Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 193–206

MÖBIUS MODULUS OF RING DOMAINS IN \overline{R}^{n}

Zair Ibragimov

University of Michigan, Department of Mathematics Ann Arbor, MI 48109, U.S.A.; ibragim@umich.edu

Abstract. We introduce a new Möbius invariant modulus for ring domains R in \overline{R}^{n} which coincides with the usual modulus whenever R is a Möbius annulus, i.e.,

f(R) = {x ∈ R^{n} : 1 < |x| < t}

for some Möbius transformation f of \overline{R}^{n} and some t > 1. We obtain a sharp upper bound for the Möbius modulus of a ring R which separates two pairs {a, b} and {c, d} of distinct points in \overline{R}^{n}. Our result proves a conjecture made by M. Vuorinen in 1992 [14].

1. Introduction

Notation. We denote by R^{n} the n-dimensional Euclidean space and by {e_{1}, e_{2}, …, e_{n}} its standard basis. The one-point compactification R^{n} ∪ {∞} of R^{n} is denoted by R^{n}. The open and closed balls of radius r > 0 and centered at x ∈ R^{n} are denoted by B^{n}(x, r) and \overline{B}^{n}(x, r), respectively. S^{n−1}(x, r) is a sphere of radius r > 0 and centered at x ∈ R^{n}. The closed segment between x ∈ R^{n} and y ∈ R^{n} is denoted by [x, y]. For x ∈ R^{n}, x ≠ 0, we set

[x, ∞] = {tx : t ≥ 1} ∪ {∞}.

The group of all Möbius transformations of R^{n} is denoted by Möb(R^{n}).

A ring is a domain R ⊂ R^{n} whose complement is the union of two disjoint non-degenerate compact connected sets. A ring with complementary components C_{1} and C_{2} is denoted by R(C_{1}, C_{2}). A ring R(C_{1}, C_{2}) is said to separate the sets E and F if E ⊂ C_{1} and F ⊂ C_{2} . Hence a ring R(C_{1}, C_{2}) separates the complementary components of a ring R(E, F) if E ⊂ C_{1} and F ⊂ C_{2} .

If R is a ring in R^{2} , then R can be mapped conformally onto a circular annulus

{z ∈ C : 1 < |z| < t}

and the modulus of R is defined to be log t.

In R^{n}, n > 2, by Liouville’s theorem the Möbius transformations are the only conformal mappings in R^{n}. A ring R is said to be a Möbius annulus if

f(R) = {x ∈ R^{n} : 1 < |x| < t} for some f ∈ Möb(R^{n}) and t > 1.

2000 Mathematics Subject Classification: Primary 30C35; Secondary 30C75.

The points f^{−1}(0) and f^{−1}(∞) are called relative centers of the Möbius annulus R. The modulus of such a ring R is defined as log t.

In general, the modulus of a ring R = R(C_{1}, C_{2}) is defined as follows. Let Γ be the family of all curves joining C_{1} and C_{2} in R and let F (Γ) be the set of all non-negative Borel functions

ϱ: R^{n} → R^{1} such that ∫_{γ} ϱ ds ≥ 1

for every locally rectifiable curve γ ∈ Γ. Then the modulus of the curve family Γ is defined as

M(Γ) = inf_{ϱ∈F (Γ)} ∫_{R^{n}} ϱ^{n} dm.

Observe that since C_{1} and C_{2} are non-degenerate, 0 < M(Γ) < ∞ by [13, 11.5 and 11.10]. The modulus of R is defined as

mod R = [\frac{ω_{n−1}}{M(Γ)}]^{1/(n−1)}.

See, for instance, [1, 8.30]. Here ω_{n−1} is the surface area of the unit sphere in R^{n}.

The rings of Grötzsch and Teichmüller play an important role in the theory of quasiconformal mappings. The complementary components of Grötzsch ring R_{G}(s), s > 1, are \overline{B}^{n}(0, 1) and [se_{1}, ∞] while those of Teichmüller ring R_{T} (t), t > 0, are [−e_{1}, 0] and [te_{1}, ∞].

For the convenience of the reader we recall some properties of the modulus of Grötzsch and Teichmüller rings. The following functional relation holds.

