Two-sided bounds for the logarithmic capacity of multiple intervals
aa r X i v : . [ m a t h . C V ] M a y Two-sided bounds for the logarithmic capacity of multiple intervals
V.N. Dubinin ∗ and D. Karp † Abstract.
Potential theory on the complement of a subset of the real axisattracts a lot of attention both in function theory and applied sciences. Thepaper discusses one aspect of the theory - the logarithmic capacity of closedsubsets of the real line. We give simple but precise upper and lower bounds forthe logarithmic capacity of multiple intervals and a lower bound valid also forclosed sets comprising an infinite number of intervals. Using some known meth-ods to compute the exact values of capacity we demonstrate graphically howour estimates compare with them. The main machinery behind our results areseparating transformation and dissymmetrization developed by V.N. Dubininand a version of the latter by K. Haliste as well as some classical symmetriza-tion and projection result for logarithmic capacity. The results of the paperimprove some previous achievements by A.Yu. Solynin and K. Shiefermayr.
Keywords:
Logarithmic capacity, multiple intervals, symmetrization, separating transformation
MSC2000: 31A15, 30C85
1. Introduction
Let E denote a compact subset of the complex plane C and write g B ( z, ∞ )for the Green function of the connected component B of C \ E containing the point at infinity. Logarithmic capacity of E is defined bycap E = exp { lim z →∞ [log | z | − g B ( z, ∞ )] } . If B does not admit the Green function we set cap E = 0. Logarithmic capacity cap E is equalto the Chebyshev constant of E and its transfinite diameter [13, 15]. Since logarithmic capacityis not an easy quantity to compute, its lower and upper estimates are of considerable interest(see, for instance [15]). In this paper we will be concerned with estimating logarithmic capacityof closed subsets of the real line in particular those comprising a finite number of intervals. Thesetype of subsets are obtained, for instance, by Steiner or circular symmetrization of most one-dimensional sets and hence are extremal for many problems of function theory. Potential theoryon the complement of such set attracts significant attention [5, 18, 20]. Since cap aE = | a | cap E for any complex a we may restrict our attention to the subsets of the interval [ − , E ) / ≤ cap E ≤ / , where mes E denote the linear Lebesgue measure of E . These inequalities albeit simple are toorough especially for the sets consisting of many intervals. In this connection the question ariseshow to obtain more precise estimates taking account of the structure of E and its dispersion in[ − ,
1] in terms of elementary functions. In the recent work [17] Schiefermayr established the upper ∗ Institute of Applied Mathematics, Vladivostok, Russia, e-mail: [email protected] † Institute of Applied Mathematics, Vladivostok, Russia, e-mail: [email protected] E α,β = [ − , α ] ∪ [ β, − < α < β <
1, some of them interms of elementary functions. He also gives a survey of some known and new lower bounds. Letus mention some of these bounds together with our comments and amendments. According to [17,Theorem 3] the inequalitycap E α,β ≥ p (1 − α )(1 − β ) p (1 − α )(1 + β ) + p (1 + α )(1 − β ) (1)holds true with equality attained when α + β = 0. It is indicated in [17] that Solynin [19, Section 2.2]proved the lower bound for the logarithmic capacity of multiple intervals which in the case of twointervals takes the formcap E α,β ≥
12 max "(cid:18) sin (cid:18) πθ ( β )2 θ ( δ ) (cid:19)(cid:19) θ ( δ ) /π (cid:18) sin (cid:18) π ( π − θ ( α ))2( π − θ ( δ )) (cid:19)(cid:19) π − θ ( δ )) /π , (2)where here and henceforth θ ( γ ) = arccos( γ ) and the maximum is taken over all δ ∈ [ α, β ]. Inview of the well-known Robinson formula (Lemma 1 below), we notice that Solynin’s inequality isa particular case of the earlier result of the first author (see, for instance, [7, Corollary 1.3] andrelated comments). Directly from polarization [7, Corollary 1.2] we get the following simple upperbound: cap E α,β ≤ cap E − γ,γ = 12 p − γ = 14 p − ( α − β ) , (3)where γ = ( β − α ) /
2. Another upper bound follows from an inequality due to Gillis [12]:cap E α,β ≤ (cid:20) log((1 + α ) /
