Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 419–436

AN ISOPERIMETRIC INEQUALITY FOR LOGARITHMIC CAPACITY

Roger W. Barnard, Kent Pearce, and Alexander Yu. Solynin

Texas Tech University, Deparment of Mathematics Lubbock, TX, U.S.A.; barnard@math.ttu.edu; pearce@math.ttu.edu Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences Fontanka 27, St. Petersburg, 191011, Russia; solynin@pdmi.ras.ru and Texas Tech University, Lubbock, TX, U.S.A.

Abstract. We prove a sharp lower bound of the form

cap E ≥ ( \frac{1}{2} ) diam E · Ψ(area E/( \frac{1}{4} π diam^{2} E))

for the logarithmic capacity of a compact connected planar set E in terms of its area and diameter. Our lower bound includes as special cases G. Faber’s inequality cap E ≥ diam \frac{1}{4} E and G. Pólya’s inequality cap E ≥ (area E/π)^{1/2} . We give explicit formulations, functions of \frac{1}{2} diam E, for the extremal domains which we identify.

1. Introduction

The logarithmic capacity, cap E, of a continuum (= compact connected set) E in the complex plane C is defined by

(1.1)  − log cap E = lim_{z→∞}(g(z) − log |z|),

where g(z) denotes Green’s function of the unbounded component Ω(E) of C \ E having singularity at z = ∞.

The measure of a set described by the logarithmic capacity is very important in potential theory, analysis, and PDE’s. It combines several characteristics of a compact set, among which are the geometric concept of transfinite diameter due to M. Fekete, the concept of Chebyshev’s constant from polynomial approximation, and the concept of the outer radius from conformal mapping, see [D], [Du], [G], [H].

2000 Mathematics Subject Classification: Primary 30C70, 30C85.

This paper was finalized during the third author’s visit at Texas Tech University, Fall 2001. This author thanks the Department of Mathematics and Statistics of this university for the wonderful atmosphere and working conditions during his stay in Lubbock. The research of the third author was supported in part by the Russian Fund for Fundamental Research, grant no. 00-01-00118a.

In general, computation of cap E is a difficult problem but there are several estimates of cap E in terms of geometric characteristics of E that are very useful in applications, see [PSz]. For instance,

(1.2)  \frac{1}{4} diam E ≤ cap E ≤ \frac{1}{2} diam E,
(1.3)  (\frac{1}{π} area E)^{1/2} ≤ cap E < ∞.

The first inequality in (1.2) was found by G. Faber [F] in a different form. The inequalities in (1.3) and the second inequality in (1.2), which is valid for any (not necessary connected) compact set, were proved by G. Pólya [Po]. Equality occurs only for rectilinear segments in Faber’s inequality and for disks in Pólya’s inequalities. The case of equality in the right inequality in (1.2) was studied by J. Jenkins [J] and A. Pfluger [Pf].

We will employ the following notation throughout this paper: let U = {z : |z| < 1} and U_{r}(c) = {z : |z − c| < r}, so that U = U_{1}(0). Finally, let U^{∗} = C \ U.

In this paper, we prove the following theorem that contains the left inequalities in (1.2) and (1.3) as special cases.

Theorem 1. Let E be a continuum in C. Then

(1.4)  cap E ≥ \frac{1}{2} diam E · Ψ(area E/(\frac{1}{4}π diam^{2} E)),

where 1/Ψ(s) is a decreasing function from [0, 1] onto [1, 2] that is the inverse function to s = p^{−2}[β^{2}(p) − 2p(β(p) − 1)], with 1 ≤ β(p) ≤ 2 defined by equation (1.7), with d replaced by p.

Equality occurs in (1.4) if and only if E = a(C\f_{d}(U^{∗}))+b for some a, b ∈ C, a ≠ 0 and 1 ≤ d ≤ 2, where the function f_{d}(z) is defined for |z| > 1 by (1.8).

The graph of Ψ is plotted in Figure 1. Figure 2 displays the shape of the extremal continua for a = 1, b = 0 and some typical values of d.

Let s = area E/(\frac{1}{4}π diam^{2} E). In the case s = 0, (1.4) gives Faber’s inequality and in the case s = 1, it gives Pólya’s inequality. Combined with the right-hand side inequality in (1.2) and the classical area-diameter inequality 0 ≤ area E/(\frac{1}{4}π diam^{2} E) ≤ 1, (1.4) describes the range of one of the quantities cap E, diam E, or area E if the other two are fixed. Several similar sharp inequalities linking three characteristics of a set are known in geometry. But to prove such a sharp inequality is a difficult task even for purely geometric quantities, see [JBo]. From this perspective, (1.4) might be the first known sharp inequality for these three quantities that includes a functional characteristic.

Since cap E, diam E, and (area E)^{1/2} all change linearly with respect to scaling we can fix one of them, say cap E, and then study the region of variability

Figure 1. Graph of Ψ(s).

of the other two. In this way, we can reformulate the problem as finding the maximal omitted area for the class Σ of univalent functions

(1.5)  f(z) = z + a_{0}(f) + a_{1}(f)z^{−1} + ⋯

which are analytic in U^{∗}, except for a simple pole at z = ∞. For f ∈ Σ, let E_{f} = C \ f(U^{∗}). It is well known, as a consequence of the normalization in (1.5), that for f ∈ Σ that 1 ≤ \frac{1}{2} diam E_{f} ≤ 2. Therefore, for 1 ≤ d ≤ 2, we will consider Σ_{d} = {f ∈ Σ : diam E_{f} = 2d}. For f ∈ Σ_{d}, define A_{f} = area E_{f} and A(d) = sup_{f∈Σ_{d}} A_{f}. It is well known that

A_{f} = π(1 − ∑_{n=1}^{∞} n|a_{n}(f)|^{2})

—this relation will be often used in what follows. Theorem 1 is equivalent to

Theorem 2. Let f ∈ Σ_{d}, 1 ≤ d ≤ 2. Then,
(1.6)  A(f) ≤ π[β^{2} − 2d(β − 1)],

where 1 ≤ β ≤ 2 is the unique solution to the equation

(1.7)  d = β − (β − 1) log(β − 1).

