Extremal problem on (2n, 2m 1)-system points on the rays.

Transcription

1 DOI: 0.55/auom An. Ş. Univ. Ovidius Consanţa Vol. 4,06, Exremal problem on n, m -sysem poins on he rays. Targonskii A. Absrac In his work derivaion of accurae esimae he producion inner radius non-overlapping domains and open se. The problems arise such ype in he firs ime in work []. I is lae resul his work generalize and srenghen in works [ 3]. In works [7, 8, 0] inroduce he general sysems poins, he name n-radial sysems poins. In his work a success he draw generalize some resuls he work [7]. Le N, R he ses naural and real numbers conformiy, C he plain complex numbers, C = C { } he Riemannian sphere, R + = 0,. For fix number n N sysem poins A n, = {a k,p C : k =, n, p =, m }, we will called on he n, -sysem poins on he rays, if a all k =, n, p =, m he relaions are execued: 0 < a k, <... < a k, < ; arg a k, = arg a k, =... = arg a k, =: θ k ; 0 = θ < θ <... < θ n < θ n+ := π. For such sysems of poins we will consider he following sizes: α k = π [θ k+ θ k ], k =, n, α n+ := α, α 0 := α n, α k =. Key Words: inner radius of domain, quadraic differenial, piecewise-separaing ransformaion, he Green funcion, radial sysems of poins, logarihmic capaciy, variaional formula. 00 Mahemaics Subjec Classificaion: Primary 30C70, 30C75. Received: Acceped:

2 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 84 M Le s consider sysem of angular domains: P k = {w C : θ k < arg w < θ k+ }, k =, n. Le s consider he following operaing funcionaliies for arbirary n, m -sysem poins on he rays A n, = M M A n, = M p= [ ak,p p= A 3 n, = A 4 n, = [ ak,p p= p= [ ak,p [ ak,p α k α k α k α k ak,p ] α k ak,p, ak,p ] α k ak,p, ak,p ] α k ak,p, ak,p ] α k ak,p, where χ = +, R +. Le {B 0, B k,p }, {B k,p, B } arbirary non-overlapping domains such ha a k,p B k,p, B k,p C, k =, n, p =, m. Le D, D C arbirary open se and w = a D, hen Da he define conneced componen D, he conain poin a. For arbirary n, m - sysem poins on he rays A n, = {a k,p C : k =, n, p =, m } and open se D, A n, D he define D k a s,p conneced componen se D a s,p P k, he conain poin a s,p, k =, n, s = k, k +, p =, m, a n+,p := a,p. The open se D, A n, D saisfied condiion mees he condiion of unapplied in relaion o he sysem of poins n, m -sysem poins on he rays A n, if a condiion is execued D k a k,s D k a k+,p =, k =, n, p, s =, m on all corners P k. The define rb; a inner radius domain B C wih respec o a poin a B look [4 7, 4]. Subjec of sudying of our work are he following problems. Problem. Le n, m N, n, m, α R +. Maximum funcional be found J = r α B k,p, a k,p r B k,p, a k,p p= p=

3 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 85 p= r α B k,p, a k,p p= r B k,p, a k,p, where A n, arbirary n, m -sysem poins on he rays, saisfied condiion, {B k,p } arbirary se non-overlapping domains, a k,p B k,p C, and all exremal he describe k =, n, p =, m. Problem. Le n, m N, n, m, α R +. Maximum funcional be found I = p= p= r α D, a k,p r α D, a k,p p= p= r D, a k,p r D, a k,p, where A n, arbirary n, m -sysem poins on he rays, saisfied condiion, D arbirary open se, he saisfied condiion, a k,p D C, and all exremal he describe k =, n, p =, m. Theorem. Le n, m N, n, α R +. Then for all n, m - sysem poins on he rays A n,, and arbirary se non-overlapping domains {B k,p }, a k,p B k,p C, k =, n, p =, m be saisfied inequaliy M p= r α B k,p, a k,p p= 4 m n r α B k,p, a k,p nα+ M A n, M A 4 n, p= p= r B k,p, a k,p r B k,p, a k,p A n, M α α A 3 n, α α α α + α+ n The equaliy obain in his inequaliy, when poins a k,p and domains B k,p are, conformiy, he poles and he circular domains of he quadraic differenial Qwdw = w n + w n m 3 iα w n + i 4m w n i 4m + α w n + w n + i 4m + w n i 4m dw.. 3

