Generalization of some extremal problems on non-overlapping domains with free poles

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1 doi: 0478/v A N N A L E S U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N P O L O N I A VOL LXVII, NO, 03 SECTIO A IRYNA V DENEGA Generalization of some extremal problems on non-overlapping domains with free poles Abstract Some results related to extremal problems with free poles on radial systems are generalized They are obtained by applying the nown methods of geometric function theory of complex variable Sufficiently good numerical results for γ are obtained Introduction In geometric function theory of complex variable extremal problems on non-overlapping domains form the well-nown classic direction In the paper [] Lavrent ev posed and solved a problem of maximizing the product of conformal radii of two non-overlapping simply connected domains Topics connected with the study of problems on nonoverlapping domains was developed in papers [] [] This paper summarizes some results obtained in [5], [] Let N, R be the set of natural and real numbers, respectively, C be the complex plane, C = C { } be the one point compactification and R + = 0, Let rb, a be an inner radius of a domain B C with respect to a point a B see [6], [3], [3] and χt = t + t Let n N A set of points A n := { a C : =, n } is called n-radial system if a R +, =, n, and 0 = arg a < arg a < < arg a n < π 00 Mathematics Subject Classification 30C75 Key words and phrases Extremal problems on non-overlapping domains, inner radius, n-radial system of points, separating transformation

2 I V Denega Denote P A n := {w : arg a < arg w < arg a + }, θ := arg a, a n+ := a, θ n+ := π, α := π arg a +, α n+ := α, =, n a This wor is based on application of separating transformation developed in [4] [6] For specific use of this method we consider a special system of conformal mappings By ζ = π w = i e iθ w α, =, n we denote unique branch of multi-valued analytic function π w performing univalent and conformal mapping P A n onto the right half plane Re ζ > 0 For an arbitrary n-radial system of points A n = {a } and γ R + we assume that L γ A n := = [ a ] γα α χ a + L 0 A n := = a + 4 γα +α, = [ a ] α χ a a + The class of n-radial systems of points for which L γ A n = L 0 A n = automatically includes all systems with n different points located on the unit circle The main purpose of this wor is to obtain exact upper estimates for the functionals J n γ = r γ B 0, 0 r B, a, = I n γ = [r B 0, 0 r B, ] γ n = r B, a, where γ R +, A n = {a } n = is n-radial system of points, a 0 = 0, and {B } n =0 is a system of non-overlapping domains ie B p B j = if p j such that a B, = 0, n Main results Theorem Let n N, n and γ 0, ] Then for any n-radial system of points A n = {a } n = such that Lγ A n = and any system of non-overlapping domains B, a B C, =, n, a 0 = 0 B 0 we have the inequality γ J n γ 4n+ n γ γ n n n n γ γ n γ n+ γ n n + γ

3 Generalization of some extremal problems 3 Equality in is achieved when a and B, = 0, n are, respectively, poles and circular domains of the quadratic differential Qwdw = n γw n + γ w w n dw Theorem Let n N, n and γ = Then for any n-radial system of points A n = {a } n = such that L0 A n = and any system of nonoverlapping domains B, B 0, B, a B C, =, n, a 0 = 0 B 0 C, a = B C we have the inequality 3 [r B 0, 0 r B, ] n+ n n r B, a n n + n n + = Equality in 3 is achieved, when a and B are, respectively, poles and circular domains of the quadratic differential Qwdw = wn + w n n + w w n dw Theorem 3 Let γ 0; γ 0], γ0 =, ε = 05 Then for any -radial system of points A = {a } = such that Lγ A =, ε < a < + ε, =, and any system of non-overlapping domains {B } =0, a B, = 0,, a 0 = 0 B 0, we have the inequality r γ B 0, 0 rb, a r γ D 0, 0 r D, d, = where D, d, = 0,, d 0 = 0, are circular domains and poles of the quadratic differential = Qwdw = 4 γw + γ w w dw Proof of Theorem We use the method due to Bahtin [] [3] and properties of separating transformation see [4] [7], [3], [8] We mae separating transformation of domains {B } n = Suppose P := P A n := {w C\{0} : θ < arg w < θ + } Consider the introduced system of functions ζ = π w = i e iθ w α, =, n Let Ω, =, n be a domain of the plane C ζ obtained by combining the connected component π B P containing a point π a, with its symmetrical reflection with respect to the imaginary axis By Ω, =, n, we denote the domain of the plane C ζ, obtained by combining the connected component π B + P, containing the point

