: ak,1 = : ak , k = 1, n , An,1 = : An . Pust\
αk : = 1
1 1 1π
arg arg, ,a ak k+ −( ),
Λk : = { }: arg arg arg, ,w a w ak k∈ < <+C 1 1 1 ,
z wk ( ) — odnoznaçnaq vetv\ funkcyy – i e w
i ak k( )
arg /,− 1 1 α
, kotoraq realyzuet od-
nolystnoe komformnoe otobraΩenye oblasty Λk , k = 1, n , na ploskost\
Re z > 0. Pust\
χ ( t ) : = 1
2
1( )t t+ − , UR : = { }:w w R∈ <C , R > 0.
Pry n, m ∈ N , n = 2 oboznaçym
L An( ),2 : =
k
n
p
k p
k p
k p
a
a
a
k
= = +
∏ ∏
1 1
2
1
1 2
χ
α
,
,
/
, ,
L Ap n( ),2 : =
k
n
k p
k p
k p
a
a
a
k
= +
∏
1 1
1 2
χ
α
,
,
/
, , p = 1, 2 .
∗
V¥polnena pry çastyçnoj fynansovoj podderΩke Hosudarstvennoj prohramm¥ Ukrayn¥
#50107U002027.
© A. K. BAXTYN, V. E. V|GN, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10 1313
1314 A. K. BAXTYN, V. E. V|GN
Pry m = 1 oboznaçym L An1 1( ), = : L An1( ) = : L An( ) ,
L An( ) : =
k
n
k
k
k
a
a
a
k
= +
∏
1 1
1 2
χ
α/
.
Pust\ zadana proyzvol\naq ( , )n 2 -luçevaq systema toçek An,2 = { },ak p , k =
= 1, n , p = 1, 2, n ∈ N , n ≥ 2. Çetverka toçek
{ }( ), ( ), ( ), ( ), , , ,z a z a z a z ak k k k k k k k1 1 1 2 1 2+ + , k = 1, n ,
raspoloΩena na mnymoj osy, pryçem ynterval ( )( ), ( ), ,z a z ak k k k1 1 1+ soderΩyt
naçalo koordynat. Konformn¥j avtomorfyzm ploskosty kompleksn¥x çysel
vyda z �
1
1
+
−
z
z
preobrazuet ukazannug çetverku v çetverku toçek na edynyç-
noj okruΩnosty. Pust\ z1 , z2 , z3 , z4 — razlyçn¥e toçky edynyçnoj okruΩnos-
ty. Oboznaçym çerez z0 toçku otkr¥toho edynyçnoho kruha, v kotoroj perese-
kagtsq neevklydov¥ heodezyçeskye, soedynqgwye sootvetstvenno par¥ toçek
z1 , z3 y z2 , z4 . Tohda konformn¥j avtomorfyzm kompleksnoj ploskosty Cz
vyda z � e
z z
z z
iθ −
−
0
01
, θ ∈ R , preobrazuet zadannug çetverku toçek v verßyn¥
nekotoroho nev¥roΩdennoho prqmouhol\nyka so storonamy, parallel\n¥my
koordynatn¥m osqm. S uçetom yzloΩennoho v¥ße netrudno ukazat\ pry
kaΩdom k = 1, n , n ∈ N , n ≥ 2, konformn¥e avtomorfyzm¥ T zk ( ) kompleks-
noj ploskosty Cz takye, çto
– i k
kρ α1/ = T z ak k k( ( )),1 , – i k
kρ α−1/ = T z ak k k( ( )),2 ,
i k
kρ α1/ = T z ak k k( ( )),+1 1 , i k
kρ α−1/ = T z ak k k( ( )),+1 2 , k = 1, n .
Sovokupnost\ çysel { }ρk k
n
=1 odnoznaçno opredelqetsq systemoj toçek An,2
(sm. takΩe [13]). Pust\
t0 : = t An0 2( ), : =
k
n
k
k
n
k
k
=
∏ −
+
1
2
2
1
1
1
ρ
ρ
α
α
/
/
/
, R
0 : =
1
1
0
0
1−
+
t
t
n/
.