4 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 86. Theorem. Le n, m N, n, α R +. Then for all n, m - sysem poins on he rays, arbirary open se D, he saisfied condiion, a k,p D C, k =, n, p =, m be saisfied inequaliy p= r α D, a k,p p= M A n, p= r D, a k,p M r D, a k,p 4 m n p= nα+ r α D, a k,p α A 3 n, M A n, M A 4 n, α α α α α + α+ The equaliy obain in his inequaliy, when D = k,p B k,p, n. and poins a k,p and domains B k,p are, conformiy, he poles and he circular domains of he quadraic differenial 3. Proof heorem. The heorem of he proof leans on a mehod of he piece-dividing ransformaion developed by Dubinin look [4 7]. Funcion z k w = i e iθ k w α k 4 realizes univalen and conformal ransformaions of domain P k o he righ half-plane Rez > 0, for all k =, n. Then funcion ζ k w := z kw 5 + z k w maps univalen and conformal domain P k on he uni circle U = {z : z }, k =, n. The define ω k,p := ζ k a k,p, ω k,p := ζ k a k,p, a n+,p := a,p, ω 0,p := ω n,p, ζ 0 := ζ n k =, n, p =, m. Family of funcions {ζ k w} n, se by equaliy 5, i is possible for by piece-dividing ransformaion look [4 7] domains { B k,p : k =, n, p =, m } in relaion o he sysem of corners {P k } n. For any domain C he define := { w C : w }. Le Ω k,p he define conneced

5 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 87, componen ζ k Bk,p P k ζk Bk,p P k conaining a poin ω k,p, Ω k,p he define conneced componen, ζ k Bk,p P k ζk Bk,p P k conaining a poin ω k,p, k =, n, p =, m, P 0 := P n, Ω 0,p := Ω n,p. I is clear, ha, Ω s k,p generally speaking, domains are muliconneced domains, k =, n, p =, m, s =,. Pair of domains Ω k,p and Ω k,p grows ou of piece-dividing ransformaion domains B k,p concerning families {P k, P k }, {ζ k, ζ k } in poin a k,p, k =, n, p =, m. From a formula 5 we receive he following asympoic expressions ζ k w ζ k a k,p [ ak,p α k ζ k w ζ k a k,p α k a k,p ] w a k,p, w a k,p, w P k. [ ak,p α k α k a k,p ] w a k,p, w a k,p, w P k, k =, n, p =, m. 6 From he heorem.9 [6] look also [4, 5] and formulas 6 we receive inequaliies { [ r B k,p, a k,p r Ω k,p, ω k,p r Ω ak,p k,p, ] ω k,p α k χ α k a k,p [ ak,p ]} α k χ α k a k,p, k =, n, p =, m. 7 Using formulas 7, i is received he following raio: p= r α B k,p, a k,p p= p= r α B k,p, a k,p [ α k α k ak,p p= p= p= r B k,p, a k,p r B k,p, a k,p ak,p α k [ α k α k α k a k,p ] α ak,p α k

6 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 88 p= p= ak,p [ ak,p α k α k [ ak,p α k α k p= α k α k α k a k,p ] ak,p ak,p α k α k a k,p ] α a k,p ] [ ] r Ω k,p, ω k,p r Ω α k,p, ω k,p p= p= p= [ ] r Ω k,p, ω k,p r Ω k,p, ω k,p [ ] r Ω k,p, ω k,p r Ω α k,p, ω k,p [ ] r Ω k,p, ω k,p r Ω k,p, ω k,p. 8 Le s noe ha [ ] r Ω k,p, ω k,p r Ω α k,p, ω k,p p= = p= p= p= { m p= p= [ ] r Ω k,p, ω k,p r Ω k,p, ω k,p [ ] r Ω k,p, ω k,p r Ω α k,p, ω k,p [ ] r Ω k,p, ω k,p r Ω k,p, ω k,p = r α Ω k,p, ω k,p r α Ω k,p, ω k,p p= p= r Ω k,p, ω k,p r Ω k,p, ω k,p