4 4 I V Denega π a +, with its symmetrical reflection with respect to the imaginary axis, B n+ := B, π n a n+ := π n a Besides, by Ω 0 we denote the domain of C ζ, obtained by combining the connected component π B 0 P, containing the point ζ = 0, with its symmetrical reflection with respect to the imaginary axis Denote π a : = ω, π a + := ω, =, n, π na n+ := ω n The definition of π implies that π w ω α a α w a, w a, w P, π w ω α a + α w a+, w a +, w P, π w w α, w 0, w P Then, using results of papers [4] [7], [3], we obtain the inequalities r 4 r B, a Ω, ω α a α Ω α a, ω α =, n, Ω 0 := Ω n, ω 0 := ω n, [ n ] 5 r B 0, 0 r α Ω 0, 0 = Repeating arguments given in [3] in the proof of Theorem 5 and taing into account introduced sets of domains {P } n =, functions {π } n = and numbers {θ } n =, we obtain the following inequality for the investigated functional : [ ] α J n γ r Ω 0, 0 γ r Ω, ω r Ω, ω α = = α α a a α 6 a = α = = a a + α [ n r γα Ω 0 n, 0 ] r Ω, ω r Ω, ω = = Expression in parentheses of the last formula in 6 is a product of the functional r β Ω 0 r, 0 Ω, ω r Ω, ω on triples of domains Ω 0, of the plane C ζ Ω, Ω,

5 Generalization of some extremal problems 5 It is nown [5] that the functional Y 3 σ, σ, σ 3 = r σ D, d σ D, d σ 3 D 3, d 3 d d σ +σ σ 3 d d 3 σ σ +σ 3 d d 3 σ +σ +σ 3, σ R +, d D C, D D p =, =,, 3, p =,, 3, p, is invariant under all conformal automorphisms of the complex plane C With this relation in mind, the following estimate holds: n a J n γ α = = [ n = = r γα a a + α Ω 0, 0 Ω, ω ω ω ω γα ω ω ω γα ω ω γα Ω γα ], ω Note that ω = a α, ω = a + α, ω ω = a α + a + α Taing into account these equalities, we obtain n a J n γ α a a + α = n = = r γα = n = n α = γ = n ω ω = Ω 0, 0 = ω ω ω ω Ω, ω ω ω ω γα ω a α χ a = [ n α χ r γα = a + a a + Ω 0, 0 γα ] α Ω, ω ω ω ω γα ω γα Ω γα n = Ω γα, ω a + a, ω γα

6 6 I V Denega = n γ = n γ α = n α = = a + 4 γα +α = = α = = r γα Ω 0, 0 [ a χ a + Ω, ω ω ω ω γα ω n α L γ A n r γα = Ω 0, 0 Ω, ω ω ω ω γα ω γα ] α Ω γα Ω γα, ω, ω For each =, n we can easily define conformal automorphism ζ = T z of complex numbers of the plane C such that T 0 = 0, T ω s = s i, G q := T Ω q, =, n, s =,, q = 0,, Then = r γα = = Ω 0, 0 Ω, ω ω ω ω γα ω r α γ G 0, 0 G, i γα Then using results of [3], [5], we obtain J n γ n γ = = n γ α = n α L γ A n = r α γ G 0, 0 = Ω γα G, ω, i G, i G, i γα α n α L γ A n n+ γ = α =

7 Generalization of some extremal problems 7 [ n = n [ n α L γ A n = ] r α γ G 0, 0 G, i G, i = Following the paper [5], we have ] r α γ G 0, 0 G, i G, i [ n / 7 J n γ γ n Ψ β ], where Ψβ = β +6 β β + β β + β +β, β [0, ] Similarly to [5], we consider the next extremal problem: Ψβ sup; = = β = γ, β = α γ, 0 < β = Necessary conditions have the form Ψ β Ψβ = λ, =, n Ψβ = We will show that all β are equal We investigate behavior of the function F β = Ψ β Ψβ = β lnβ+ β + β ln β +β ln+β on interval β [0, ] It is strictly decreasing on the interval 0; β 0 ], β 0 3; 33 and increasing on [β 0 ; Then we use the method of the proof of Theorem 4 [5] and obtain that unique solution of the extremal problem is the point n,, n Estimates 4, 5, 7 yield inequality of Theorem The case of equality is verified directly and Theorem is proved Dubinin proved Theorem if γ = and for any distinct points a that lie on the unit circle and any non-overlapping domains B see [5], [8] Proof of Theorem We retain all notation for separating transformation of domains introduced in the proof of Theorem for domains B, = 0, n By Ω we denote the domain of plane C ζ, obtained by combining the connected component π B E containing the point ζ = with its{ symmetrical } reflection with respect to the imaginary axis The n family is a result of separating transformation of an arbitrary Ω = domain B, B C with respect to the families {P } n = and {π } n = at the point ζ =