Equality occurs in (1.6) if and only if f(z) = e^{iτ} f_{d}(e^{−iτ} z) + b with τ ∈ R, b ∈ C, and

(1.8)  f_{d}(z) = d + ∫_{1}^{z} z^{−1}φ(z; d) dz,

Figure 2. Extremal continua.

where the function φ(z; d) = zf′_{d}(z) is defined by the equation

(1.9)  φ(z; d) = A\frac{z^{2} − 1}{z} \frac{\sqrt{1 + Bz^{2} + \sqrt{r(z)}} \sqrt{B + z^{2} + \sqrt{r(z)}}}{c(1 + z^{2}) + \sqrt{r(z)}}

with principal branches of the radicals and

c = β − 1, A = \frac{1 + c}{2c}, B = 2c^{2} − 1, r(z) = 1 + 2Bz^{2} + z^{4}.

The function φ maps U^{∗} conformally onto the complement of the “double anchor”

\begin{aligned}F (β, ψ) & = [−iβ, iβ] ∪ \{βe^{it} : \frac{1}{2}π − ψ ≤ t ≤ \frac{1}{2}π + ψ\} \\ & ∪ \{βe^{it} : \frac{3}{2}π − ψ ≤ t ≤ \frac{3}{2}π + ψ\}, \end{aligned}

where β is defined by (1.7) and ψ = \frac{1}{2} cos^{−1}(8β^{−1} − 8β^{−2} − 1).

The graph of the maximal omitted area A(d) = π[β^{2}(d) − 2d(β(d) − 1)] is shown in Figure 3.

To prove Theorem 2, we apply techniques developed in [BS], which were based on symmetrization transformations and some elementary local variations. Section 2 contains preliminary results and necessary definitions. In Section 3, we identify the extremal function by solving a specific boundary value problem for analytic functions. Section 4 completes the proofs of Theorems 1 and 2.

Note that some similar sharp estimates for the area of f(U) for problems with analytic side conditions instead of geometric constraints, as imposed in the present paper, were found in [ASS1], [ASS2] using a different method.

Figure 3. Graph of A(d).

2. Preliminaries

First we show the existence of an extremal function and describe some simple properties of the maximal omitted area.

Lemma 1. For every 1 ≤ d ≤ 2 there is a function f ∈ Σ_{d} such that A_{f} = A(d).

The maximal area A(d) is continuous and strictly decreases from π to 0 as d increases from 1 to 2.

Proof. For a fixed d, Σ_{d} is compact in the topology of uniform convergence on compact subsets of U^{∗}. Since A_{f} is upper semi-continuous, the existence of an extremal function follows.

Let 1 < d_{1} < d_{2} ≤ 2 and let f ∈ Σ_{d} be extremal for A(d_{2}). Note that f has at least one non-zero coefficient a_{k}(f) for some k ≥ 1. The function

(2.1)  f_{t}(z) = t^{−1}f(tz) = z + a_{0}(t)t^{−1} + a_{1}(t)t^{−2}z^{−1} + ⋯,

as well as the area A_{f_{t}} and diameter d(f_{t}) = diam E_{f_{t}} depend continuously on t, 1 ≤ t < ∞. Since f_{t}(z) → z as t → ∞, one can easily show that d(f_{t}) → 2 as t → ∞. Hence, there is t_{1} > 1 such that d(f_{t_{1}}) = 2d_{1} . Therefore

(2.2)  \begin{aligned}A(d_{1}) & ≥ A_{f_{t_{1}}} = π(1 − ∑_{n=1}^{∞} nt_{1}^{−2(n+1)}|a_{n}(f)|^{2}) \\ & > π(1 − ∑_{n=1}^{∞} n|a_{n}(f)|^{2}) = A(d_{2}), \end{aligned}

with strict inequality since a_{k}(f) ≠ 0. Equation (2.2) proves the strict monotonicity of A(d).

Finally, the compactness of Σ_{d} and continuity of A_{f_{t}} imply the continuity of A(d).

Since the class Σ_{d} is invariant under the rigid motions of C, i.e., e^{−iθ}f(e^{iθ}z)+ b ∈ Σ_{d} if f ∈ Σ_{d} and θ ∈ R, b ∈ C, we may restrict ourselves to functions f ∈ Σ_{d} such that the points w_{1} = d, w_{2} = −d belong to E_{f}. Thus, the condition diam E_{f} = |w_{1} − w_{2}| = 2d will be assumed if a different condition is not imposed explicitly.

To prove symmetry properties of an extremal continuum E_{f} we shall apply Steiner symmetrization defined as follows:

The Steiner symmetrization of a compact set E w.r.t. the real axis R is a compact set E^{∗} such that for every u ∈ R, E^{∗} ∩ l(u) = ∅ if E ∩ l(u) = ∅ and E^{∗}∩l(u) = \{w = u+it : −\frac{1}{2}m ≤ t ≤ \frac{1}{2}m\} if E∩l(u) ≠ ∅. Here l(u) = \{w = u+it : −∞ < t < ∞} and m = meas(E ∩ l(u)) denotes the linear Lebesgue measure. Steiner symmetrization w.r.t. the imaginary or other axis is defined similarly.

It is well known that Steiner symmetrization preserves area and diminishes diameter and logarithmic capacity [H], [D].

Lemma 2. For 1 < d < 2, let f ∈ Σ_{d} be an extremal function normalized as above. Then, E_{f} possesses Steiner symmetry w.r.t. the real and imaginary axes. Moreover, the boundary of E_{f}, ∂E_{f}, consists of a Jordan rectifiable curve L_{f} plus, possibly, some added segments I_{+} = [d_{0}, d], I_{−} = [−d, −d_{0}], 0 < d_{0} = d_{0}(d) ≤ d, of the real axis.