7 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 89 p= p= p= p= [α k α k ] α r α Ω k,p, ω k,p r Ω k,p, ω k,p r α Ω k,p, ω k,p p= [ ak,p p= = M [ ak,p p= p= p= p= p= } r Ω k,p, ω k,p, 9 [α k α k ] [α k α k ] = α k [ ak,p [ ak,p A n, M α k α k α k ak,p ak,p ak,p ak,p p= [α k α k ] α α α+ k, 0 α k α k α k α k a k,p ] α a k,p ] a k,p ] α a k,p ] = α A 3 n, M A n, M A 4 n,. Then from 8 using formulas 9, 0,, i is received he following raio p= r α B k,p, a k,p p= α α+ k M r α B k,p, a k,p p= p= r B k,p, a k,p r B k,p, a k,p A n, M A 3 n, αm A n, M A 4 n,

8 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 90 { m p= p= r α Ω k,p, ω k,p r α Ω k,p, ω k,p p= p= Considering ha p= p= r Ω k,p, ω k,p r Ω k,p, ω k,p r α Ω k,p, ω k,p r Ω k,p, ω k,p r α Ω k,p, ω k,p from he previous raio we receive n p= α k p= p= } r Ω k,p, ω k,p. n, n r α B k,p, a k,p p= r α B k,p, a k,p p= p= r B k,p, a k,p r B k,p, a k,p nα+ M A n, M A 3 n, αm A n, M A 4 n, { m p= p= r α Ω k,p, ω k,p r α Ω k,p, ω k,p p= p= p= p= r Ω k,p, ω k,p r Ω k,p, ω k,p r α Ω k,p, ω k,p r Ω k,p, ω k,p r α Ω k,p, ω k,p p= p= } r Ω k,p, ω k,p. 3

9 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 9 from he heorem 4.. [7] inequaliies follow r α Ω p= p= k,p, ω k,p p= r p= Ω k,p, ω k,p r α Ω p= k,p, ω k,p r Ω k,p, ω k,p r α G p, g p r G p, g p, r α Ω k,p, ω k,p p= p= p= r Ω k,p, ω k,p p= r α Ω k,p, ω k,p r Ω k,p, ω k,p r G p, g p r α G p, g p, 4 p= where G p, G p, G p, G p he poles of he quadraic differenial g p Q ζ k dζk = ζ m 3 αζ4m i k k From 4 we receive, using inequaliies 3 r α B k,p, a k,p p= p= nα+ n M r α B k,p, a k,p A n, p= sysem circular domains, g p, g p, g + + αζ k + iα p, ζ 4m k + dζ k, k =, n p= p= M A 3 n, r B k,p, a k,p r B k,p, a k,p αm A n, M 5 A 4 n, r α G p, g p r G p, g p n, 6 where G p, G p sysem circular domains, g p, g p he poles of he quadraic differenial 5. From he las raio, he approval of he heorem follows, using he heorem 4.. [7]. The heorem is proved. Proof heorem. A once we will noe ha from he condiion of unapplying follows ha cap C\D > 0 and se D possesses Green s generalized funcion g Da z, a, z Da, g D z, a, where g D z, a = 0, z C\Da, Green s lim g Daζ, a, ζ Da, z Da ζ z

10 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 9 generalized funcion open se D concerning a poin a D, and g Da z, a Green s funcion domain Da concerning a poin a Da. Furher, we will use mehods of works [6, 7]. Ses we will consider E 0 = C\D; Ea k,p, = {w C : w a k,p }, k =, n, p =, m, n 3, n, m N, R +. The condenser we will ener ino consideraion for raher small > 0 C, D, A n, = {E 0, E, E }, where E = E = n p= n m n Ea k,p, p= m n Ea k,p, p= p= Ea k,p,, Ea k,p,. Capaciy of he condenser C, D, A n, is called as look [5] [G capc, D, A n, = inf x + G y ] dxdy, where an infimum underakes on all coninuous and o he lipschicevym in C funcions G = Gz, such ha G = 0, G = α, G = E0 E E Le is named he module of condenser C, reverse he capaciy of condenser From a heorem [6] ge C = [capc] C, D, A n, = π nm α + log +MD, An,+o, 0, 7 where MD, A n, = π [ n m α + α + p= m log rd, a k,p + +α p= p= m log rd, a k,p + p= log rd, a k,p + log rd, a k,p +