8 8 I V Denega By Theorem in [5] we have [ n 8 rb, = r α Ω ], Using 4, 5, 8, we obtain n [r B 0, 0 r B, ] r B, a n α L 0 A n 9 = = r α γ Ω 0, 0 Theorem 6 in [5] gives r α γ Ω, = Ω, ω ω Ω ω [r B 0, 0 r B, ] α γ B, a r B, a a a Ψβ = β β β β + β +β, 0 < β, ω Inequality 9 was obtained by Dubinin using the results of Kolbina [5] Similarly to [5], we consider the extremal problem: Ψβ sup; β = = = We introduce a function F β = Ψ β Ψβ Calculations show that this function is decreasing on the interval 0; β 0 ] and increases on [β 0 ;, 085 < β 0 < Further, as in the proof of Theorem 6 in [5], we verify that the unique solution of the extremal problem is the point n,, n Estimates 4, 5, 8, 9 yield the inequality 3 of Theorem The case of equality is verified directly Theorem is proved Proof of Theorem 3 The proof is based on application of separating transformation, developed in details in [6] According to the conditions of Theorem 3, a 0 = 0, ε < a < + ε, =, Assume 0 = arg a < arg a < π Let α := π arg a arg a, α := π π arg a, P := {w : arg a < arg w < arg a + }, =,, arg a 3 = π, P 0 := P, P 3 := P The family of two symmetrical domains {D ; D } with respect to the imaginary axis is called a result of separating transformation of the domain B

9 Generalization of some extremal problems 9 Further, as in Theorem 5 in [3], using the separating transformation we obtain [ ] 0 J γ L γ A α rγ α D0, 0 D, D, = We will prove that for α 0 γ, α 0 = max{α, α } extremal configurations different than those referred in Theorem 3 do not exist For this we find a value of the functional see Theorem 53 in [3] J 0 γ = rγ D 0, 0 rd, d = 4 = γ γ γ 4 + γ γ + γ γ if γ = We have that J Denote rb 0, 0 = r 0, rb, a = r, rb, a = r The Lavrent ev s theorem [7] gives r 0 r < a, r 0 r < a, r0 r r < a a = r r < a a r0 Then r γ B 0, a 0 J 0 γ r 0 rb, a = r γ 0 = a a J 0 γ γ If r 0 = rb, a r γ 0 a a r 0 a a J 0 γ γ configurations do not exist Consider the case r 0 J γ r γ 0 a a = r γ 0 a a J 0γ, then the extremal a a J 0 γ γ a a + 4 a a sin α 0 π γ γ a a + 4 a a sin α 0 π J 0 Substituting ε = 05, γ =, n =, a = ε, a = + ε, J 0 = 0835 in the last inequality, we obtain + ε, 4ε ε sin π J 0 + Performing calculations of right and left sides of the inequality, we have < From this it follows that if ε = 05, then inequality is true Hence J = J 0736 J = <, ie if α 0 > γ, J γ < J 0γ then the maximum value of the functional J γ for such domains is not attained Then α 0 γ and we can apply inequality 0 Using a result obtained in the proof of Theorem 4 in [5], we can write the following inequality [ ] J γ γ σ +6 σ σ + σ σ + σ +σ, =

10 0 I V Denega where σ = γ α Consider the function Ψσ = σ +6 σ σ + σ σ + σ +σ, σ [0, ] and we will conduct detailed investigation of its graph on this interval see Fig Ψσ is logarithmically convex on interval [0; x 0 ], x 0 3 On [0; x ], x 05 the function increases from Ψ0 = 0 to Ψx 095, and it decreases on interval [x ; x ], x 6049 to Ψx 086, and on [x ; ] it increases to Ψ = x 3 9, Ψx 3 = Ψx 095 Figure Using equality σ + σ = γ, we will prove that Ψσ Ψσ Ψx For σ [0; x 0 ] we mae appropriate conclusion from the logarithmic convexity of the function Ψσ For σ [x 0 ; x 3 ] from properties of the graph of the function Ψσ, we have Ψσ Ψx and Ψσ Ψx and thus Ψσ Ψσ Ψx If σ [x 3 ; ] then Ψσ < Ψ =, Ψσ < Ψ0, 04, and hence Ψσ Ψσ < 0, 4 < Ψx So, J γ J 0 γ Inequality is true and Theorem 3 is proved Acnowledgement The author is grateful to Prof Bahtin for suggesting problems and useful discussions References [] Bieberbach, L, Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitsreises vermitteln, Sitzungsber Preuss Aad Wiss Phys- Math Kl 38 96,

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12 I V Denega Iryna V Denega Department of Complex Analysis and Potential Theory Institute of Mathematics of National Academy of Sciences of Uraine Tereshchenivsa St Kyiv Uraine iradenega@yandexru Received September 0, 0