8) log((1 − β ) / α )(1 − β ) / (cid:21) . The main result of [17] is the estimatecap E α,β ≤ α β ) exp (cid:18) EK − (1 + α )(1 − β )(1 − α )(1 + β ) (cid:19) " log √ √ − α √ α , (4)where K = K ( k ), E = E ( k ) are Legendre’s complete elliptic integrals of the first and second kinds,respectively, k = 2( β − α )(1 − α )(1 + β ) , and it is assumed that α + β ≥
0, otherwise one has to replace E α,β with E − β, − α having thesame logarithmic capacity. In order to reduce this bound to elementary functions one may applytwo-sided estimates for elliptic integrals and their ratios from [3, 4].In this note we give rather general upper and lower estimates for the logarithmic capacity ofa subset E of [ − ,
1] (Theorems 1-3 below). In particular, if E consists of n intervals Theorem 2gives the lower bound which coincides with (2) for n = 2 but is stronger than the correspondingresult from [19] for n >
2. This bounds is also stronger than (1) for n = 2. Our upper bound fromTheorem 3 is both very simple and more precise than (4) except for very narrow neighbourhood of β = 1 where (4) becomes asymptotically precise. Our bound remains very good for n >
2, whereit seems to be the only known non-trivial upper bound.The main results of the paper proved in section 3 are based on the Robinson formula and theestimates for the capacity of subsets of the unit circle given in section 2. In the final section 4 wecompare our estimates with exact values computed using the formulas due to Akhieser [1, 2] (for n = 2) and Widom [21] (in a modified form for n > . Auxiliary results. Let Γ = { z : | z | = 1 } . The following statement can be derived from theproperties of conformal mappings and symmetry considerations. Lemma 1 (Robinson [16]) . Suppose that F is a closed subset of the unit circle Γ symmetricwith respect to the real axis and let E be its orthogonal projection onto the real axis. Then cap E = 12 (cap F ) . A particular case of the principle of circular symmetrization (see [7]) is the following
Lemma 2 (Beurling [6, p.35-36]) . The logarithmic capacity of a closed subset of Γ having thelength l attains its minimal value sin( l/ for a subarc of Γ . Define the infinite sectors D k = { z : α k < arg z < α k +1 } , k = 1 , , . . . , n , α < α < · · · < α n <α n +1 = α + 2 π and let ζ = p k ( z ) = − i ( e − iα k z ) π/ ( α k +1 − α k ) , k = 1 , , . . . , n . The function p k ( z )effects univalent conformal mapping of D k onto the right half-plane ℜ ζ >
0. For a compact set F satisfying F ∩ D k = ∅ , k = 1 , , . . . , n , denote by F k the union of p k ( F ∩ D k ) with its reflectionwith respect to imaginary axis. According to the terminology of [7] the family of sets { F k } nk =1 isthe result of separating transformation of F with respect to the family of functions { p k } nk =1 . Aparticular case of [7, Corollary 1.3] is Lemma 3
The following inequality holds: cap F ≥ n Y k =1 (cap F k ) ( α k +1 − α k ) / (2 π ) . Introduce the notation F ( l, n ) = { z ∈ Γ : | arg z n | ≤ l/ } , 0 < l < π . The next statement wasproved using dissymmetrization (see [7]): Lemma 4 (Haliste [14])
Suppose that F is a union of n closed arcs on the unit circle Γ havingtotal length l . Then cap F ≤ cap F ( l, n ) = [sin( l/ /n . The last two lemmas also allow for the complete description of equality cases.
3. Main results
Given a closed subset e of the interval [ − ,
1] putm e = Z e dx √ − x . The meaning of this formula in the present context is that 2m e gives the length of the symmetricpre-image of e on Γ under orthogonal projection. Denote by E ( l, n ) the orthogonal projection of F ( l, n ) onto the real axis. Theorem 1
Let { e k } nk =1 be a partitioning of the interval [ − , by closed intervals having nocommon inner points, n ∪ k =1 e k = [ − , . Then for any closed set E ⊂ [ − , the inequality cap E ≥ n Y k =1 (cid:20) sin π m ( e k ∩ E )2m e k (cid:21) e k ) /π (5) holds true. Equality is attained for the sets E = E ( l, n ) , < l < π , and the partitioning { e k } nk =1 by the points cos( πk/n ) , k = 0 , , . . . , n . roof. First let us consider a partitioning the of unit circle Γ by closed arcs σ k having no commoninner points, m ∪ k =1 σ k = Γ. Denote by D k the open infinite sector formed by two rays passing throughthe endpoints of σ k with the vertex at the origin and write β k π for the angular span of D k (i.e.Lebesgue measure of σ k ). The function p k ( z ) = α k z /β k , | α k | = 1, effects univalent conformalmapping of D k onto the right half plane. According to Lemma 3 if a closed subset F of the unitcircle Γ intersects all sectors D k , thencap F ≥ m Y k =1 (cap F k ) β k / , where { F k } mk =1 is the result of separating transformation of F with respect to the family { p k } mk =1 .According to Lemma 2cap F k ≥ sin[(mes F k ) /