11 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 93 +α +α k,p q,s k,p q,s g D a k,p, a q,s + g D a k,p, a q,s + g D a k,p, a q,s + α k,p q,s g D a k,p, a q,s + +g D a k,p, a q,s + g D a k,p, a q,s + g D a k,p, a q,s + + g D a k,p, a q,s + g D a k,p, a q,s + k,p q,s + k,p q,s ] g D a k,p, a q,s. 8 Funcion 5 and definiion ω k,p, ω k,p, a n+,p, ω 0,p, ζ 0,,, using, by us he heorems enered a proof. Le, oo, Ω k,p define conneced componen ζ k D P k ζk D P k, conaining a poin ω k,p, Ω k,p, conneced componen ζ k D P k ζk D P k conaining a poin ω k,p, k =, n, p =, m, P 0 := P n, Ω 0,p := Ω n,p. I is clear, ha Ω s k,p generally speaking, domains are muliconneced domains, k =, n, p =, m, s =,. Pair of domains Ω k,p Ω k,p grows ou of piecedividing ransformaion open se D concerning families {P k, P k }, {ζ k, ζ k } in poin a k,p, k =, n, p =, m. Le s consider condensers C k, D, A n, = E k 0, Ek, Ek, where E k s = ζ k E s P k [ ζ k E s P k ], k =, n, s = 0,,, {P k } n he sysem of corners corresponding o sysem of poins A n, ; operaion [A] compares o any he se A C a se, symmeric a se A is relaive uni circle w =. From his i follows ha o he condenser C, D, A n,, a dividing ransformaion is relaive {P k } n and {ζ k } n, here corresponds a se of condensers he sysem of corners corresponding o sysem of poins A n, ; operaion [A] compares o any he se A C a se, symmeric a se A is relaive uni circle w =. From his i follows ha o he condenser C, D, A n,, a dividing ransformaion is relaive {P k } n and {ζ k} n, here corresponds a se of condensers

12 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 94 {C k, D, A n, } n, symmeric relaively {z : z = }. According o works [6, 7], we will receive capc, D, A n, From here follows C, D, A n, capc k, D, A n,. 9 C k, D, A n,. 0 The formula 7 gives a module asympoics C, D, A n, a 0, and M D, A n, is he given module of a se D relaively A n,. Using formulas 6 and ha fac ha a se D mees he condiion of unapplied in relaion o he sysem of poins A n,, for condensers we will receive similar asympoic represenaions C k, D, A n,, k =, n C k, D, A n, = where α πm α + log + M k D, A n, + o, 0, M k D, A n, = πm α + m r Ω k,p, ω k,p log [α k χ a k,p α k a k,p ] + p= r Ω k,p, ω k,p log [α k χ a k,p α k a k,p ] + +α + p= m r Ω k,, ω k, + log = [α k χ a k, α k a k, ] + r Ω k,, ω k, log [α k χ a k, α k a k, ], = M k D, A n, = = πm α + r Ω k,p α, ω k,p log [α k χ a k,p α k a k,p ] + p=

13 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 95 + m r Ω k,p, ω k,p +α log p= [α k χ a k+,p α k a k+,p ] + r Ω k,, ω k, + log = [α k χ a k+, α k a k+, ] + m r Ω k,, ω k, log [α k χ a k, α k a k, ], k =, n. = By means of, we receive = + C k, D, A n, = πm α + log πm α + log M k D, A n,m + o log = πm α + log M k D, A n, +o log, 0. 3 πm α + log Furher, from, follows ha C k, D, A n, = 4πnm α + log πm α + log M k D, A n, +o log, 0. 4 In urn, allows 4 o receive he following asympoic represenaion C k, D, A n, = πm α + n log log 4πnm α + M k D, A n, + o log =

14 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 96 = log 4πnm α + + 4n M k D, A n, + o, 0. 5 Inequaliies, 9 and 0 aking ino 7 and 5 allow o noice ha π nm α + log + MD, A n, + o log πnm α + + n From 6 a 0 we receive ha MD, A n, n M k D, A n, + o. 6 M k D, A n,. 7 Formulas 8, and 7 lead o he following expression +α π [ n m α + α + p= k,p q,s + α m log rd, a k,p + +α p= p= p= m log rd, a k,p + log rd, a k,p + log rd, a k,p + g D a k,p, a q,s + g D a k,p, a q,s + +α k,p q,s k,p q,s g D a k,p, a q,s + g D a k,p, a q,s + g D a k,p, a q,s + +g D a k,p, a q,s + g D a k,p, a q,s + + g D a k,p, a q,s + g D a k,p, a q,s + + k,p q,s k,p q,s ] g D a k,p, a q,s 4πn m α +