Proof. Suppose that E_{f} does not possess Steiner symmetry w.r.t. R. Let E^{∗} be the Steiner symmetrization of E_{f} w.r.t. R. Note that

(2.3)  2d = diam E_{f} = diam E^{∗}  and  cap E_{f} > cap E^{∗}

since the points ±d ∈ E_{f} and since E^{∗} is not a rigid motion of E_{f} (see [D]). Let

(2.4)  F (z) = αz + α_{0} + α_{1}z^{−1} + ⋯,  α > 0,

map U^{∗} conformally onto Ω(E^{∗}). The inequality in (2.3) shows that α < 1. Let F_{α} = α^{−1}F. Then, F_{α} ∈ Σ_{d/α}. Therefore, we have

A(d/α) ≥ A_{F_{α}} = πα^{−2}(1 − ∑_{n=1}^{∞} nα^{−2}|α_{n}|^{2}) ≥ π(1 − ∑_{n=1}^{∞} n|a_{n}(f)|^{2}) = A(d).

Since d/α > d, the latter contradicts the strict monotonicity of A(d) in Lemma 1.

The same arguments show that E_{f} possesses Steiner symmetry w.r.t. the imaginary axis.

Let Lf^{+} = {w ∈ ∂E_{f} : Im w > 0} and Lf^{−} = {w ∈ ∂E_{f} : Im w < 0}.

The Steiner symmetries of E_{f} w.r.t. the real and imaginary axes can be used to show that Lf^{+} and Lf^{−} are rectifiable Jordan arcs; a similar argument was used in [ASS2, Lemma 4]. Indeed, let L^{+} = {w ∈ Lf^{+} : Re w > 0} and let d_{0} = sup{Re w : w ∈ L^{+}}, m_{0} = sup{Im w : w ∈ L^{+}}. The function

τ(w) = u + m_{0} − v, where w = u + iv,

is continuous on L^{+} and maps the closure L̄^{+} one-to-one onto the segment {τ : 0 ≤ τ ≤ d_{0} + m_{0}}. Therefore, L̄^{+} is Jordan. Since Re w and Im w are both monotonic on L^{+} , it follows that L̄^{+} is rectifiable. This implies that ∂E_{f} consists of a rectifiable Jordan curve L_{f} plus, possibly, some added horizontal segments [−d, −d_{0}], [d_{0}, d] and vertical segments [−im, −im_{0}], [im_{0}, im] with 0 ≤ m_{0} ≤ m < ∞.

The presence of vertical segments, i.e., the segments [−im, −im_{0}], [im_{0}, im] with m > m_{0} easily leads to a contradiction: shortening the vertical slits and expanding the horizontal ones we can find a continuum Ẽ such that area Ẽ = area E_{f}, cap Ẽ = cap E_{f} = 1, and diam Ẽ > diam E_{f} that contradicts the strict monotonicity property of A(d) in Lemma 1.

Let f ∈ Σ_{d} be an extremal function for A(d). By Lemma 2, ∂f(U^{∗}) =

Lf^{+} ∪ Lf^{−} ∪ [−d, −d_{0}] ∪ [d_{0}, d] with 0 < d_{0} ≤ d. If d_{0} < d then there is 0 < θ_{0} < \frac{1}{2}π

such that Lf^{+} = {f(z) : z ∈ lf^{+}}, [d_{0}, d] = {f(z) : z ∈ ln^{++}}, [−d, −d_{0}] = {f(z) :

z ∈ ln^{+−}}, where lf^{+} = {e^{iθ} : θ_{0} < θ < π − θ_{0}}, ln^{++} = {e^{iθ} : 0 ≤ θ ≤ θ_{0}},

ln^{+−} = {e^{iθ} : π − θ_{0} ≤ θ ≤ π}. The corresponding arcs in the lower half-plane will be denoted by lf^{−}, l_{n}^{−+} , and ln^{−−}. The image curves Lf^{+} and Lf^{−} are called the free boundary, the preimages lf^{+} and lf^{−} are called the free arcs. Respectively, [d_{0}, d], [−d, −d_{0}] and ln^{++} , ln^{+−}, l_{n}^{−+} , ln^{−−} are called the non-free boundary and the non-free arcs.

To study the behavior of f′ on the non-free arcs we shall use two lemmas from [BS], which are limiting cases of Theorem 1 in [S2].

Let Hτ^{+} and Hτ^{−} be the left and right half-planes w.r.t. the vertical line

l(τ) = {w = u + iv : u = τ}. For D ⊂ C , let Dτ^{+} = D ∩ Hτ^{+} , Dτ^{−} = D ∩ Hτ^{−}

and let D_{τ}^{∗} denote the set symmetric to D w.r.t. l(τ), i.e., D_{τ}^{∗} = {w = u + iv : 2τ − u + iv ∈ D}.

We say that D possesses the polarization property in the interval τ_{1} < τ < τ_{2} if (Dτ^{−})_{τ}^{∗} ⊂ Dτ^{+} for all τ_{1} < τ < τ_{2} . Extremal configurations shown in Figure 2 give an example of domains possessing the polarization property in the corresponding intervals 0 < τ < d.

Lemma 3 ([BS, Lemma 4]). Let f ∈ Σ, D = f(U^{∗}), and let f map a boundary arc {e^{iθ} : θ_{1} < θ < θ_{2}} onto a horizontal interval {w : Im w = v_{0}, τ_{1} < Re w < τ_{2}}. Let D possess the polarization property in τ_{1} < τ < τ_{2} . Then, |f′(e^{iθ})| strictly increases in θ_{1} < θ < θ_{2} if f(e^{iθ_{1}}) = τ_{2}+iv_{0} and strictly decreases if f(e^{iθ_{1}}) = τ_{1} + iv_{0} .

If f is extremal for A(d), the joint symmetry of E_{f} assures that the polarization property of D = f(U^{∗}) holds in d_{0} < τ < d. Thus, we obtain from Lemma 3,

Corollary 1. Let f ∈ Σ_{d} be extremal for A(d), 1 < d < 2 and suppose that the non-free arc ln^{++} is not degenerate, i.e., 0 < θ_{0} < \frac{1}{2}π. Then, |f′(e^{iθ})| strictly increases in 0 < θ < θ_{0} and π < θ < π + θ_{0} and strictly decreases in π − θ_{0} < θ < π and 2π − θ_{0} < θ < 2π.