4] = sin[mes ( σ k ∩ F ) / (2 β k )] , k = 1 , . . . , m. Hence, cap F ≥ m Y k =1 (cid:20) sin mes ( σ k ∩ F )2 β k (cid:21) β k / . (6)This estimate reduces to trivially valid inequality cap F ≥ F ∩ D k = ∅ for some k , 1 ≤ k ≤ m .Now let E and { e k } nk =1 be the set and the partitioning from the hypothesis of the theorem. Let F be the symmetric subset of the unit circle Γ such that it’s orthogonal projection onto the realaxis is E . Similarly, the let { σ k } nk =1 be the pre-image of n ∪ k =1 e k on Γ (the points +1 and − σ k ). Now as we mentioned earlier if e l is the projectionof σ k then the linear measure of σ k is m e l and mes ( σ k ∩ F ) = m ( e l ∩ E ). Inequality (6) appliedto the set F and the partitioning { σ k } nk =1 takes the formcap F ≥ n Y k =1 (cid:20) sin π mes ( σ k ∩ F )2mes σ k (cid:21) (mes σ k ) / (2 π ) = n Y k =1 (cid:20) sin π m ( e k ∩ E )2m e k (cid:21) (m e k ) /π . It is left to apply Lemma 1 which says that cap E = (cap F ) /
2. The case of equality can be verifieddirectly. (cid:3)
Let us notice that the survey paper [15, page 257] cites a particular case of inequality (6) with β k = 2 /m with incorrectly attributed authorship of this result (see [7, Theorem 2.9]).If the set E comprises n intervals and the partitioning { e k } sk =1 is tailored so that e k ∩ E consistsof one interval having one common endpoint with e k then inequality (5) reproduces inequality (2.7)from [19]. The latter is essentially obtained using the approach similar to that given in [7, §
4] butin terms of reduced moduli of triangles. The following theorem gives a strengthening of this resultwhen n > Theorem 2
Suppose E = n ∪ k =1 [ a k , b k ] , − a < b < a < b < · · · < a n < b n = 1 , n ≥ .Then cap E ≥
12 max n Y k =1 (cid:26) (cid:20) cos π ( θ ( b k ) − θ ( δ k )) θ ( δ k − ) − θ ( δ k ) − cos π ( θ ( a k ) − θ ( δ k )) θ ( δ k − ) − θ ( δ k ) (cid:21)(cid:27) ( θ ( δ k ) − θ ( δ k − )) /π , (7) where δ = − , δ n = 1 and the maximum is taken over all δ k satisfying b k < δ k < a k +1 , k =1 , . . . , n − . The equality is attained for E = E ( l, n − , < l < π , where E ( l, s ) is definedbefore Theorem 1, and the values δ k = cos( π ( n − k ) /n ) , k = 0 , , . . . , n . roof. Let F be the symmetric (with respect to the real axis) subset of the unit circle Γ suchthat its orthogonal projection to the real axis is E . Let us introduce the notation θ k = θ ( δ n − k +1 ), k = 1 , . . . , n + 1 and θ k = 2 π − θ n − k +2 , k = n + 2 , . . . , n + 1, where the numbers δ k satisfy thehypotheses of the Theorem. Applying Lemmas 1 and 3 with the values of α k = θ k and n replacedby 2 n , we will have cap E = 12 (cap F ) ≥ n Y k =1 (cap F k ) ( θ k +1 − θ k ) /π . (8)Each set F k , k = 1 , , . . . , n comprises one or two subarcs of Γ symmetric with respect to theimaginary axis. The orthogonal projection of F k onto the imaginary axis coincides with that of F n − k +1 and the length of this projection is equal to l k = cos π ( θ ( b n − k +1 ) − θ ( δ n − k +1 )) θ ( δ n − k ) − θ ( δ n − k +1 ) − cos π ( θ ( a n − k +1 ) − θ ( δ n − k +1 )) θ ( δ n − k ) − θ ( δ n − k +1 ) ,k = 1 , , . . . , n . In view of Lemma 1 we obtain:cap F k = cap F n − k +1 = p l k / , k = 1 , . . . , n. Substituting these values of capacities into (8) and changing n − k + 1 k we arrive at (7). Theequality case is straightforward to verify. (cid:3) As we mentioned earlier for n = 2 inequalities (2) and (7) coincide while for n ≥ { e k } nk =1 . See more details insection 4 below. Theorem 3
Suppose E = n ∪ k =1 [ a k , b k ] , − a < b < a < b < · · · < a n < b n = 1 , n ≥ .Then cap E ≤ ( cos " n − X k =1 ( θ ( a k +1 ) − θ ( b k )) / ( n − (9) The equality is attained for E = E ( l, n − , < l < π . Proof.