Rassmotrym otkr¥toe mnoΩestvo D ⊂ C , A Dn,2 ⊂ . Svqznug komponentu
mnoΩestva D k∩ Λ , soderΩawug toçku a, budem oboznaçat\ D ak ( ) . Esly
mnoΩestva D ak k( ),1 , D ak k( ),2 , D ak k( ),+1 1 , D ak k( ),+1 2 (sootvetstvenno D ak k( ),
D ak k( )+1 ) ne pust¥ y poparno ne peresekagtsq pry vsex k = 1, n , to budem ho-
voryt\, çto mnoΩestvo D udovletvorqet obobwennomu uslovyg nenalehanyq
otnosytel\no system¥ toçek An,2 (sootvetstvenno otnosytel\no system¥
toçek An = { }ak k
n
=1) (sm., naprymer, [14, 15]). Na mnoΩestve par celoçys-
lenn¥x yndeksov opredelym ravenstvo ( k, p ) = ( q, s ) ⇔ k = q y p = s .
Yspol\zuem¥e v dal\nejßem opredelenyq vnutrenneho radyusa r ( B , a ) ob-
lasty B otnosytel\no soderΩawejsq v nej toçky a, kvadratyçnoho dyffe-
rencyala, obobwennoj funkcyy Hryna g z wB( , ) oblasty B, kondensatora y
svqzann¥e s nym ponqtyq eho emkosty y modulq soderΩatsq, naprymer, v [2, 3, 8,
10]. Esly B — otkr¥toe mnoΩestvo, a ∈ B , to B ( a ) — svqznaq komponenta
mnoΩestva B, soderΩawaq toçku a. PoloΩym r ( B , a ) : = r ( B ( a ) , a ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
PRYMENENYE RAZDELQGWEHO PREOBRAZOVANYQ K OCENKAM … 1315
g w aB( , ) : =
g w a w B a
g a w B a B a
w B a
B a
w
B a
( )
( )
( , ), ( ),
lim ( , ), ( ) ( ( )),
, ( ).\
∈
∈ ∈
∈
→ζ
ζ ∂ ζ
0 C
Vo vvedenn¥x v¥ße oboznaçenyqx spravedlyva sledugwaq teorema.
Teorema. Pust\ n ≥ 2. Tohda kakov¥ b¥ ny b¥ly ( , )n 2 -luçevaq systema
toçek An,2 takaq, çto L An( ),2 = 1, y otkr¥toe mnoΩestvo D ⊂ C , udov-
letvorqgwee obobwennomu uslovyg nenalehanyq otnosytel\no zadannoj An,2 ,
ymeet mesto neravenstvo
k
n
p
k pr D a
= =
∏ ∏
1 1
2
( , ), ≤ 2
1
1
2
1
2
1
2
2
2
n
k
n
k
k
n
k
k
D k p q s
k p q s
k
k
g a a
= = ≠
∏ ∏ ∏
−
+
−{ }α
ρ
ρ
α
α
/
/ , ,
( , ) ( , )
exp ( , ) , (1)
znak ravenstva v kotorom dostyhaetsq, kohda toçky ak p, , k = 1, n , p = 1, 2,
y mnoΩestvo D qvlqgtsq sootvetstvenno polgsamy y obæedynenyem vsex
kruhov¥x oblastej kvadratyçnoho dyfferencyala
Q w dw( ) 2 = –
w w
w R R w
dw
n n
n n n n
− +
− −
2 2
0 2 0 2
21
1
( )
( ( ) ) ( ( ) )
.
∏ta teorema obobwaet rezul\tat¥ o nenalehagwyx oblastqx, ustanovlenn¥e
v rabotax [12, 13].
Dokazatel\stvo sleduet sxeme, predloΩennoj v rabotax [14, 15], y ys-
pol\zuet ydey y metod¥ rabot [8 – 13].
Yz uslovyj, naloΩenn¥x na mnoΩestvo D , sleduet, çto funkcyq g z wD( , )
opredelena pry vsex z, w ∈ D y koneçna pry vsex z ≠ w. Pust\ E0 = C \ D ,
E ak p( , ), ε = { }: ,w w ak p∈ − ≤C ε , ε > 0, k = 1, n , p = 1, 2.
Dlq dostatoçno mal¥x ε rassmotrym kondensator
C D An( , , ),ε 2 = { }, ( )E E0 1 ε , E1( )ε =
k
n
p
k pE a
= =1 1
2
∪ ∪ ( , ), ε .