We need a similar result concerning angular polarization. Let γ_{φ} = {w = te^{iφ}, t ≥ 0} and let Hφ^{+} denote the right half-plane w.r.t. the line determined by γ_{φ} and Hφ^{−} denote the left half-plane w.r.t. the line determined by γ_{φ}. For D ⊂ C , let Dφ^{+} = D ∩ Hφ^{+} , Dφ^{−} = D ∩ Hφ^{−} and let D_{φ}^{∗} denote the set symmetric to D w.r.t. the line determined by γ_{φ}.

We say that a domain D possesses the angular polarization property in φ_{1} < φ < φ_{2} if (Dφ^{−})_{φ}^{∗} ⊂ Dφ^{+} for all φ_{1} < φ < φ_{2} . For example, domain G depicted in Figure 4 possesses the angular polarization property in 0 < φ < \frac{1}{2}π.

Lemma 4 ([BS, Lemma 5]). Let g map U^{∗} conformally onto D and map a boundary arc {e^{iθ} : θ_{1} < θ < θ_{2}} onto a circular arc L = {w = ρe^{iφ} : φ_{1} < φ < φ_{2}}. Let g(∞) ∈ Hφ_{1}^{+} ∩ Hφ_{2}^{+} and let D possess the angular polarization property in φ_{1} < φ < φ_{2} . Then, |g′(e^{iθ})| strictly increases in θ_{1} < θ < θ_{2} if g(e^{iθ_{1}}) = ρe^{iφ_{1}} and strictly decreases in θ_{1} < θ < θ_{2} if g(e^{iθ_{1}}) = ρe^{iφ_{2}} .

Remark. The domains D in Lemmas 3 and 4 possess the polarization property for a horizontal interval and the angular polarization property w.r.t. rays issuing from the origin, respectively. One can easily reformulate these lemmas for arbitrary intervals and for rays issuing from arbitrary centers. For instance, Lemma 4 in [BS] is formulated for vertical intervals.

The term “polarization property” comes from the proofs of Theorem 1 in [S2] and Lemmas 4 and 5 in [BS] that use the polarization transformation.

To find a boundary condition for an extremal function f ∈ Σ_{d} on the free arcs, we apply, in a suitable form, the local variation used in [BS]. First, we recall the relations linking the logarithmic capacity of a continuum E with the outer radius and reduced module of Ω(E). Let

g(w) = w + b_{0} + b_{1}w^{−1} + ⋯

map Ω(E) conformally onto U_{R}^{∗} = C \ U_{R}, where U_{R} = {ζ : |ζ| < R}. The radius R = R(E) of the omitted disk is uniquely determined and is called the outer radius of Ω(E); it is well known that cap E = R(E), see [Du, Section 10.2], [G, Chapter 7]. The quantity

(2.5)  m(Ω(E), ∞) = −\frac{1}{2π} log R(E) = −\frac{1}{2π} log cap E

is called the reduced module of Ω(E) at w = ∞.

The variation used in [BS] and in the present paper is rather complicated. Let Ω ⊂ C be a simply connected domain which contains ∞ such that its boundary arcs lying in the vicinities of two of its boundary points, w_{1} and w_{2} , are Jordan and rectifiable. Let ∂Ω have a tangent l at w_{1} and let n_{1} be a unit inward normal at w_{1} . For ε_{1} > 0 small enough, let c_{ε_{1}}^{0} and c_{ε_{1}} be open and closed crosscuts of D at the boundary point w_{1} , i.e., c_{ε_{1}}^{0} and c_{ε_{1}} are respectively the biggest open and closed arcs of C_{ε_{1}}(w_{1}), where C_{r}(w_{0}) = {w : |w − w_{0}| = r}, such that w_{1} + ε_{1}n_{1} ∈ c_{ε_{1}}^{0} ⊂ Ω and w_{1} + ε_{1}n_{1} ∈ c_{ε_{1}} ⊂ Ω , respectively. Let Uε_{1}^{+}(w_{1}) denote the connected component (half-disk) of U_{ε_{1}}(w_{1}) \ l that contains the point w_{1}+ε_{1}n_{1} on its boundary. Let c′_{ε_{1}} denote the maximal open circular arc contained in the intersection c_{ε_{1}}^{0} ∩ ∂Uε_{1}^{+}(w_{1}). Let Ω̂_{ε_{1}} be a connected component of Ω \ \overline{U}_{ε_{1}}(w_{1}) containing ∞ and let

(2.6)  Ω_{ε_{1}} = Ω̂_{ε_{1}} ∪ Uε_{1}^{+}(w_{1}) ∪ c′_{ε_{1}}.

Let I(ε_{1}) = \{w = w_{1} − itn_{1} : −ε_{1} < t < ε_{1}\}. For 0 < φ_{1} ≤ \frac{1}{2} , let M(ε_{1}, φ_{1}) denote the open lune in U_{ε_{1}}(w_{1}) \ Uε_{1}^{+}(w_{1}) bounded by I(ε_{1}) and a circular arc γ(ε_{1}, φ_{1}) that forms angles of opening πφ_{1} with the interval I(ε_{1}) at its end points. Let

(2.7)  Ω(ε_{1}, φ_{1}) = Ω_{ε_{1}} ∪ M(ε_{1}, φ_{1}) ∪ I(ε_{1}).

Let g(w) = g(w; ε_{1}, φ_{1}) map Ω(ε_{1}, φ_{1}) conformally onto U^{∗} such that g(∞) = ∞, g(w_{2}) = 1. Let 0 < φ_{2} ≤ \frac{1}{2} and ε_{2} > 0 be small enough. Let U^{ε_{2},φ_{2}} ⊂ U^{∗} be the simply connected domain which contains ∞ and which is bounded by the arc L(ε_{2}) = {e^{iθ} : ε_{2} ≤ |θ| ≤ π} and by the circular arc L(ε_{2}, φ_{2}) with ends at the points e^{iε_{2}} and e^{−iε_{2}} that forms an angle of opening π(1 − φ_{2}) with the arc L(ε_{2}) at the points e^{iε_{2}} and e^{−iε_{2}} .