(1.1)  mod R_{T} (t) = 2 mod R_{G}(\sqrt{t + 1} ).

See [4] and [1, 8.32 and 8.37(1)]. The function

(1.2)  mod R_{T} (t) − log(t + 1)

is a nondecreasing function in (0, ∞) and

(1.3)  lim_{t→∞}(mod R_{T} (t) − log(t + 1)) = log λ_{n}^{2} < ∞.

Here λ_{n} is a constant which depends only on n. See, for instance, [4] and [1, 8.38].

Our main focus in this paper is the following extremal problem of Teichmüller. Let a, b, c, d be distinct points in R^{n}. Among all the rings which separate the sets {a, b} and {c, d} it is required to find one with the largest modulus.

Möbius modulus of ring domains in R^{n} 195

For n = 2, this problem was considered by O. Teichmüller [12] in 1938 and a complete solution was given by M. Schiffer [11] in 1946. In this case the points a, b, c, d can be normalized so that a = −1, b = 1, c = ξ, d = −ξ, where ξ ∈ \overline{B}^{2}(0, 1) is a unique point with

\frac{c − a}{c − b} · \frac{d − b}{d − a} = (\frac{ξ + 1}{ξ − 1})^{2}.

If ℛ(ξ) = R(C_{1}(ξ), C_{2}(ξ)) is an extremal ring, i.e., a ring with the largest modulus, then

h_{1}(C_{1}(ξ)) = C_{1}(ξ), h_{1}(C_{2}(ξ)) = C_{2}(ξ)

and

h_{2}(C_{1}(ξ)) = C_{2}(ξ), h_{2}(C_{2}(ξ)) = C_{1}(ξ),

where

h_{1}(z) = −z  and  h_{2}(z) = \frac{ξ}{z} .

In particular,

(1.4)  0 ∈ C_{1}(ξ)  and  ∞ ∈ C_{2}(ξ).

See [8, pp. 199–200].

For n > 2, Teichmüller’s problem is solved only when the points a, b, c, d lie on a circle or a line in this order. In this case the points a, b, c, d can be normalized so that

a = −e_{1}, b = 0, c = te_{1}, d = ∞, where t = \frac{|b − c| |d − a|}{|b − a| |d − c|}.

Then by means of a spherical symmetrization one shows that Teichmüller’s ring R_{T} (t) is an extremal ring. See, for instance, [4], [10] and [1, Theorem 8.46].

When the points a, b, c, d do not lie on a circle or a line in this order, the problem is still open. In general the points a, b, c, d can be normalized so that

a = 0, b = e_{1}, c = x, d = ∞ where x ∈ R^{n} and |x| = \frac{|a − c| |b − d|}{|a − b| |c − d|}.

M. Vuorinen has considered a ring R_{0} whose complementary components are some circular arc joining the points 0 and e_{1} and some ray emanating from the point x [14]. This ring coincides (up to a Möbius transformation) with Teichmüller’s ring when the points a, b, c, d lie on a circle or a line in this order. It follows from Theorem 3.16 [14] that there exists a Möbius annulus A separating the complementary components of R_{0} and such that

mod A = arccosh (|x| + |x − e_{1}|).

Due to the monotonicity of the modulus we then have

arccosh (|x| + |x − e_{1}|) = mod A ≤ mod R_{0}.

In order to assure that arccosh (|x| + |x − e_{1}|) is the sharpest lower bound for mod R_{0} obtained in this manner, Vuorinen was led to the following conjecture.

Conjecture 1.5. For x ∈ R^{n} \ [0, e_{1}],
max_{A} mod A = arccosh (|x| + |x − e_{1}|),

where the maximum is taken over all Möbius annuli A which separate the sets {0, e_{1}} and {x, ∞}.

In this paper we consider a new measure for ring domains called the Möbius modulus. Our main result, Theorem 3.8 below, shows that Teichmüller’s problem has a complete solution when considered with respect to the Möbius modulus. As a corollary to Theorem 3.8 we settle the conjecture of Vuorinen. After this paper was submitted the referee pointed out that an alternative proof of the conjecture was also given in [3].