Emkost\ kondensatora C D An( , , ),ε 2
cap C D An( , , ),ε 2 : = inf ∫∫ ∂
∂
+ ∂
∂
ϕ ϕ
x x
dx dy
2 2
,
hde toçnaq nyΩnqq hran\ beretsq po vsem vewestvenn¥m neprer¥vn¥m lypßy-
cev¥m na C funkcyqm ϕ = ϕ ( z ) takym, çto ϕ E0
= 0, ϕ εE1( ) = 1. Velyçy-
na C D An( , , ),ε 2 : = ( )( , , ),cap C D Anε 2
1−
naz¥vaetsq modulem kondensatora
C D An( , , ),ε 2 .
Rassmotrym razdelqgwee preobrazovanye kondensatora C D An( , , ),ε 2 otno-
sytel\no system¥ funkcyj { }( )z wk k
n
=1 y system¥ oblastej { }Λk k
n
=1. Pust\
C D Ak n( , , ),ε 2 = { }( ) ( ), ( )E Ek k
0 1 ε , hde E k
0
( )
— obæedynenye obraza mnoΩestva
E k0 ∩ Λ pry otobraΩenyy z = z wk ( ) s symmetryçn¥m emu mnoΩestvom otno-
sytel\no mnymoj osy, a E k
1
( )( )ε — obæedynenye obraza mnoΩestva E k1( )ε ∩ Λ ,
k = 1, n , pry tom Ωe otobraΩenyy s symmetryçn¥m emu mnoΩestvom otnosy-
tel\no mnymoj osy. Tohda (sm. [9, 10]) v¥polnqetsq osnovnoe neravenstvo meto-
da kusoçno-razdelqgweho preobrazovanyq
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1316 A. K. BAXTYN, V. E. V|GN
cap C D An( , , ),ε 2 ≥ 1
2 1
2
k
n
k nC D A
=
∑ ( , , ),ε . (2)
Yz (2) poluçaem neravenstvo, yhragwee klgçevug rol\ v dal\nejßyx ocenkax:
C D An( , , ),ε 2 ≤ 2
1
2
1
1
k
n
k nC D A
=
−
−
∑
( , , ),ε . (3)
Yz teorem¥551 rabot¥ [9] sleduet asymptotyçeskoe ravenstvo
C D An( , , ),ε 2 = 1
4
1 12π εn
M D A onlog ( , ) ( ),+ + , ε → 0, (4)
hde M D An( , ),2 — pryvedenn¥j modul\ mnoΩestva D otnosytel\no system¥
toçek An,2 :
M D An( , ),2 = 1
8 2
1 1
2
πn
r D a g a a
k
n
p
k p
k p q s
D k p q s
= = ≠
∑ ∑ ∑+
log ( , ) ( , ),
( , ) ( , )
, , . (5)
Obæedynenye svqznoj komponent¥ mnoΩestva z Dk k( )Λ ∩ , soderΩawej
toçku ωk p,
( )1 : = z ak k p( ), , s ee symmetryçn¥m otraΩenyem otnosytel\no mnymoj
osy, oboznaçym çerez Ωk p,
( )1
. Analohyçno, obæedynenye svqznoj komponent¥
mnoΩestva z Dk k( )Λ ∩ , soderΩawej toçku ωk p,
( )2 : = z ak k p( ),+1 , s ee symmet-
ryçn¥m otraΩenyem otnosytel\no mnymoj osy, oboznaçym çerez Ωk p,
( )2
. V¥pol-
nqgtsq sledugwye ravenstva:
z w z ak k s p( ) ( ),− = 1 1
1 1
α
α
k
s p s pa w a ok
,
/
, ( )
− − + , w → as p, ,
(6)
k = 1, n , p = 1, 2, s = k, k + 1.
Netrudno zametyt\, çto
ω ωk p k p,
( )
,
( )1 2− = a ak p k p
k k
,
/
,
/1
1
1α α+ + , k = 1, n , p = 1, 2. (7)
Prymenqq teoremu551 yz rabot¥ [9] y yspol\zuq (6), poluçaem sootnoßenyq
C D Ak n( , , ),ε 2 = 1
8
1 12π ε
log ( , ) ( ),+ +M D A ok n , ε → 0, k = 1, n , (8)
hde
M D Ak n( , ),2 = 1
2
1
16 1
2 1 1
1 1 1
2 2
1
1
1 1π
ω
α
ω
αα α
p
k p k p
k k p
k p k p
k k p
r
a
r
ak k
=
− − −
+
−∑ log
, ,( ) ( ),
( )
,
( )
,
/
,
( )
,
( )
,
/
Ω Ω
. (9)
V svog oçered\, yz (8) sledugt ravenstva
C D Ak n( , , ),ε 2
1−
=
8 8
1 1
2
2
π
ε
π
εlog log
( , ),− −+
M D Ak n +
+ o 1
1
2
log ε−
, ε → 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
PRYMENENYE RAZDELQGWEHO PREOBRAZOVANYQ K OCENKAM … 1317
k
n
k nC D A
=
−∑
1
2
1
( , , ),ε =
8 8
1 1
2
1
2
π
ε
π
ε
n
M D A
k
n
k nlog log
( , ),− −
=
+
∑ +
+ o 1
1
2
log ε−
, ε → 0.
