Ahlfors-Beurling conformal invariant and relative capacity of compact sets

For a given domain $D$ in the extended complex plane $\bar{\mathbb C}$ with an accessible boundary point $z_0 \in \partial D$ and for a subset $E \subset {D},$ relatively closed w.r.t. $D,$ we define the relative capacity $\rc E$ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $r(D\setminus E,z)/r(D, z)$ when $z$ approaches the point $z_0.$ Here $r(G,z)$ denotes the inner radius at $z$ of the connected component of the set $G$ containing the point $z.$ The asymptotic behavior of this quotient is established. Further, it is shown that in the case when the domain $D$ is the upper half plane and $z_0=\infty$ the capacity $\rc E$ coincides with the well-known half-plane capacity ${\hc} E.$ Some properties of the relative capacity are proven, including the behavior of this capacity under various forms of symmetrization and under some other geometric transformations. Some applications to bounded holomorphic functions of the unit disk are given.


Introduction
Let D 1 and D 2 be domains having Green functions in the extended complex plane C, and let the point z ∈ D 1 ⊂ D 2 . We denote by r(D k , z) the inner radius of the domain D k , k = 1, 2, at the point z (see e.g. [H], [D1]). The quotient r(D 1 , z)/r(D 2 , z) is conformal invariant in the sense that for every conformal map f of D 2 we have the equality r(D 1 , z) r(D 2 , z) = r(f (D 1 ), f (z)) r(f (D 2 ), f (z)) .
The study of this kind of invariant expressions goes back to the works of Ahlfors and Beurling [A, p.436]. Some significant applications to geometric theory of functions are given in ( [A, AB, O, Pom]). In this paper we study the behavior of the invariant r(D 1 , z)/r(D 2 , z) when the point z tends to a given common boundary point of the domains D 1 and D 2 . More precisely, we investigate the following situation. Let D be a domain in C, and let z 0 be an accessible boundary point of the domain D . Suppose that in a neighborhood of the point z 0 the boundary ∂D is represented by an analytic arc γ (in the case z 0 = ∞ it is required that the image of the arc γ under the mapping z → 1/z be analytic.) Consider an arbitrary set E ⊂ D, relative closed with respect to D such that the inner distance ρ(E, z 0 ) from the point z 0 to the set E with respect to the domain D is positive. In the case of a finite point z 0 , the relative capacity relcap E of the set E is defined via the File: abc120310.tex, printed: 2012-3-13, 5.22 †The research of this author was supported by the Russian Foundation for Basic Research, project 11-01-00038.
‡ Supported by the Academy of Finland, project 2600066611.
following asymptotic expansion (1.1) r(D \ E, z) r(D, z) where the approach of z to the point z 0 takes place along an arbitrary arc in D perpendicular to γ at the point z 0 . Here r(G, z) denotes the inner radius of the connected component of the set G containing the point z . If the point z 0 = ∞, then the parameter |z − z 0 | in the definition of the relative capacity relcap E is replaced by |1/z| . We say that relcap E is the relative capacity of the set E with respect to the domain D at the point z 0 . We shall establish the asymptotic expansion (1.1) in Section 2. It will also be shown that in the case when the domain D is the upper half plane H and the point where hcap E is the half-plane capacity of the set E [L1] . In the case when the set H \ E is a simply-connected domain, the equality of the relative and half-plane capacities was proven in [D2], with essential use of conformal mapping (cf. [D2, formula (3)]). We recall further that the notion of half-plane capacity arises naturally in statistical physics when examining the Schramm-Löwner equations (see G. Lawler [L1, L2, L3]). There are several definitions of this capacity in the literature. For example, let G = H \ E and where B t is a standard Brownian motion and τ G = inf{t : B t / ∈ G} , and E z is the mathematical expectation. Then φ G is a positive harmonic function on G that vanishes at the regular points of the boundary ∂G and such that The half-plane capacity (at infinity) of E is defined by (see [L2,Lecture 2]). Some new geometric properties of the half-plane capacity were proved in the recent paper [D2]. Following this paper we shall examine the properties of the relative capacity relcap E under various geometric transformations of the set E in the case when the domain D is the disk and the point z 0 = 1 (Section 3). Our results do not reduce to the corresponding statements of [D2] in the particular case D = H, z 0 = ∞ , because the symmetrization procedures applied here are not invariant under Möbius transformations. Finally, in Section 4 there are given some applications of the introduced capacity to holomorphic functions. The results obtained here turn out to be effective in the study of a boundary version of the Schwarz lemma involving the Schwarzian derivative (see, e.g. [S,TV,D3]). After the completion of the writing of this paper, we learned about a very recent paper of S. Rohde and C. Wong, also studying half-plane capacity [RW].