2. Some results on the cross-ratio

The main result of this section is Theorem 2.16. The cross-ratio of a quadruple a, b, c, d of points in R^{n} with a ≠ b and c ≠ d is defined as follows. If a, b, c, d ∈ R^{n}, then

(2.1)  |a, b, c, d| = \frac{|a − c| |b − d|}{|a − b| |c − d|}.

Otherwise we omit the terms containing ∞. For example,

(2.2)  |a, b, c, ∞| = \frac{|a − c|}{|a − b|}.

A homeomorphism f: R^{n} → R^{n} belongs to Möb(R^{n}) if and only if

(2.3)  |f(a), f(b), f(c), f(d)| = |a, b, c, d|

for all quadruples a, b, c, d in R^{n}. See [2, Theorem 3.2.7]. For a quadruple a, b, c, d in R^{n} we put

(2.4)  σ(a, b, c, d) = |a, b, c, d| + |b, a, c, d|.

Hence

(2.5)  σ(a, b, c, d) = \frac{|a − c| |b − d| + |a − d| |b − c|}{|a − b| |c − d|} ≥ 1

with equality if and only if the points a, c, b, d lie on a circle or a line in this order. A simple computation shows that

(2.6)  |a, b, c, d| = \frac{σ(a, b, c, d) + 1}{σ(a, c, b, d) + 1}  and  σ(a, b, c, d) = \frac{|a, d, c, b| + 1}{|d, a, c, b|}.

It follows from (2.3) and (2.6) that a homeomorphism f: R^{n} → R^{n} belongs Möb(R^{n}) if and only if

(2.7)  σ(f(a), f(b), f(c), f(d)) = σ(a, b, c, d)

for all quadruples a, b, c, d in R^{n}.

The following two lemmas are used in the proof of Theorem 2.16. The first lemma is an immediate consequence of Corollary 7.25 [1].

Lemma 2.8. Let a, b, c, d ∈ R^{n} be distinct points. Let f be a Möbius transformation such that

f(a) = −e_{1}, f(b) = e_{1}, f(c) = −w and f(d) = w

for some w ∈ R^{n} with |w| ≥ 1. Then

|w| = p + \sqrt{p^{2} − 1} , where p = σ(a, b, c, d).

Proof. Let g ∈ Möb(R^{n}) be the inversion in S^{n−1}(0, 1). Then by applying Corollary 7.25 [1] to the composition g ∘ f and using (2.6) we obtain

\frac{1}{|w|} = \frac{|d, a, c, b|}{1 + |a, d, c, b| + \sqrt{(1 + |a, d, c, b|)^{2} − (|d, a, c, b|)^{2}}} = \frac{1}{p + \sqrt{p^{2} − 1}}

as required.

The next lemma is an extension of a special case of Lemma 2.12 [6].

Lemma 2.9. If a_{1} , a_{2} , a_{3} , a_{4} are distinct points in R^{n} with

(2.10)  |a_{1}| = |a_{2}| = s < t = |a_{3}| = |a_{4}|,

then

(2.11)  \frac{s^{2} + t^{2}}{2st} ≤ σ(a_{1}, a_{2}, a_{3}, a_{4}).

Equality holds if and only if

(2.12)  a_{1} + a_{2} = a_{3} + a_{4} = 0.

Proof. Define c_{ij} as the cosine of the angle ∠(a_{i}0a_{j}) and set

g(u) = \frac{N(u)}{D}, where D = (1 − c_{12})^{1/2}(1 − c_{34})^{1/2}

and

N(u) = (u − c_{13})^{1/2}(u − c_{24})^{1/2} + (u − c_{14})^{1/2}(u − c_{23})^{1/2}.

Note that D ≤ 2 and that D = 2 if and only if (2.12) holds. Next

\frac{d}{du}((u − c_{ij})^{1/2}(u − c_{kl})^{1/2}) = \frac{1}{2}(\frac{(u − c_{kl})^{1/2}}{(u − c_{ij})^{1/2}} + \frac{(u − c_{ij})^{1/2}}{(u − c_{kl})^{1/2}} ) ≥ 1,

whence

(2.13)  g′(u) = \frac{N′(u)}{D} ≥ 1  for u > 1.

Since

g(1) = σ(a_{1}, a_{2}, \frac{s}{t}a_{3}, \frac{s}{t}a_{4}) ≥ 1,

we have

(2.14)  g(u) = ∫_{1}^{u} g′(r) dr + g(1) ≥ u  for all u ≥ 1.