Otsgda poluçaem v¥raΩenye, dagwee asymptotyku pravoj çasty neravenst-
va5(3):
k
n
k nC D A
=
−
−
∑
1
2
1
1
( , , ),ε =
= 1
8
1 1
8 1
1
1
2 1
1
π ε
π
ε εn n
M D A o
k
n
k nlog
log
( , )
log,+ +
−
=
−
−
∑ =
= 1
8
1 1 12
1
2π εn n
M D A o
k
n
k nlog ( , ) ( ),+ +
=
∑ , ε → 0. (10)
Yz neravenstva (3) s uçetom (4) y (10) sleduet, çto
1
4
1 12π εn
M D A onlog ( , ) ( ),+ + ≤ 1
4
1 2 12
1
2π εn n
M D A o
k
n
k nlog ( , ) ( ),+ +
=
∑ .
Sokrawaq osobennosty v poslednem neravenstve y ustremlqq ε k nulg, polu-
çaem neravenstvo dlq pryvedenn¥x modulej
M D An( , ),2 ≤ 2
2
1
2
n
M D A
k
n
k n
=
∑ ( , ), . (11)
V¥raΩenyq (5) y (11) pryvodqt k neravenstvu
1
8 2
1 1
2
πn
r D a g a a
k
n
p
k p
k p q s
D k p q s
= = ≠
∑ ∑ ∑+
log ( , ) ( , ),
( , ) ( , )
, , ≤
≤ 2 1
2
1
162
1 1
2 1 1
1 1 1
1
2 2 2
1
1
1 1n
r
a
r
ak
n
p
k p k p
k k p p
k p k p
k k p
k kπ
ω
α
ω
αα α
= =
− −
=
−
+
−∑ ∑ ∑+
log
,
log
,( ) ( ),
( )
,
( )
,
/
,
( )
,
( )
,
/
Ω Ω
.
Otsgda s uçetom (7) poluçaem
k
n
p
k p D k p q s
k p q s
r D a g a a
= = ≠
∏ ∏ ∏
1 1
2
( , ) exp ( , ), , ,
( , ) ( , )
≤
≤
k
n
k
k
n
p
k p
k p k p
a
a a k
= = = +
∏ ∏ ∏
( )1
2
1 1
2
1
1 2α α
,
, ,
/ ×
×
k
n
p
k p k p k p k pr r
= =
∏ ∏
1 1
2
1 1 2 2
1 2
( ) ( ),
( )
,
( )
,
( )
,
( )
/
, ,Ω Ωω ω =
=
k
n
k
k
n
p
k p k p
k p k p
k p
a a
a a
a
k k
k
= = =
+
+
∏ ∏ ∏
+
( )1
2
1 1
2 1
1
1
1
1 2α
α α
α
,
/
,
/
, ,
/ , ×
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1318 A. K. BAXTYN, V. E. V|GN
×
k
n
p
k p k p k p k p
k p k p
r r
= =
∏ ∏
−
1 1
2 1 1 2 2
1 2 2
1 2
( ) ( ),
( )
,
( )
,
( )
,
( )
,
( )
,
( )
/
, ,Ω Ωω ω
ω ω
=
= 22
1
2
2
1 1
2 1 1 2 2
1 2 2
1 2
n
k
n
k n
k
n
p
k p k p k p k p
k p k p
L A
r r
= = =
∏ ∏ ∏
−
α
ω ω
ω ω
( )
, ,
,
,
( )
,
( )
,
( )
,
( )
,
( )
,
( )
/
( ) ( )Ω Ω
.