Existence and properties of relative capacity
Following the proof of Lemma 1 in the paper [D2] we first establish the existence of the asymptotic expansion (1.1).
Theorem 2.1. Let D be a domain in the extended complex plane C, and let z 0 be a finite accessible boundary point of the domain D . Suppose that in some neighborhood of z 0 the boundary ∂D is represented by an analytic arc γ . Then for every set E relatively closed with respect to D and with ρ(E, z 0 ) > 0 , the following expansion holds where the convergence of z to the point z 0 takes place along a path, perpendicular to the arc γ at z 0 and where c ≥ 0 is a constant, depending only on the set E , the domain D , and on the point z 0 .
Proof. We consider the function f mapping D conformally and univalently onto a domain f (D) lying in the upper half plane H . Extending f to the arc γ in the sense of the boundary correspondence we may assume that f (γ) is a finite interval of the real axis and that f (z 0 ) = 0, |f (z 0 )| = 1 . Let g(w, ζ) be the Green function of the connected component of the symmetric set that contains the origin and let h(w, ζ) = log |w − ζ| + g(w, ζ) be the regular part of this function. By the symmetry of G , we have Adding log |w − f (z)| to both sides and letting w → f (z) , this relation gives We consider the function h(w, ζ) as a function of four real arguments h(u, v, ξ, η) and introduce the notation f (z) = ∆u + i∆v . It is clear that when z → z 0 along an arc perpendicular to γ , we have ∆v → 0 and ∆u = o(∆v) , ∆v → 0 . The symmetric difference enclosed within the square brackets in (2.3) has the following form in the new notation In view of the symmetry of the set G with respect to the real axis we see that Therefore at the point (0, 0, 0, 0) Applying this fact and applying Taylor's formula in a neighborhood of the point (0, 0, 0, 0) , we have Taking into account the expression in (2.4) and substituting into (2.3) we arrive at the identity Here the constant c 1 = 4 ∂ 2 h ∂v∂η does not depend on the choice of the arc γ because the left hand side is independent of this arc.
Repeating the preceding argument and replacing f (D) \ f (E) with f (D) we arrive at a similar relation Because the expression on the left hand side does not depend on the choice of the function f and does not exceed 1 , we see that the constant c := − 1 2 (c 1 − c 2 ) ≥ 0 is independent of f . The theorem is proved.
The asymptotic expansion in the case z 0 = ∞ is contained in what was proved above with the change of variable z − z 0 → 1/z . Note that we did not only prove the existence of the expansion (1.1), but also established a representation of the relative capacity in terms of the Green functions of the domains in question.
Remark 2.5. From the above proof it is clear that the requirement of the analyticity of the arc γ can be weakened. It is enough to require that the arc γ have a tangent at the point z 0 and the function f has an extension the arc γ and the following expansion holds where a is some constant.
Next we prove that in the case D = H and z 0 = ∞ the relative capacity and the half-plane capacity coincide.
Theorem 2.6. The capacity relcap E of a bounded set, relatively closed in the half-plane H , at the point z 0 = ∞ is equal to the half plane capacity hcap E .
Proof. The expansions (1.1) and (1.2) take the following form after the change of variable At the regular boundary points of the boundary of the domainG we have According to the maximum principle we have This relation together with (2.8) give which in view of (2.7) gives relcap E = hcap E . The theorem is proved.
We next record some immediate properties of the relative capacity that follow from the properties of the inner radius and the expansion (1.1) (cf. [L1]). In what follows the domain D and the point z 0 are fixed.
Proof. The proof is a consequence of the monotonicity of the inner radius holds.