In particular,

(2.15)  σ(a_{1}, a_{2}, a_{3}, a_{4}) = g(\frac{s^{2} + t^{2}}{2st} ) ≥ \frac{s^{2} + t^{2}}{2st}

which completes the proof of (2.11).

Assume next that (2.11) holds with equality and set

v = \frac{s^{2} + t^{2}}{2st} .

Then (2.15) implies that g(v) = v. Using the differentiability of g along with (2.14) we obtain

g′(v) = lim_{u→v} \frac{g(v) − g(u)}{v − u} ≤ lim_{u→v} \frac{v − u}{v − u} = 1

and hence using (2.13) we conclude that g′(v) = 1. Since N′(v) ≥ 2 and D ≤ 2, the equality g′(v) = 1 implies D = 2 which, as noted above, implies (2.12) as required.

Finally, a simple computation shows that (2.12) implies that (2.11) holds with equality.

Theorem 2.16. Let a, b, c, d ∈ R^{n} be distinct points and p = σ(a, b, c, d). Then for all distinct pairs {u, v} of points in R^{n} \ {a, b, c, d}

(2.17)  min\{|u, a, c, v|, |u, a, d, v|, |u, b, c, v|, |u, b, d, v|\} ≤ p + \sqrt{p^{2} − 1} .

Moreover, if p > 1, then there exists a unique pair {u, v} for which the equality holds.

Proof. Let u, v ∈ R^{n} \ {a, b, c, d} be distinct points. Since (2.17) is invariant under the elements of Möb(R^{n}), we can assume that u = 0 and v = ∞. Then

min\{|u, a, c, v|, |u, a, d, v|, |u, b, c, v|, |u, b, d, v|\} = \frac{min\{|c|, |d|\}}{max\{|a|, |b|\}} = \frac{t}{s}.

Since p + \sqrt{p^{2} − 1} ≥ 1, there is nothing to prove if s ≥ t. Hence we may assume that s < t. Then (2.17) is equivalent to

(2.18)  \frac{s^{2} + t^{2}}{2st} ≤ σ(a, b, c, d).

Let

S^{n−1}(v_{1}, r_{1}) ⊂ \overline{B}^{n}(0, s) and S^{n−1}(v_{2}, r_{2}) ⊂ R^{n} \ B^{n}(0, t)

be spheres such that a, b ∈ S^{n−1}(v_{1}, r_{1}) and c, d ∈ S^{n−1}(v_{2}, r_{2}). Then we have R_{1} ⊂ R_{2} , where

R_{1} = R(\overline{B}^{n}(0, s), R^{n} \ B^{n}(0, t))

and

R_{2} = R(\overline{B}^{n}(v_{1}, r_{1}), R^{n} \ B^{n}(v_{2}, r_{2})).

In particular,

(2.19)  mod R_{1} ≤ mod R_{2}.

Choose h ∈ Möb(R^{n}) that maps S^{n−1}(v_{1}, r_{1}) and S^{n−1}(v_{2}, r_{2}) onto concentric spheres S^{n−1}(0, s′) and S^{n−1}(0, t′), respectively. Then

0 < |h(a)| = |h(b)| = s′ < t′ = |h(c)| = |h(d)|

and we have

(2.20)  log \frac{t}{s} = mod R_{1} ≤ mod R_{2} = log \frac{t′}{s′} .

Using (2.11) we now have

\frac{s^{2} + t^{2}}{2st} ≤ \frac{s′^{2} + t′^{2}}{2s′t′} ≤ σ(h(a), h(b), h(c), h(d)) = σ(a, b, c, d)

which proves (2.18) and hence the first part of the theorem.

Notice that equality in (2.18) implies that |a| = |b| = s and |c| = |d| = t.

To prove the second part of the theorem, we let f ∈ Möb(R^{n}) be such that

f(a) = −e_{1}, f(b) = e_{1}, f(c) = −w and f(d) = w

for some w ∈ R^{n} with |w| ≥ 1. Since p > 1, we have |w| = p + \sqrt{p^{2} − 1} > 1 by Lemma 2.8. Then the equality in (2.17) holds if we take u = f^{−1}(0) and v = f^{−1}(∞). This establishes the existence of the pair {u, v}.