Let the set F ⊂ Γ be the same as in the proofs of Theorems 1 and 2. This set comprises2( n −
1) subarcs of Γ with total length l = 2 θ ( a n ) + n − X k =2 θ ( a k ) − θ ( b k )) + 2( π − θ ( b )) = 2 π + 2 n − X k =1 ( θ ( a k +1 ) − θ ( b k )) . An application of Lemmas 1 and 4 yields:cap E = 12 (cap F ) ≤
12 (cap F ( l, n − = 12 (sin( l/ / ( n − . The equality case is clear. (cid:3)
4. Numeric comparison of estimates.
In order to compare the presicion of various capacityestimates we need a method to compute the exact values of the logarithmic capacity of severalintervals. In case n = 2 this is provided by the well known formulas due to Akhieser [1, 2]:cap ( E α,β ) = 12 (cid:20) ϑ (0; q ) ϑ (0; q ) ϑ ( ω ; q ) ϑ ( ω ; q ) (cid:21) , (10)5here Jacobi’s theta functions are ϑ ( z ; q ) = 1 + 2 ∞ X n =1 q n cos(2 nz ) , (11) ϑ ( z ; q ) = 1 + 2 ∞ X n =1 ( − n q n cos(2 nz ) , (12)and parameters are found from: k = 2( β − α )(1 − α )(1 + β ) , k ′ = 1 − k , (13) q = exp (cid:18) − π K ( k ′ ) K ( k ) (cid:19) , ω = πF ( p (1 − α ) / , k )2 K ( k ) , (14)where F ( λ, k ) is the first incomplete elliptic integral of Legendre.Formula (10) was generalized to three intervals by Falliero and Sebbar [10, 11] in terms of genus2 theta functions. For arbitrary n a mehod to compute the capacity via Schwarz-Christoffel mapwas first given by Widom in [21]. We will use a slightly different guise of his formula (see furtherdevelopment of Widom’s idea in [9]). Indeed, one can verify directly that the Green function of C \ E with pole at infinity is (see [5, 9, 18, 21]): g ( z ) = ℜ F ( z ) , F ( z ) = z Z a p ( t ) dt p q ( t ) , (15)where q ( t ) = n Y i =1 ( t − a i )( t − b i )and the branch of square root p q ( t ) is chosen so that it is asymptotically t n near infinity. Thepolynomial p ( t ) = t n − + c n − t n − + · · · + c is chosen so that the Schwarz-Christoffel map F ( z ) maps ( a i , b i ) into imaginary axis. Since q ( t ) > R \ E we have the linear system of equations F ( b i ) = F ( a i +1 ), i = 1 , . . . , n −
1, or a i +1 Z b i p ( t ) dt p q ( t ) = 0 , i = 1 , . . . , n − , for the definition of the coefficients c k , k = 0 , , . . . , n −
2. Further, the Green function has theexpansion g ( z ) = ln | z | + R + o (1) , | z | → ∞ , where R is the Robin constant of E and cap E = e − R . Since the asymptotic expansion of the Greenfunction is true no matter from which direction we approach infinity, we can move along the realaxis: R = lim x →∞ x Z b n p ( t ) dt p q ( t ) − ln( x ) = ∞ Z b n " p ( t ) p q ( t ) − t dt = ∞ Z b n tp ( t ) − p q ( t ) t p q ( t ) dt, ℜ b n Z a p ( t ) dt p q ( t ) = 0 . So, finally cap E = exp ∞ Z b n " t − p ( t ) p q ( t ) dt = exp ∞ Z b n p q ( t ) − tp ( t ) t p q ( t ) dt . (16)We will make some numerical comparisons between various estimates and precise capacity valuescomputed using formula (10) for n = 2 and formula (16) for n = 3. For convenience let us recordthe lower bound from [19] and our bounds from Theorems 2 and 3 for these values of n . For n = 2, we have as before E α,β = [ − , α ] ∪ [ β, − < α < β <
1, and the lower bounds due toSchiefermayr [17] and Solynin [19] are given by formulas (1) and (2), respectively. One can verify bystraightforward computation that for n = 2 our lower bound (7) reduces to (2). We demonstratethese bounds in two typical situations - that of a moving gap and that of a spreading gap onFigure 1. Clearly in both situation Solynin’s bound (2) or equivalently our bound (7) provide abetter estimate. Note also that for the moving gap situation the capacity is monotone and maximalwhen α = − β . A proof of this and similar facts and their generalizations can be found in [8].The upper bound of Schiefermayr [17] and our simple upper bound obtained by polarizationare given by (4) and (3). The upper bound (9) for n = 2 reads (recall that θ ( α ) ≡ arccos( α )):cap E ≤