The domain

(2.8)  Ω̃ = g^{−1}(U^{ε_{2},φ_{2}}, ε_{1}, φ_{1})

will be called the two point variation of Ω centered at w_{1} and w_{2} with radii ε_{1} and ε_{2} and inclinations φ_{1} and φ_{2} .

The following lemma is a reformulation of Lemma 10 in [BS] for conformal mappings f of U^{∗} normalized by condition f(∞) = ∞; in [BS] this result is formulated for conformal mappings f of the unit disk U with normalization f(0) = 0.

Lemma 5 ([BS, Lemma 10]). Let w = f(z) map U^{∗} conformally onto Ω defined above such that f(∞) = ∞, f(e^{iθ_{1}}) = w_{1} , f(e^{iθ_{2}}) = w_{2} and let there exist the limits

(2.9)  f′(e^{iθ_{k}}) = lim_{z→e^{iθ_{k}} , z∈\overline{U}^{∗}} \frac{f(z) − w_{k}}{z − e^{iθ_{k}}} ≠ 0, ∞  for k = 1, 2.

Let |f′(e^{iθ_{k}})| = α_{k}, k = 1, 2. Let Ω̃(ε_{1}, ε_{2}, φ_{1}, φ_{2}) be the two-point variation of Ω defined by (2.8) with ε_{2} replaced by ε_{2}/α_{2} . Then, for fixed 0 < φ_{1} ≤ \frac{1}{2} and 0 < φ_{2} ≤ \frac{1}{2} ,

m(Ω̃(ε_{1}, ε_{2}, φ_{1}, φ_{2}), ∞) − m(Ω, ∞) = \frac{φ_{1}(2 + φ_{1})}{12πα_{1}^{2}(1 + φ_{1})^{2}} ε_{1}^{2}

− \frac{φ_{2}(2 − φ_{2})}{12πα_{2}^{2}(1 − φ_{2})^{2}} ε_{2}^{2} + o(ε_{1}^{2}) + o(ε_{2}^{2}) (2.10)

and

area(C \ Ω̃(ε_{1}, ε_{2}, φ_{1}, φ_{2})) − area(C \ Ω) = −\frac{2πφ_{1} − sin 2πφ_{1}}{2 sin^{2} πφ_{1}} ε_{1}^{2}

+ \frac{2πφ_{2} − sin 2πφ_{2}}{2 sin^{2} πφ_{2}} ε_{2}^{2} + o(ε_{1}^{2}) + o(ε_{2}^{2}) (2.11)

as ε_{1} → 0 and ε_{2} → 0.

To prove uniqueness of the extremal function in Σ_{d}, we need the following modification of Lemma 1 in [S1] where a similar result is proved for domains with one axis of symmetry.

Lemma 6 (cf. [S1, Lemma 1]). For k = 1, 2, let Ω_{k} ⊂ C be simply connected domains which contain ∞ and which have double symmetry w.r.t. the coordinate axes. Let there be a point ζ ∈ ∂Ω_{1} , Re ζ ≥ 0, Im ζ > 0 such that the points ζ, ζ̄, −ζ, and −ζ̄ split ∂Ω_{1} into four boundary arcs lr^{+} , lr^{−}, li^{+} , and li^{−}, where lr^{+} lies in the closed right half-plane and connects ζ and ζ̄, li^{+} lies in the closed upper half-plane and connects ζ and −ζ̄. Let

g_{k}(z) = z + a_{1}(g_{k})z^{−1} + ⋯

map U^{∗} conformally onto Ω_{k}.

If lr^{+} ⊂ Ω_{2} , li^{+} ⊂ C \ Ω_{2} , then

(2.12)  g_{1}(r) ≤ g_{2}(r),  |g_{2}(ir)| ≤ |g_{1}(ir)|

for all r > 1. Equality can occur in (2.12) if and only if Ω_{1} = Ω_{2} .

3. Boundary value problem for extremal functions

In this section f will denote the extremal function in Σ_{d} with 1 < d < 2 and D = f(U^{∗}).

Lemma 7. There is β > 0 such that

(3.1)  |f′(e^{iθ})| = β  for a.e. e^{iθ} ∈ l_{f} := lf^{+} ∪ lf^{−}

and

(3.2)  |f′(e^{iθ})| < β  for all e^{iθ} ∈ l_{n} := T \ l̄_{f}.

Proof. Since ∂D = L_{f} ∪ I_{+} ∪ I_{−} by Lemma 2, where L_{f} is Jordan and rectifiable, the non-zero finite limit

(3.3)  f′(ζ) = lim_{z→ζ, z∈\overline{U}^{∗}} \frac{f(z) − f(ζ)}{z − ζ} ≠ 0, ∞

exists for almost all ζ ∈ T. This easily follows from Theorem 6.8 in [P] applied to the univalent function 1/f(1/z). Assume that

(3.4)  0 < β_{1} = |f′(e^{iθ_{1}})| < |f′(e^{iθ_{2}})| = β_{2} < ∞

for some e^{iθ_{1}}, e^{iθ_{2}} ∈ l_{f}. Note that (3.3) and (3.4) allow us to apply the two point variation of Lemma 5.

For positive k_{1} , k_{2} such that

(3.5)  0 < k_{1} < 1 < k_{2}  and  k_{1}β_{1}^{−1} > k_{2}β_{2}^{−1}

and for fixed φ > 0 small enough consider the two point variation D̃ of D centered at w_{1} = f(e^{iθ_{1}}) and w_{2} = f(e^{iθ_{2}}) with inclinations φ and radii ε_{1} = k_{1}ε, ε_{2} = k_{2}ε, respectively. Computing the change in the omitted area by formula (2.11), we find

area(C \ D̃) − area(C \ D) = \frac{2πφ − sin 2πφ}{2 sin^{2} πφ}ε^{2}(k_{2}^{2} − k_{1}^{2}) + o(ε^{2}).