To prove the uniqueness, we assume that {u_{1}, v_{1}} is another pair for which the equality in (2.17) holds and show that u_{1} = u and v_{1} = v. Let g ∈ Möb(R^{n}) be such that

g(u_{1}) = 0, g(v_{1}) = ∞ and g(a) = −e_{1}.

By our assumption we have

min{|0, −e_{1}, g(c), ∞|,|0, −e_{1}, g(d), ∞|, |0, g(b), g(c), ∞|, |0, g(b), g(d), ∞|}

= \frac{min\{|g(c)|, |g(d)|\}}{max\{|g(a), |g(b)|\}} = p + \sqrt{p^{2} − 1} .

Hence

|g(a)| = |g(b)| < |g(c)| = |g(d)|

as we have noted above. Then using (2.12) we have

g(a) + g(b) = g(c) + g(d) = 0.

Hence

g(a) = −e_{1}, g(b) = e_{1}, g(c) = −z and g(d) = z

for some z ∈ R^{n} and by Lemma 2.8 we have |z| = p + \sqrt{p^{2} − 1} = |w|. Since

|f(a), f(b), f(c), f(d)| = |a, b, c, d| = |g(a), g(b), g(c), g(d)|

and

|f(b), f(a), f(c), f(d)| = |b, a, c, d| = |g(b), g(a), g(c), g(d)|,

we have

|w − e_{1}| = |z − e_{1}| and |w + e_{1}| = |z + e_{1}|.

Hence the angle ∠(w0e_{1}) is equal to the angle ∠(z0e_{1}). By means of a preliminary rotation about the e_{1} -axis if necessary, we can assume that w = z. Hence f^{−1} and g^{−1} agree on a set {−e_{1}, e_{1}, −w, w} and consequently they agree on a 2dimensional (1-dimensional, if w lies on the e_{1} -axis) subspace of R^{n} containing these points. In particular,

u = f^{−1}(0) = g^{−1}(0) = u_{1}  and  v = f^{−1}(∞) = g^{−1}(∞) = v_{1}

as required.

Remark 2.21. The hypothesis p > 1 in the uniqueness part of Theorem 2.16 cannot be removed. We now show that if p = 1 the uniqueness part fails.

Indeed, if a = −e_{1} , b = te_{1} , c = −te_{1} , d = e_{1} for some 0 < t < 1, then σ(a, b, c, d) = 1. But equality in (2.17) holds for all points {u, v} with

|u − a| = |v − a|, |u − b| = |v − b|  and  |u − c| = |v − c|, |u − d| = |v − d|.

3. Möbius modulus of ring domains

In this section we define the Möbius modulus of rings. Our main result is Theorem 3.8 which gives a solution to Teichmüller’s extremal problem for the Möbius modulus. As a corollary to Theorem 3.8 we settle the conjecture of Vuorinen. We will then establish a relationship between the usual and the Möbius moduli of rings and compute the Möbius modulus of Grötzsch and Teichmüller rings.

Definition 3.1. Let R = R(C_{1}, C_{2}) be a ring in R^{n}. The quantity

(3.2)  mod_{M}R = max_{u,v∈\overline{R}^{n}} min_{x∈C_{1}, y∈C_{2}}|log \frac{|u − y| |x − v|}{|u − x| |y − v|}|

is called the Möbius modulus of R.

Observe that if R is any ring and A is a Möbius annulus separating the complementary components of R, then

(3.3)  mod_{M}R ≥ mod A > 0.

Indeed, we can assume that

A = B^{n}(0, t) \ \overline{B}^{n}(0, 1)

for some t > 1. Then

mod_{M}R ≥ min_{x∈C_{1}, y∈C_{2}}|log \frac{|0 − y| |x − ∞|}{|0 − x| |y − ∞|}| = min_{x∈C_{1}, y∈C_{2}}|log \frac{|y|}{|x|}| ≥ mod A.

On the other hand, if mod_{M}R > 0, then there exists a Möbius annulus A separating the complementary components of R such that

(3.4)  mod A = mod_{M}R.