15 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 97 m α log p= = r Ω k,p, ω k,p [α k χ a k,p α k a k,p ] + r Ω k,p, ω k,p +α log p= [α k χ a k,p α k a k,p ] + m r Ω k,, ω k, + log = [α k χ a k, α k a k, ] + r Ω k,, ω k, + log = [α k χ a k, α k a k, ] + r Ω k,p, ω k,p +α log p= [α k χ a k,p α k a k,p ] + m r Ω k,p, ω k,p +α log p= [α k χ a k+,p α k a k+,p ] + r Ω k,, ω k, + log [α k χ a k+, α k a k+, ] + + = m r Ω k,, ω k, log [α k χ a k, α k a k, ]. Thus, we receive. Furher, he proof of he heorem comes o an end in he same way, as well as he proof of he heorem. The heorem is proved. Acknowledgemen. The auhor is graeful o Prof. Bakhin for suggesing problems and useful discussions. References [] M. A. Lavren ev, On he heory of conformal mappings, Tr. Fiz.-Ma. Ins. Akad. Nauk SSSR, 5, [] G. M. Goluzin, Geomeric Theory of Funcions of a Complex Variable [in Russian], Nauka, Moscow 966.

16 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 98 [3] G. P. Bakhina, Variaional Mehods and Quadraic Differenials in Problems for Disjoin Domains [in Russian], Auhor s Absrac of he Candidae-Degree Thesis Physics and Mahemaics, Kiev 975. [4] V. N. Dubinin, Separaing ransformaion of domains and problems of exremal division, Zap. Nauchn. Sem. Leningrad. Odel. Ma. Ins. Ros. Akad. Nauk, 68, P [5] V. N. Dubinin, Mehod of symmerizaion in he geomeric heory of funcions of a complex variable, Usp. Ma. Nauk, 49, No. 95, [6] V. N. Dubinin, Asympoic represenaion of he modulus of a degeneraing condenser and some is applicaions, Zap. Nauchn. Sem. Peerburg. Odel. Ma. Ins., 37, [7] Bakhin, A. K., Bakhina, G. P., Zelinskii, Yu. B., Topological-algebraic srucures and geomeric mehods in complex analysis, Proceedings of he Insiue of Mahemaics of NAS of Ukraine , 308 pp. Russian. [8] O. K. Bakhin, Inequaliies for he inner radii of nonoverlapping domains and open ses, Ukr. Ma. Zh., 6, No. 5, ; English ranslaion: Ukr. Mah. J., 6, No. 5, [9] Dubinin, V. N., Capaciies of condensers and symmerizaion in geomeric funcion heory of complex variables, Dalnayka, Vladivosok, 009 Russian. [0] A. K. Bakhin and A. L. Targonskii, Exremal problems and quadraic differenials, Nelin. Kolyvannya, 8, No. 3, ; English ranslaion: Nonlin. Oscillaions, 8, No. 3, [] A. L. Targonskii, Exremal problems of parially nonoverlapping domains on a Riemann sphere, Dopov. Nas. Akad. Nauk Ukr., No. 9, [] G. V. Kuz mina, Problem of exremal division of a Riemann sphere, Zap. Nauchn. Sem. Leningrad. Odel. Ma. Ins. Ros. Akad. Nauk, 76, [3] E. G. Emel yanov, On he problem of he maximum of he produc of powers of conformal radii of disjoin domains, Zap. Nauchn. Sem. Leningrad. Odel. Ma. Ins. Ros. Akad. Nauk, 86, [4] W. K. Hayman, Mulivalen Funcions, Cambridge Universiy, Cambridge 958.

17 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 99 [5] J. A. Jenkins, Univalen Funcions and Conformal Mapping, Springer, Berlin 958. Andrey Targonskii, Deparmen of Mahemaics, Zhiomir Sae Universiy, Veluka Berdichivska, 40, 0008 Zhiomir, Ukraine. argonsk@zu.edu.ua, argonsk@mail.ru

18 EXTREMAL PROBLEM ON n, m -SYSTEM POINTS ON THE RAYS 300