Therefore,

(3.6)  area(C \ D̃) > area(C \ D)

for all ε > 0 small enough. Applying the variation (2.10) of Lemma 6, we get

(3.7) \begin{aligned}m(D̃, ∞) − m(D, ∞) & = \frac{1}{12π} [\frac{φ(2 + φ)}{(1 + φ)^{2}} \frac{k_{1}^{2}}{β_{1}^{2}} − \frac{φ(2 − φ)}{(1 − φ)^{2}} \frac{k_{2}^{2}}{β_{2}^{2}} ]ε^{2} + o(ε^{2}) \\ & = [\frac{φ}{6π} (\frac{k_{1}^{2}}{β_{1}^{2}} − \frac{k_{2}^{2}}{β_{2}^{2}} ) + o(φ)]ε^{2} + o(ε^{2}), \end{aligned}

which together with (3.5) implies that

(3.8)  m(D̃, ∞) > m(D, ∞)

for all ε > 0 small enough if φ is chosen such that the expression in the brackets in (3.7) is positive. Let E = C \ D, Ẽ = C \ D̃. Equations (3.8) and (2.5) show that

(3.9)  cap Ẽ < cap E.

Since area Ẽ > area E and diam Ẽ ≥ diam E, (3.9) contradicts the monotonicity property of the function A(d) in Lemma 1.

Assume that l_{n} ≠ ∅. Then f′(1) = f′(−1) = 0. To prove that |f′(e^{iθ})| < β for all e^{iθ} ∈ l_{n} \ {±1}, we assume that β = |f′(e^{iθ_{1}})| < |f′(e^{iθ_{2}})| = β_{2} with e^{iθ_{1}} ∈ l_{f} and some e^{iθ_{2}} ∈ l_{n} \ {±1}. Then applying the two point variation as above we get (3.6) and (3.9), again contradicting the monotonicity property of A(d). Hence, |f′(e^{iθ})| ≤ β for all e^{iθ} ∈ l_{n}, which combined with the strict monotonicity property of Corollary 1 leads to the strict inequality in (3.2).

Lemma 8. If 1 < d < 2, then l_{n} = {e^{iθ} : −θ_{0} < θ < θ_{0}} ∪ {e^{iθ} : π − θ_{0} < θ < π + θ_{0}\} with some 0 < θ_{0} = θ_{0}(d) < \frac{1}{2}π; f′ is continuous on \overline{U}^{∗} and for all z ∈ \overline{U}^{∗}

(3.10)  |f′(z)| ≤ |f′(e^{iθ})| = β,

where e^{iθ} ∈ l̄_{f} and β > 1.

Proof. Consider the function g(z) = 1/f(1/z). By Lemma 2, g maps U onto a Jordan rectifiable domain possibly slit along two symmetric radial segments lying on the real axis. The double symmetry of D = f(U^{∗}) implies that G = g(U) is starlike w.r.t. w = 0. Since G is rectifiable and starlike, it follows from classical results of Lavrent′ev, see [P, p. 163], that G is a Smirnov domain (non-Jordan in general). This shows that log |g′(z)| = log |f′(1/z)|−2 log |zf(1/z)|, and therefore log |f′(1/z)|, can be represented by the Poisson integral

(3.11)  log |f′(1/z)| = \frac{1}{2π} ∫_{0}^{2π} P (r, θ − t) log |f′(e^{−it})| dt

with boundary values defined a.e. on T, see [P, p. 155]. Equation (3.11) along with (3.1) and (3.2) shows that 1 = |f′(∞)| ≤ β with equality only for the function f(z) ≡ z.

If l_{n} = ∅, then (3.11) and (3.1) show that |f′| = β identically on U^{∗}. Therefore, f(z) ≡ z contradicting the condition d = \frac{1}{2} diam f(U^{∗}) > 1. Hence,

Figure 4. Domain G for β = 1.7.

l_{n} ≠ ∅. The latter implies that f is analytic in vicinities of the points z = 1 and z = −1 and f′(z) has a simple zero at z = 1, z = −1. Consider the function h(z) = log |f′(1/z)/(z^{2} − 1)|, which can be represented by the Poisson integral

(3.12)  h(z) = \frac{1}{2π} ∫_{0}^{2π} P (r, θ − t) log |f′(e^{it})/(e^{2it} − 1)| dt.

Equation (3.12) and the previous analysis show that h is a bounded harmonic function on U. Let h_{1} be a bounded harmonic function on U with boundary values log(β/|z^{2} − 1|) on l_{f} and h(z) on l_{n}. Then h_{1} − h has nontangential limit 0 a.e. on T. Therefore, h_{1} − h ≡ 0 in U. Hence, |f′| = β everywhere on l_{f}.

Since D possesses the double symmetry we need to show only that f′ is continuous at e^{iθ_{0}} . By the symmetry principle, f can be continued analytically through l_{n} and f′ can be continued analytically through l_{f}. This implies that f can be considered as a function analytic in a slit disk Δ_{0} = U_{ε}(e^{iθ_{0}}) \ [(1 − ε)e^{iθ_{0}}, e^{iθ_{0}}] with ε > 0 small enough.

Using the Julia–Wolff lemma, see [P, Proposition 4.13], boundedness of f′, and well-known properties of the angular derivatives, see [P, Propositions 4.7, 4.9], one can prove that f′ has a finite limit f′(e^{iθ_{0}}), |f′(e^{iθ_{0}})| = β, along any path in \overline{U}^{∗} ending at e^{iθ_{0}} . The details of this proof are similar to the arguments in Lemma 13 in [BS].

Lemma 9. Let f be extremal in Σ_{d} for 1 < d < 2 and let φ(z) = zf′(z). Then φ maps U^{∗} univalently onto the complement Ω(β, ψ) = C \ F (β, ψ) of the double anchor F (β, ψ) defined in Theorem 2 where β is defined by (1.7) and ψ = \frac{1}{2} cos^{−1}(8β^{−1} − 8β^{−2} − 1).

Proof. (1) Let g(z) = f′(\sqrt{z} ). Since g(z̄) = \overline{g(z)}, the symmetry principle implies that g is analytic in U^{∗}. We will show that g is univalent there.