Proof. The proof follows from an inequality of Renggli (see [R]) and the formula (1.1).
Let the domain D be symmetric with respect to the imaginary axis and let z 0 be a point on this axis, z 0 ∈ ∂D . For a given set E ⊂ D we define the set where A * denotes a set symmetric to A with respect to the imaginary axis, whereas A + (A − ) is the intersection of A with the right (left) closed half plane.

Proof. Consider the set
It is easy to see that P c (D \ E) = D \ P E. According to Corollary 1.2 of [D1] r(P c (D \ E), z) ≥ r(D \ E, z) for every point of the imaginary axis, contained in D \ E . It remains just to apply the formula (1.1).
Property 4. (Composition principle) Under the hypotheses and notations introduced before Property 3 we have the following inequality Proof. According to Theorem 1.9 of [D2] for the points of the imaginary axis. For both parts of the inequality for r 2 (D, z) we apply the formula (1.1), and arrive at the desired conclusion. Proof. The proof is essentially contained in the proof of a particular case (Lemma 3 of the paper [D2]) noting the representation of the relative capacity in terms of the Green function (see the end of the proof of Theorem 2.1 ).
Property 6. Suppose that f maps a domain D onto a domain f (D) conformally and univalently such that a boundary point z 0 is mapped in the sense of boundary correspondence to a point w 0 ∈ ∂f (D) . We assume further that in a neighborhood of the point z 0 the boundary ∂D is an analytic arc and the boundary ∂f (D) in a neighborhood of w 0 is also an analytic arc. If, furthermore, z 0 = w 0 = ∞ and lim z→∞ f (z)/z = a , then relcap f (E) = |a| 2 relcap E , and if z 0 and w 0 are finite points, then for every relatively closed subset E ⊂ D , ρ(E, z 0 ) > 0 .
The proof follows from the formula (1.1) .

The behavior of the relative capacity under geometric transformations of subsets in the disk
It would be an interesting problem to study the behavior of the relative capacity under simultaneous geometric transformations of the domain D and the set E ⊂ D . However, this problem appears to be very difficult. Therefore we restrict ourselves here only to the case when the domain D is fixed. Furthermore, throughout this section, the domain D is the unit disk U = U z = {z : |z| < 1} , the point z 0 = 1, E is a relatively closed subset of U, with ρ(E, 1) > 0 , and relcap E stands for the relative capacity of E with respect to the disk U at the point z = 1 .
We recall the definition of the circular symmetrization of closed and open set with respect to a given ray (see [PS,H,D1]). Given a real number a let γ r (a) be the circle |z − a| = r (for 0 < r < ∞), which degenerates to the point a (or ∞) if r = 0 (r = ∞, resp.). By the circular symmetrization of a closed set F ⊂ C with respect to the ray [−∞, a] we mean the transformation of this set onto a symmetric set Cr − a F which is defined as follows. If, for a given 0 ≤ r ≤ ∞ , the 'circle' γ r (a) does not meet the set F , then it has empty intersection with the set Cr − a F as well. If γ r (a) ⊂ F , then γ r (a) ⊂ Cr − a F . In the remaining cases, the set Cr − a F intersects γ r (a) along a closed arc with center at the ray [−∞, a] , whose linear measure agrees with the measure of the intersection of F with γ r (a) . It is readily verified that the set Cr − a F is closed. In the same way we define the the result of the circular symmetrization Cr + a F with respect to the ray [a, ∞]. The only difference consists of the fact that the center of the closed arc is now contained on the ray [a, ∞]. In the same way, we define the circular symmetrizations Cr + a , Cr − a of open sets, now taking an open arc in place of a closed arc. Theorem 3.1. The following inequalities hold Proof. All the three inequalities can be proved in a unified way, applying formula (1.1) and Polya's result on the behavior of the inner radius of a domain under circular symmetrization (see [H, D1]). For instance, from Polya's theorem it follows that for every x, 0 < x < 1 , . It remains to apply the formula (1.1) with D = U . Analogously, for a ≤ 0 . and for a ≥ 1 we have . The theorem is proved.
where λ(x) is the line Re z = x, and µ(·) is the linear Lebesgue measure.