Uçyt¥vaq uslovye teorem¥ L An( ),2 = 1, pryxodym k neravenstvu
k
n
p
k p D k p q s
k p q s
r D a g a a
= = ≠
∏ ∏ ∏
1 1
2
( , ) exp ( , ), , ,
( , ) ( , )
≤
≤ 22
1
2
1 1
2 1 1 2 2
1 2 2
1 2
n
k
n
k
k
n
p
k p k p k p k p
k p k p
r r
= = =
∏ ∏ ∏
−
α
ω ω
ω ω
( ) ( ),
( )
,
( )
,
( )
,
( )
,
( )
,
( )
/
, ,Ω Ω
.
Otmetym, çto oblasty Ωk,
( )
1
1 , Ωk,
( )
1
2 , Ωk,
( )
2
1 , Ωk,
( )
2
2
vzaymno ne peresekagtsq.
Pust\ ∆k p
s
,
( ) : = Tk k p
s( ),
( )Ω , ζ ωk k p
s( ),
( ) : = ( ) ( )− − − −
1 1 1
s
ki
p
kρ α , 0 < ρ k < 1, k =
= 1, n , p, s = 1, 2 (hde T zk ( ) , k = 1, n , opredelen¥ vo vvedenyy k rabote).
Yspol\zuq ynvaryantnost\ funkcyonala otnosytel\no konformn¥x avto-
morfyzmov ploskosty kompleksn¥x çysel
J =
r r r rk k k k k k k k
k k k k
( ) ( ) ( ) ( ),
( )
,
( )
,
( )
,
( )
,
( )
,
( )
,
( )
,
( )
,
( )
,
( )
,
( )
,
( )
, , , ,Ω Ω Ω Ω1
1
1
1
1
2
1
2
2
1
2
1
2
2
2
2
1
1
1
2 2
2
1
2
2 2
ω ω ω ω
ω ω ω ω− −
,
pryxodym k v¥vodu, çto
k
n
p
k p D k p q s
k p q s
r D a g a a
= = ≠
∏ ∏ ∏
1 1
2
( , ) exp ( , ), , ,
( , ) ( , )
≤
≤ 2
4
2
1
2
1
1
1 1
1
2 1
2
n
k
n
k
k
n
k k k k
k
r i r ik k
k
= =
∏ ∏
−
α
ρ ρ
ρ
α α
α
( ) ( ),
( ) /
,
( ) /
/
, ,∆ ∆
×
×
r i r ik k k k
k
k k
k
( ) ( ),
( ) /
,
( ) /
/
/
, ,∆ ∆2
1 1
2
2 1
2
1 2
4
−
− −
−
ρ ρ
ρ
α α
α . (12)
Dlq dal\nejßyx ocenok yspol\zuem sledugwyj rezul\tat.
Lemma. Pry k = 1, n v¥polnqetsq neravenstvo
r E i r E i r E i r E ik k k k
k
k k k k
k
k k
k
k k
k
( ) ( ) ( ) ( ),
( ) /
,
( ) /
/
,
( ) /
,
( ) /
/
, , , ,1
1 1
1
2 1
2
2
1 1
2
2 1
24 4
− − − −
−
ρ ρ
ρ
ρ ρ
ρ
α α
α
α α
α ≤
≤
1
1
2
2
4
−
+
ρ
ρ
α
α
k
k
k
k
/
/ ,
znak ravenstva v kotorom dostyhaetsq tohda y tol\ko tohda, kohda Ek p
s
,
( ) , s ,
p = 1, 2, pry kaΩdom k = 1, n qvlqgtsq kruhov¥my oblastqmy kvadratyçnoho
dyfferencyala
Q w dwk ( ) 2 =
( )
( ) ( )
1
1
2 2
2 2 2 2 2 2
2−
+ +
w
w w
dw
k kρ ρ
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
PRYMENENYE RAZDELQGWEHO PREOBRAZOVANYQ K OCENKAM … 1319
V odnosvqznom sluçae πta lemma poluçena v rabote E.5H.5Emel\qnova [12],
dlq proyzvol\n¥x mnohosvqzn¥x oblastej πtot rezul\tat dokazan v rabote
V.5N.5Dubynyna [9].
Uçyt¥vaq lemmu y neravenstvo (12), pryxodym k sootnoßenyg
k
n
p
k pr D a
= =
∏ ∏
1 1
2
( , ), ≤
≤ 2
1
1
2
1
2
1
2
2
2
n
k
n
k
k
n
k
k
D k p q s
k p q s
k
k
g a a
= = ≠
∏ ∏ ∏
−
+
−{ }α ρ
ρ
α
α
/
/ , ,
( , ) ( , )
exp ( , ) .