By Corollary 1, |g(e^{iθ})| = |f′(e^{iθ/2})| strictly increases from 0 to β as θ runs from 0 to 2θ_{0} . Since arg g(e^{iθ}) = arg f′(e^{iθ/2}) = \frac{1}{2}(π − θ) strictly decreases from \frac{1}{2}π to φ_{0} = \frac{1}{2}π − θ_{0} as θ runs from 0 to 2θ_{0} , it follows that g maps the arc {e^{iθ} : 0 ≤ θ ≤ 2θ_{0}} one-to-one onto an analytic Jordan arc δ_{1} lying in the domain Uβ^{+} = {w ∈ U_{β} : Re w > 0, Im w > 0} and connecting the points w = 0 and w = βe^{iφ_{0}} = f′(e^{iθ_{0}}).

Since f(U^{∗}) is starlike w.r.t. w = 0,

Re \frac{e^{iθ}f′(e^{iθ})}{f(e^{iθ})} ≥ 0

for 0 ≤ θ ≤ 2π. Since f(U^{∗}) is symmetric w.r.t. the coordinate axes, the latter inequality shows that −π ≤ arg f′(e^{iθ}) ≤ π − θ_{0} for 0 ≤ θ ≤ θ_{0} . This combined with (3.10) implies that g maps the arc {e^{iθ} : 2θ_{0} ≤ θ ≤ π} one-to-one onto the circular arc δ_{2} = {βe^{iφ} : 0 ≤ φ ≤ φ_{0}} such that g(e^{2iθ_{0}}) = βe^{iφ_{0}} , g(−1) = β. By symmetry, g maps the arc \{e^{iθ} : −2θ_{0} ≤ θ ≤ 0\} onto δ̄_{1} = \{w : \overline{w} ∈ δ_{1}\} and the arc \{e^{iθ} : π ≤ θ ≤ 2π −2θ_{0}\} onto δ̄_{2} = \{w : \overline{w} ∈ δ_{2}\}. Thus, g maps the unit circle T one-to-one onto a closed Jordan arc δ composed by δ_{1} , δ_{2} , δ̄_{2} , and δ̄_{1} . Since g(∞) = f′(∞) = 1 the argument principle implies that g maps U^{∗} conformally and one-to-one onto a simply connected domain G which contains 1 and which is bounded by δ. The domain G for β = 1.7 is plotted in Figure 4.

The above-mentioned properties of δ_{1} show that G is circularly symmetric w.r.t. the positive real axis. Therefore by Lemma 4, |g′(e^{iθ})| = |f′′(e^{iθ/2})| strictly decreases in 2θ_{0} < θ < π.

(2) Considering boundary values of φ we have

Re φ(e^{iθ}) = Re e^{iθ}f′(e^{iθ}) = 0 for 0 ≤ θ ≤ θ_{0}

since Im f(e^{iθ}) = 0 for such θ. Since Im φ(e^{iθ}) = |f′(e^{iθ})| strictly increases in 0 ≤ θ ≤ θ_{0} , φ maps ln^{++} continuously and one-to-one onto the vertical segment {w : Re w = 0, 0 ≤ Im w ≤ β}.

For θ_{0} ≤ θ ≤ \frac{1}{2}π, |φ(e^{iθ})| = β and

\frac{∂}{∂θ} arg φ(e^{iθ}) = \frac{∂}{∂θ} Im log(e^{iθ}f′(e^{iθ})) = 1 + \frac{e^{iθ}f′′(e^{iθ})}{f′(e^{iθ})} = 1 − β^{−1}|f′′(e^{iθ})|,

since e^{iθ}f′′(e^{iθ})/f′(e^{iθ}) is real non-positive for θ_{0} ≤ θ ≤ \frac{1}{2}π.

(∂/∂θ) arg φ(e^{iθ}) changes its sign at most once in the interval θ_{0} < θ < \frac{1}{2}π since |f′′(e^{iθ})| strictly decreases in θ_{0} ≤ θ ≤ \frac{1}{2}π.

We have shown in (1) that arg f′(e^{iθ}) decreases from \frac{1}{2}π − θ_{0} to 0 when θ runs from θ_{0} to \frac{1}{2}π. Since arg φ(e^{iθ_{0}}) = \frac{1}{2}π, the latter implies that

(3.13)  0 < θ_{0} < arg φ(e^{iθ}) = θ + arg f′(e^{iθ}) < π − θ_{0}

for θ_{0} < θ < \frac{1}{2}π.

We claim that there is θ_{1} , θ_{0} < θ_{1} < \frac{1}{2}π such that

(3.14)

\frac{∂}{∂θ} arg φ(e^{iθ}) < 0 if θ_{0} < θ < θ_{1},
\frac{∂}{∂θ} arg φ(e^{iθ}) > 0 if θ_{1} < θ < \frac{1}{2}π.

Suppose to the contrary that (∂/∂θ) arg φ(e^{iθ}) ≤ 0 for all θ_{0} < θ < \frac{1}{2}π. Then, we would have that arg φ(e^{iθ}) monotonically decreases over θ_{0} < θ < \frac{1}{2}π. Since φ(e^{iθ_{0}}) = φ(i) = iβ, we would have

Δ arg φ(e^{iθ}) |_{θ_{0}}^{π/2}= −2πk for some k ∈ N

contradicting (3.13). The assumption (∂/∂θ) arg φ(e^{iθ}) ≥ 0 for all θ_{0} < θ < \frac{1}{2}π leads to the same contradiction. Since |f′′(e^{iθ})| strictly decreases in θ_{0} ≤ θ ≤ \frac{1}{2}π, the claim follows.

Let ψ = arg φ(e^{iθ_{1}}). The previous analysis shows that θ_{0} < ψ < \frac{1}{2}π and φ maps each of the arcs \{e^{iθ} : θ_{0} ≤ θ ≤ θ_{1}\} and \{e^{iθ} : θ_{1} ≤ θ ≤ \frac{1}{2}π\} continuously and one-to-one onto the arc \{βe^{it} : ψ ≤ t ≤ \frac{1}{2}π\} such that φ(e^{iθ_{1}}) = βe^{iψ}. The symmetry principle now yields that φ maps the unit circle T continuously and one-to-one in the sense of boundary correspondence onto the boundary of the domain Ω(β, ψ). Hence by the argument principle, φ maps U^{∗} conformally and univalently onto Ω(β, ψ).