12 cos ([ θ ( β ) − θ ( α )] / . (17)Figure 2 illustrates these two bounds and the value of capacity computed by (10). Again wehave chosen two typical situation - the moving gap and the spreading gap. Since Schiefermayr’sinequality (4) is asympotically precise when one of the intervals vanishes his bound becomes moreprecise in a narrow neighbourhoods of the right end of the α and β ranges. Our bound providesalmost uniform and tight fit in the whole parameter ranges. Α Cap E Α , Β Capacity of two intervals - moving gap Β=Α+
Inequality H L = Inequality H L Inequality H L Cap E Α , Β Β Cap E Α , Β Capacity of two intervals - spreading gap Α=-
Inequality H L = Inequality H L Inequality H L Cap E Α , Β Figure 1: The capacity of two intervals and its lower estimates in two situationsFor n > .3 0.4 0.5 0.6 0.7 0.8 Α Cap E Α , Β Capacity of two intervals - moving gap Β=Α+
Inequality H L Inequality H L Cap E Α , Β Β Cap E Α , Β Capacity of two intervals - moving gap Β=Α+
Inequality H L Inequality H L Cap E Α , Β Figure 2: The capacity of two intervals and its upper estimates in two situations - - - - Capacity of three intervals - one spreading gap Inequality H L Inequality H L Inequality H L Cap E = @ - b D U @ D U @ D - - - - Capacity of three intervals - two moving gaps Inequality H L Inequality H L Inequality H L Cap E = @ - b D U @ b + b + D U @ b + D Figure 3: The capacity of three intervals and its lower and upper estimates in two situations n = 3 [19, formula (2.7)] can be written ascap E ≥
12 max (cid:18) sin π ( π − θ ( b ))2( π − θ ( δ )) (cid:19) π − θ ( δ )) /π (cid:18) sin π ( θ ( a ) − θ ( γ ))2( θ ( δ ) − θ ( γ )) (cid:19) θ ( δ ) − θ ( γ )) /π × (cid:18) sin π ( θ ( γ ) − θ ( b ))2( θ ( γ ) − θ ( δ )) (cid:19) θ ( γ ) − θ ( δ )) /π (cid:18) sin πθ ( a )2 θ ( δ ) (cid:19) θ ( δ ) /π , (18)where the maximum is taken over δ ∈ ( b , a ), γ ∈ ( a , b ) and δ ∈ ( b , a ), while the bound (7)takes the formcap E ≥
12 max (cid:18) (cid:20) π ( θ ( b ) − θ ( δ )) π − θ ( δ ) (cid:21)(cid:19) ( π − θ ( δ )) /π × (cid:18) (cid:20) cos π ( θ ( b ) − θ ( δ )) θ ( δ ) − θ ( δ ) − cos π ( θ ( a ) − θ ( δ )) θ ( δ ) − θ ( δ ) (cid:21)(cid:19) ( θ ( δ ) − θ ( δ )) /π (cid:18) (cid:20) − cos πθ ( a ) θ ( δ ) (cid:21)(cid:19) θ ( δ ) /π , (19)where the maximum is taken over δ ∈ ( b , a ) and δ ∈ ( b , a ). It can be seen from the proofof [19, formula (2.7)] that for any choice of γ the latter estimate is greater (i.e. better) than theformer if δ i , i = 1 ,
2, take the same values in both formulas. Similar statement holds for n > n = 3 readscap E ≤ { cos ([ θ ( a ) − θ ( b ) + θ ( a ) − θ ( b )] / } / . (20)8e compare all three bounds with each other and the value of capacity computed by (16) onFigure 3. Again we have chosen a spreading gap as one typical situation while as the other typicalsituation we have taken two simultaneously moving gaps. The figure confirms our prediction thatthat bound (19) is more precise than (18) in both situation.
5. Acknowledgements.
This work supported by Far Eastern Branch of the Russian Academyof Sciences (grants 09-III-A-01-008 and 09-II-CO-01-003), Russian Basic Research Fund (grant08-01-00028-a) and the Presidential Grant for Leading Scientific Schools (grant 2810.2008.1).
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