UtverΩdenye o znake ravenstva v neravenstve (1) proverqetsq neposredstvenno.
Teorema dokazana.
Yz teorem¥ neposredstvenno v¥tekagt nekotor¥e sledstvyq.
Sledstvye(1. Pust\ n ≥ 3, n ∈ N , ρ ∈ ( 0, 1 ) . Tohda dlq lgboj ( , )n 2 -lu-
çevoj system¥ toçek An,2 = { },ak p , k = 1, n , p = 1, 2, takoj, çto L An( ),2 =
= 1, R An
0
2( ), = ρ , y dlq lgboho otkr¥toho mnoΩestva D , A Dn,2 ⊂ ⊂ C ,
udovletvorqgweho obobwennomu uslovyg nenalehanyq otnosytel\no zadannoj
system¥ toçek An,2 , v¥polnqetsq neravenstvo
k
n
k p
p
r D a
= =
∏ ∏
1 1
2
( , ), ≤ 2
1
1
2
1
2 2
n
k
n
k
n
n
n
D k p q s
k p q s
g a a
= ≠
∏ ∏
−
+
−{ }α ρ
ρ
exp ( , ), ,
( , ) ( , )
.
Znak ravenstva v πtom neravenstve dostyhaetsq, kohda toçky ak p, y mnoΩe-
stvo D qvlqgtsq sootvetstvenno polgsamy y obæedynenyem kruhov¥x oblas-
tej kvadratyçnoho dyfferencyala
Q w dw( ) 2 = –
w w
w w
dw
n n
n n n n
− +
− −
2 2
2 2
21
1
( )
( ) ( )ρ ρ
.
Sledstvye(2. Pust\ n ≥ 3, n ∈ N , 0 < ρ < R . Tohda dlq lgboj ( , )n 2 -
luçevoj system¥ toçek An,2 = { },ak p , k = 1, n , p = 1, 2, takoj, çto
L An( ),2 = R n2 , R R An
0
21(( ) )/ , = ρ / R , y dlq lgboho otkr¥toho mnoΩestva D ,
A Dn,2 ⊂ ⊂ C , udovletvorqgweho obobwennomu uslovyg nenalehanyq otnosy-
tel\no system¥ toçek An,2 , v¥polnqetsq neravenstvo
k
n
k p
p
r D a
= =
∏ ∏
1 1
2
( , ), ≤ ( ) exp ( , ), ,
( , ) ( , )
2 2
1
2 2
R
R
R
g a an
k
n
k
n n
n n
n
D k p q s
k p q s= ≠
∏ ∏
−
+
−{ }α ρ
ρ
.
Znak ravenstva v πtom neravenstve dostyhaetsq, kohda toçky ak p, , k = 1, n ,
p = 1, 2, y mnoΩestvo D qvlqgtsq sootvetstvenno polgsamy y obæedyneny-
em kruhov¥x oblastej kvadratyçnoho dyfferencyala
Q w dw( ) 2 = –
w w R
w R w
dw
n n n
n n n n n
− +
− −
2 2
2 2 2
2( )
( ) ( )ρ ρ
.
Proyzvol\noj ( , )n 2 -luçevoj systeme toçek An,2 = { },ak p , k = 1, n , p = 1, 2,
sopostavym „kruhovug” ( , )n 2 -luçevug systemu An, ( )2 λ = { }, ( )bk p λ po sledu-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1320 A. K. BAXTYN, V. E. V|GN
gwemu pravylu: bk p, ( )λ =
a
a
k p
k p
p
,
,
λ , k = 1, n , p = 1, 2, hde λ1 = [ ]( ),
/L An
n
1 2
1 ,
λ2 = R2
1/ λ , L An( ),2 = R n2 .
Analohyçno, dlq n-luçevoj system¥ toçek An = { }ak k
n
=1 vvedem „kruhovug”
systemu An( )λ = { }( )bk k
nλ =1 : bk ( )λ =
a
a
k
k
λ , k = 1, n , hde λ = L A
n
n
1/ ( ) . Esly
An = { }ak k
n
=1 ⊂ UR y λ = L A
n
n
1/ ( ) < R , R > 0, to An( )λ ⊂ U R . Dlq n-lu-
çev¥x system toçek An ⊂ UR, L A
n
n
1/ ( ) < R , poloΩym t An0( ) : = t An0 2( ˆ ), , hde
ˆ
,An 2 = { },ak p , ak,1 : = ak , ak,2 : = R ak
2 / , k = 1, n .