The normalization f′(∞) = 1 leads after some work left to the interested readers to the relation ψ = \frac{1}{2} cos^{−1}(8β^{−1} − 8β^{−2} − 1).

4. Proofs of Theorems 1 and 2

To prove uniqueness of the extremal function in Σ_{d}, we assume that for some fixed d, 1 < d < 2, there are distinct extremals f_{1} and f_{2} . By Lemma 9, zf′_{k}(z) = g_{β_{k}}(z) for some 1 < β_{1} < 2, 1 < β_{2} < 2, where g_{β_{k}} maps U^{∗} conformally onto the domain Ω(β_{k}) = C \ F (β_{k}, ψ(β_{k})). To be explicit, assume that β_{1} < β_{2} . The domains Ω(β_{1}) and Ω(β_{2}) satisfy the conditions of Lemma 6. Therefore,

(4.1)  g_{β_{1}}(r) > g_{β_{2}}(r)

for all r > 1.

Since f_{k}(U^{∗}) is doubly symmetric w.r.t. the real and imaginary axes, it follows that a_{0}(f_{k}) = 0 for k = 1, 2. Hence, it follows from the normalization in (1.5) that

(4.2)  lim_{R→∞}(f_{1}(R) − f_{2}(R)) = 0.

On the other hand, since f_{1}(1) = f_{2}(1) = d, we have

f_{1}(R) − f_{2}(R) = ∫_{1}^{R} t^{−1}(g_{β_{1}}(t) − g_{β_{1}}(t)) dt

and (4.1) implies that the integrand is positive and hence, that f_{1}(R) − f_{2}(R) is a (positive) increasing function of R, which contradicts (4.2).

Let f_{d} denote the unique extremal function in Σ_{d}. To find f_{d} explicitly, we represent φ(z; d) = zf′_{d}(z) as

φ(z; d) = (F_{d^{−2}}(z^{−2}))^{−1/2},

where F_{p}(ζ) = ζ + a_{2}(F_{p})ζ^{2} + ⋯, \frac{1}{4} ≤ p ≤ 1, is the univalent function in the standard class S that maps the unit disk U onto the domain C \ ((−∞, −p] ∪ \{pe^{iτ} : |τ − π| ≤ α\}) with α = cos^{−1}(8\sqrt{p} − 8p − 1). It is well known that F_{p} is extremal in a number of problems, for instance in the problem on max |a_{2}(f)| studied by E. Netanyahu [N] and T. Suffridge [S], on the subclass of functions f ∈ S that cover the disk U_{p}. Using an explicit expression for F_{p}, see for example, [S], we get (1.9) and after an integration (1.8).

The integral in (1.8) can be evaluated in terms of elementary functions. We leave to the interested readers to check (one can use “Mathematica” or “Maple”) that f_{d}(1) coincides with the right-hand side in (1.7), which in this case is equivalent to the equality f_{d}(1) = d. Since for each 1 ≤ d ≤ 2 the extremal function is unique in Σ_{d}, (1.7) has a unique solution β = β(d) on the interval 1 ≤ β ≤ 2; this also follows easily from the monotonicity of the right-hand side of (1.7).

To evaluate the maximal omitted area A(d) = Area (E_{f_{d}}), we apply a standard line integral formula and the fact that Im(\overline{w} dw) = 0 on the non-free boundary. We have

A(d) = \frac{1}{2} Im ∫_{∂E_{f_{d}}} \overline{w} dw = \frac{1}{2} Im ∫_{L_{f_{d}}} \overline{w} dw = \frac{1}{2} Re ∫_{l_{f_{d}}} \overline{f_{d}(e^{iθ})}e^{iθ}f′_{d}(e^{iθ}) dθ.

Since |f′_{d}|^{2} = β^{2} on l_{f_{d}}, we obtain

A(d) = \frac{β^{2}}{2} Re lim_{ε→+0}\{∫_{ε}^{π−ε} \frac{f_{d}(e^{iθ})}{e^{iθ}f′_{d}(e^{iθ})} dθ + ∫_{π+ε}^{2π−ε} \frac{f_{d}(e^{iθ})}{e^{iθ}f′_{d}(e^{iθ})} dθ\}

= \frac{β^{2}}{2} Im\{∫_{T} \frac{f_{d}(z)}{z^{2}f′_{d}(z)} dz − πiRes [\frac{f_{d}(z)}{z^{2}f′_{d}(z)}, 1] − πiRes [\frac{f_{d}(z)}{z^{2}f′_{d}(z)}, −1]\},

where ∫_{T} f_{d}/(z^{2}f′_{d}) dz is understood as the Cauchy principal value. The function f_{d}/(z^{2}f′_{d}) has simple poles at z = 1 and z = −1. Computing the integral and residues, we obtain

A(d) = π[β^{2} − 2d(β − 1)],

which implies (1.6). This finishes the proof of Theorem 2.

An isoperimetric inequality for logarithmic capacity 435

To deduce (1.4), we write (1.6) in an invariant form:

(4.3)  \frac{area E}{\frac{1}{4}π diam^{2} E} ≤ p^{−2}[β^{2}(p) − 2p(β(p) − 1)]

with p = diam E/(2 cap E), where 1 ≤ β(p) ≤ 2 is defined by (1.7) with d replaced by p. Since the maximal omitted area A(d) strictly decreases, the expression in the brackets in (4.3) decreases and therefore the right-hand side of (4.3) itself decreases from 1 to 0 when p runs from 1 to 2. Therefore there is a function p = Ψ_{1}(s) inverse to s = p^{−2}[β(p)^{2} − 2p(β(p) − 1)]. Let Ψ(s) = 1/Ψ_{1}(s). Since the inverse Ψ_{1} is decreasing, (4.3) leads to the inequality

p ≤ Ψ_{1}(area E/(\frac{1}{4}π diam^{2} E)),

which is equivalent to (1.4) with equality only for the continua described in Theorem 1.

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Received 25 October 2001