Indeed, let u, v ∈ R^{n} be a pair with

mod_{M}R = min_{x∈C_{1}, y∈C_{2}}|log \frac{|u − y| |x − v|}{|u − x| |y − v|}| > 0.

We can assume that u = 0 and v = ∞. Then

mod_{M}R = min_{x∈C_{1}, y∈C_{2}}|log \frac{|y|}{|x|}| = log \frac{s}{t},

where

s = min{|x| : x ∈ C_{2}}  and  t = max{|x| : x ∈ C_{1}}.

Then the ring

A = {x ∈ R^{n} : t < |x| < s}

is the required Möbius annulus.

Thus we have the following remark to Definition 3.1.

Remark 3.5. Let R be any ring with mod_{M}R > 0. Then

(3.6)  mod_{M}R = max_{A} mod A,

where the maximum is taken over all Möbius annuli A which separate the complementary components of R. In particular, if R is a Möbius annulus, then

(3.7)  mod_{M}R = mod R.

Ring domains with separating euclidean or Möbius annuli were studied in [7] (n = 2) and [14] (n ≥ 2), respectively.

Theorem 3.8. Let a, b, c, d ∈ R^{n} be distinct points. Then

(3.9)  max_{R} mod_{M}R = arccosh (σ(a, b, c, d)),

where the maximum is taken over all rings R which separate the sets {a, b} and {c, d}.

Proof. Let R = R(C_{1}, C_{2}) be a ring with a, b ∈ C_{1} and c, d ∈ C_{2} and assume that mod_{M}R > 0. Let u, v ∈ R^{n} and x_{0} ∈ C_{1} , y_{0} ∈ C_{2} be such that

mod_{M}R = |log |u, x_{0}, y_{0}, v||.

By performing a preliminary Möbius transformation we can assume that u = 0, v = ∞ and |x_{0}| < |y_{0}|. Then the Möbius annulus

A = B^{n}(0, |y_{0}|) \ \overline{B}^{n}(0, |x_{0}|)

separates C_{1} and C_{2} . In particular, we have

max{|a|, |b|} ≤ |x_{0}| < |y_{0}| ≤ min{|c|, |d|}.

Hence by Theorem 2.16 we have

mod_{M}R = log \frac{|y_{0}|}{|x_{0}|} ≤ log(min\{\frac{|c|}{|a|}, \frac{|d|}{|a|}, \frac{|c|}{|b|}, \frac{|d|}{|b|}\}) ≤ arccosh (σ(a, b, c, d)).

The equality holds for the ring R_{0} = R(C_{1}, C_{2}) where

C_{1} = f^{−1}([−w, ∞] ∪ [w, ∞])  and  C_{2} = f^{−1}([−e_{1}, e_{1}])

and f is a Möbius transformation such that

f(a) = w, f(b) = −w, f(c) = e_{1} and f(d) = −e_{1}

for some w ∈ R^{n} with |w| ≥ 1. Indeed, for u = f^{−1}(∞) and v = f^{−1}(0) we have

mod_{M}R_{0} = mod_{M}f(R_{0}) ≥ min{|log |∞, x, y, 0|| : x ∈ C_{1}, y ∈ C_{2}}
= log |w| = arccosh (σ(a, b, c, d))

using Lemma 2.8. Hence

mod_{M}R_{0} = arccosh (σ(a, b, c, d)).

completing the proof.

Next we have the following two corollaries of Theorem 3.8. The first one settles the conjecture of Vuorinen.

Corollary 3.10. For x ∈ R^{n} \ [0, e_{1}],
(3.11)  max_{A} mod A = arccosh (|x| + |x − e_{1}|),

where the maximum is taken over all Möbius annuli A which separate the sets {0, e_{1}} and {x, ∞}.

Proof. Since

|x| + |x − e_{1}| = σ(0, e_{1}, x, ∞),

Theorem 3.8 along with (3.6) imply that

max_{A} mod A = mod_{M}R_{0} = arccosh (σ(0, e_{1}, x, ∞)) = arccosh (|x| + |x − e_{1}|),

where R_{0} = R([0, e_{1}], [x, ∞]).

The next corollary gives a characterization of Möbius transformations of R^{n} in terms of the Möbius modulus of rings. A similar type of characterization in terms of the modulus of rings is given in [5].