Sledstvye(3. Pust\ n ≥ 3, n ∈ N , 0 < λ < R . Tohda dlq proyzvol\noj
( , )n 2 -luçevoj system¥ toçek An,2 = { },ak p , k = 1, n , p = 1, 2, takoj, çto
L An( ),2 = R n2 , L An1 2( ), = λ
n
, t An0 2( ), = t An0 2( ( )), λ , y dlq lgboho otkr¥toho
mnoΩestva D, A Dn,2 ⊂ ⊂ C , udovletvorqgweho obobwennomu uslovyg nena-
lehanyq, v¥polnqetsq neravenstvo
k
n
k p
p
r D a
= =
∏ ∏
1 1
2
( , ), ≤ 4
2
R
n
R
R
g a a
n n
n n
n
D k p q s
k p q s
⋅ −
+
−{ }
≠
∏λ
λ
exp ( , ), ,
( , ) ( , )
.
Znak ravenstva v πtom neravenstve dostyhaetsq, kohda toçky ak p, y mno-
Ωestvo D qvlqgtsq sootvetstvenno polgsamy y obæedynenyem kruhov¥x ob-
lastej kvadratyçnoho dyfferencyala
Q w dw( ) 2 = –
w w R
w R w
dw
n n n
n n n n n
− +
− −
2 2
2 2 2
2( )
( ) ( )λ λ
.
Sledstvye(4. Pust\ n ≥ 3, λ y R — poloΩytel\n¥e dejstvytel\n¥e
çysla, λ < R . Tohda dlq lgboj n-luçevoj system¥ toçek An = { }ak k
n
=1 t a -
koj, çto L ( An ) = λ
n
, t0 ( An ) = t0 ( An ( λ )) , y dlq lgboho otkr¥toho mnoΩestva
D takoho, çto An ⊂ D ⊂ UR , udovletvorqgweho obobwennomu uslovyg nena-
lehanyq otnosytel\no system¥ toçek An , v¥polnqetsq neravenstvo
r D ak
k
n
( , )
=
∏
1
≤
4λ λ
λn
R
R
g a a
n n n
n n
n
D k p
k p
−
+
−{ }
≠
∏ exp ( , ) .
Znak ravenstva v πtom neravenstve dostyhaetsq, kohda toçky ak y mnoΩest-
vo D qvlqgtsq sootvetstvenno prynadleΩawymy kruhu UR polgsamy y
obæedynenyem kruhov¥x oblastej kvadratyçnoho dyfferencyala
Q w dw( ) 2 = –
w w R
w R w
dw
n n n
n n n n n
− +
− −
2
2 2 2
2( )
( ) ( )λ λ
.
Sledstvye554 obobwaet odyn rezul\tat V. N. Dubynyna yz rabot¥ [11].
1. Lavrent\ev M. A. K teoryy konformn¥x otobraΩenyj // Tr. Fyz.-mat. yn-ta AN SSSR. –
1934. – 5. – S.5159 – 245.
2. Holuzyn H. M. Heometryçeskaq teoryq funkcyj kompleksnoho peremennoho. – M.: Nauka,
1966. – 628 s.
3. DΩenkyns DΩ. A. Odnolystn¥e funkcyy y konformn¥e otobraΩenyq. – M.: Yzd-vo ynostr.
lyt., 1962. – 256 s.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
PRYMENENYE RAZDELQGWEHO PREOBRAZOVANYQ K OCENKAM … 1321
4. Lebedev N. A. Pryncyp plowadej v teoryy odnolystn¥x funkcyj. – M.: Nauka, 1975. –
3365s.
5. Tamrazov P. M. ∏kstremal\n¥e konformn¥e otobraΩenyq y polgs¥ kvadratyçn¥x
dyfferencyalov // Yzv. AN SSSR. Ser. mat. – 1968. – 32, # 5. – S.51033 – 1043.
6. Baxtyna H. P. Varyacyonn¥e metod¥ y kvadratyçn¥e dyfferencyal¥ v zadaçax o
nenalehagwyx oblastqx: Avtoref dys. … kand. fyz.-mat. nauk. – Kyev, 1975. – 115s.