Theorem 3.5. The following inequality holds Proof. In the same way as in the proof of Theorem 3.1, it is enough to apply a theorem of Polya and Szegö [PS] (3.7) and to note the equality The proof is complete.
Theorem 3.8. The following inequality holds Proof. Let {B n } ∞ n=1 be a sequence from Property 5 for the given set E, D = U , and let E n = B n ∩ U, n = 1, 2, . . . . According to a theorem of Marcus [M1] for a fixed integer n and for values of x sufficiently close to 1, 0 < x < 1 , we have On the other hand, because of the openness of B n , one can take x so close to 1 that It remains to apply formula (1.1) and Property 5. The proof is complete.
Let A = {α k } n k=1 be a family of positive numbers with n k=1 α k = 1 . We consider an averaging transformation (see [M2]) , which assigns the starlike set . For a family of open subsets of the disk containing the neighborhood {z : |z−1| < ε}∩D for some ε > 0, the set is defined but under a constraint on θ : π/2 < θ < 3π/2 . Repeating the proof of preceding theorem replacing M x with R x A and applying Marcus' theorem and Property 5, we arrive at the following result. of positive numbers with n k=1 α k = 1 , the following inequality hold Applying Theorem 3.10 to the case when the collection of sets consists of two sets {E, {z : z ∈ E}} and the family of numbers A = { 1 2 , 1 2 } , we obtain the following theorem. Theorem 3.11. The following inequality holds relcap E ≥ relcap RE , RE = 1 + r iθ : (M (θ, U \ E)M (2π − θ, U \ E)) 1/2 ≤ r < −2 cos θ, π/2 < θ < 3π/2 .

Applications to bounded holomorphic functions in the disk
We denote by B the class of functions f holomorphic in the disk U z = {z : |z| < 1} , f (U z ) ⊂ U w with the asymptotic expansion where a 1 > 0, Re(2a 2 + a 1 (1 − a 1 )) = 0 and ∠o((z − 1) 3 ) is an infinitesimal quantity in comparison with (z − 1) 3 as z → 1 in any Stolz angle in U z with vertex at z = 1 . Everywhere in what follows in this section relcap E stands for the relative capacity of E with respect to the disk U at the point z = 1 . The equality holds for functions f of class B , mapping conformally and univalently the disk U z onto the domain U w \ E .
Proof. In view of the monotonicity of the inner radius and the formula (1.1) we have for points x on the segment (0, 1) In the case f (U z ) = U w \ E, the equality holds in (4.2) . On the other hand, from the result of Hayman [H] it follows that where, furthermore, the sign of equality holds for univalent functions. We next find the asymptotic behavior of the right hand side of (4.3) where U = 2a 2 a 1 + 1 2 , V = a 1 2 + Re a 2 a 1 . Next, writing Substituting this resulting asymptotic expansion in (4.3) and adding (4.3) and (4.2) we arrive at the required inequality. The theorem is proved. Remark 4.5. It is well-known that functions of class B satisfy ImS f (1) = 0 (see [S, TV]). Therefore in the inequality of Theorem 4.1 and in the following applications, the real part of the Schwarzian could be replaced with S f (1) .
Some applications of Theorem 4.1 to the proof of geometric properties of holomorphic functions are given, for instance, in the following statements (cf. [S,TV,LLN,D3]).
Theorem 4.6. Let the function f be of class B , and let the angular Lebesgue measure of the intersection of the image of the unit disk f (U z ) with every circle |w| = r, 0 < r < 1 , be smaller than or equal to α, 0 < α < 2π . Then
Proof. By Theorems 4.1 and 3.1 (inequality (3.2)) we have for every set E relatively closed with respect to U w , lying in U w \ f (U z ) and having a positive distance from the point z = 1 . Set E = U w \f (U z ) . If the distance from U w \f (U z ) to the point z = 1 is zero, then we set for a sufficiently small ε > 0 . If β is a number with 0 < α < β < 2π , then by the hypothesis of the theorem, we can choose ε such that the set Cr − o E contains U w \ B(β) . From the monotonicity of the capacity and again by Theorem 4.1 we obtain The proof now follows by comparing the two obtained inequalities, computing the derivative of the function f β and letting β → α .