Corollary 3.12. A homeomorphism f: R^{n} → R^{n} belongs to Möb(R^{n}) if and only if

mod_{M}f(R) = mod_{M}R

for all rings R in R^{n}.

Proof. The necessity part follows from (2.3) and Definition 3.1. For the sufficiency part it is enough to show that

σ(f(a), f(b), f(c), f(d)) = σ(a, b, c, d)

for all quadruples a, b, c, d ∈ R^{n}. Given a, b, c, d, we let R = R(C_{1}, C_{2}) be a maximal ring, i.e., a ring with

mod_{M}R = arccosh (σ(a, b, c, d)).

Then using Theorem 3.8 and the fact that f(a), f(b) ∈ f(C_{1}) and f(c), f(d) ∈ f(C_{2}) we get

arccosh (σ(a, b, c, d)) = mod_{M}R(f(C_{1}), f(C_{2}))
≤ arccosh (σ(f(a), f(b), f(c), f(d)))

which implies

σ(a, b, c, d) ≤ σ(f(a), f(b), f(c), f(d)).

By applying the same argument to f^{−1} we get

σ(f(a), f(b), f(c), f(d)) ≤ σ(a, b, c, d)

as required.

We have the following relation between the modulus and the Möbius modulus of a ring R.

Lemma 3.13. For any ring R = R(C_{1}, C_{2}) in R^{n} we have
(3.14)  mod_{M}R ≤ mod R ≤ mod_{M}R + c(n),

where c(n) is a constant depending only on n.

Proof. It follows from (1.2) and (1.3) that

mod R_{T} (t) ≤ log(λ_{n}^{2}(t + 1)) for all t > 0.

The first inequality in (3.14) follows from the monotonicity of the modulus along with Corollary 3.5. To show the second inequality in (3.14), let log |x, u, v, y| = mod_{M}R and

log |x′, u′, v′, y′| = max_{u∈C_{1}, v∈C_{2}} min_{x∈C_{1}, y∈C_{2}}|log \frac{|u − y| |x − v|}{|u − x| |y − v|}|.

Then |x′, u′, v′, y′| ≤ |x, u, v, y| and by Theorem 8.46 [1] we have

mod R ≤ mod R_{T} (|x′, u′, v′, y′|) ≤ mod R_{T} (|x, u, v, y|)
≤ log(2λ_{n}^{2}|x, u, v, y|) = mod_{M}R + log(2λ_{n}^{2}).

Hence the lemma holds with c(n) = log(2λ_{n}^{2}).

Finally, we compute the Möbius modulus of Grötzsch and Teichmüller rings.

Example 3.15. For s > 1

mod_{M}R_{G}(s) = arccosh (s).

Proof. Since −e_{1}, e_{1} ∈ \overline{B}^{n}(0, 1) and se_{1}, ∞ ∈ [se_{1}, ∞], Theorem 3.8 implies that

mod_{M}R_{G}(s) ≤ arccosh (s).

Equality holds for

u_{0} = (s − \sqrt{s^{2} − 1} )e_{1}  and  v_{0} = (s + \sqrt{s^{2} − 1} )e_{1}.

See (3.2).

Example 3.16. For t > 0

mod_{M}R_{T} (t) = arccosh (2t + 1).

Proof. Since −e_{1}, 0 ∈ [−e_{1}, 0] and te_{1}, ∞ ∈ [te_{1}, ∞], Theorem 3.8 implies that

mod_{M}R_{T} (t) ≤ arccosh (2t + 1).

Equality holds for

u_{0} = (−\sqrt{t(t + 1)} + t)e_{1}  and  v_{0} = (\sqrt{t(t + 1)} + t)e_{1}.

See (3.2).

Remark 3.17. We have the same relation between the Möbius modulus of Grötzsch and Teichmüller rings as in (1.1), namely

(3.18)  mod_{M}R_{T} (t) = 2mod_{M}R_{G}(\sqrt{t + 1} ).

Acknowledgements. I would like to thank my advisor Frederick Gehring for discussions and suggestions on this paper. I also would like to acknowledge many valuable comments of the referee.

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206 Zair Ibragimov

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Received 10 June 2002