7. Kuz\myna H. V. Moduly semejstv kryv¥x y kvadratyçn¥e dyfferencyal¥. – L.: Nauka,
1980. – 241 s.
8. Dubynyn V. N. Metod symmetryzacyy v heometryçeskoj teoryy funkcyj kompleksnoho
peremennoho // Uspexy mat. nauk. – 1994. – 49, # 1 (295). – S.53 – 76.
9. Dubynyn V. N. Asymptotyka modulq v¥roΩdagwehosq kondensatora y nekotor¥e ee
prymenenyq // Zap. nauçn. sem. LOMY. – 1997. – 237. – S.556 – 73.
10. Dubynyn V. N. Emkosty kondensatorov v heometryçeskoj teoryy funkcyj: Uç. posobye. –
Vladyvostok: Yzd. Dal\nevostoç. un-ta, 2003. – 1165s.
11. Dubynyn V. N. O proyzvedenyy vnutrennyx radyusov „çastyçno nenalehagwyx” oblastej //
Vopros¥ metryçeskoj teoryy otobraΩenyj y ee prymenenye. – Kyev: Nauk. dumka, 1978. –
S.524 – 31.
12. Emel\qnov E. H. O svqzy dvux zadaç ob πkstremal\nom razbyenyy // Zap. nauçn. sem. LOMY.
– 1987. – 160. – S.591 – 98.
13. Baxtyn A. K. ∏kstremal\n¥e zadaçy o nenalehagwyx oblastqx so svobodn¥my polgsamy
na dvux okruΩnostqx // Dop. NAN Ukra]ny. – 2005. – # 7. – S.512 – 16.
14. Baxtyn A. K. Pryvedenn¥e moduly otkr¥t¥x mnoΩestv y πkstremal\n¥e zadaçy so
svobodn¥my polgsamy // Tam Ωe. – 2006. – # 5. – S.57 – 13.
15. Baxtyn A. K. Neravenstva dlq vnutrennyx radyusov nenelehagwyx oblastej y otkr¥t¥x
mnoΩestv // Tam Ωe. – # 10. – S.57 – 13.
16. Baxtyn A. K., V\gn V. E. Razdelqgwee preobrazovanye y neravenstva dlq otkr¥t¥x mno-
Ωestv // Tam Ωe. – 2007. – # 4. – S.57 – 11.
Poluçeno 21.06.2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
|
| id | umjimathkievua-article-3390 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:41:37Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ec/6362b31a60e8acc64ce4acd52fc611ec.pdf |
| spelling | umjimathkievua-article-33902020-03-18T19:53:10Z Application of a separating transformation to estimates of inner radii of open sets Применение разделяющего преобразования к оценкам внутренних радиусов открытых множеств Bakhtin, A. K. V'yun, V. E. Бахтин, А. К. Вьюн, В. Е. Бахтин, А. К. Вьюн, В. Е. We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets. Одержано розв'язки нових екстремальних задач геометричної теорії функцій комплексної змінної, пов'язаних з оцінками внутрішніх радіусів неперетинних областей. Узагальнено деякі раніш відомі результати на випадок відкритих множин. Institute of Mathematics, NAS of Ukraine 2007-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3390 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 10 (2007); 1313–1321 Український математичний журнал; Том 59 № 10 (2007); 1313–1321 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3390/3523 https://umj.imath.kiev.ua/index.php/umj/article/view/3390/3524 Copyright (c) 2007 Bakhtin A. K.; V'yun V. E. |
| spellingShingle | Bakhtin, A. K. V'yun, V. E. Бахтин, А. К. Вьюн, В. Е. Бахтин, А. К. Вьюн, В. Е. Application of a separating transformation to estimates of inner radii of open sets |
| title | Application of a separating transformation to estimates of inner radii of open sets |
| title_alt | Применение разделяющего преобразования к оценкам внутренних радиусов открытых множеств |
| title_full | Application of a separating transformation to estimates of inner radii of open sets |
| title_fullStr | Application of a separating transformation to estimates of inner radii of open sets |
| title_full_unstemmed | Application of a separating transformation to estimates of inner radii of open sets |
| title_short | Application of a separating transformation to estimates of inner radii of open sets |
| title_sort | application of a separating transformation to estimates of inner radii of open sets |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3390 |
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