Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 211–222

QUASIREGULAR MAPPINGS OF MAXIMAL LOCAL MODULUS OF CONTINUITY

Leonid V. Kovalev

Washington University, Department of Mathematics St. Louis, MO 63130, U.S.A.; lkovalev@math.wustl.edu

Abstract. We study the behavior of a K -quasiregular mapping near points where its local modulus of continuity has order 1/K. We prove that the mapping is spherically analytic at such points and is asymptotically a rotation on circles. This result is used to prove sharp distortion estimates, including a version of Schwarz’s lemma.

1. Introduction

We consider a K -quasiregular mapping f: Ω → C, where Ω is a domain in C. Recall that f ∈ W_{loc}^{1,2}(Ω; R^{2}) is K -quasiregular if |Df(z)|^{2} ≤ KJ_{f}(z) for a.e. z ∈ Ω. Here |Df(z)| is the operator norm of the 2 × 2 differential matrix Df(z) and J_{f}(z) = det Df(z). It is well known that f is locally Hölder continuous with exponent 1/K [1]. Consequently, for z_{0} ∈ Ω and small δ > 0 the local modulus of continuity

ω_{f}(z_{0}, δ) = max{|f(z) − f(z_{0})| : z ∈ Ω, |z − z_{0}| ≤ δ}

is majorized by Aδ^{1/K} for some A > 0. A number of sharper upper bounds have been established for ω_{f}(z_{0}, δ), see [2], [17] and references therein. The upper limit

ω_{f}(z_{0}) = lim sup_{δ→0} \frac{ω_{f}(z_{0}, δ)}{δ^{1/K}}

is always finite, but can be strictly positive. An example is provided by the radial stretch map f_{K}(z) = |z|^{1/K−1}z, for which ω_{f_{K}} (0) = 1.

The purpose of this paper is to study the behavior of f near points where ω_{f} > 0, which are called points of maximal stretch. Unless K = 1, the mapping f is not differentiable at a point of maximal stretch; nevertheless, Theorem 2.1 shows that it behaves in a rather regular way near it. In particular, for every z_{0} ∈ Ω

lim_{z→z_{0}} \frac{|f(z) − f(z_{0})|}{|z − z_{0}|^{1/K}} = ω_{f}(z_{0})

2000 Mathematics Subject Classification: Primary 30C62; Secondary 30C75, 30C80.

(which is trivially true when ω_{f}(z_{0}) = 0). Recall that the linear distortion function of f

H_{f}(z_{0}) = lim sup_{r→0} sup_{z_{1},z_{2}}\{\frac{|f(z_{1}) − f(z_{0})|}{|f(z_{2}) − f(z_{0})|} : |z_{1} − z_{0}| = r = |z_{2} − z_{0}|\}

is bounded by a constant depending on K and the degree of f at z_{0} [14, Theorem 4.5]. Theorem 2.1 implies that H_{f}(z_{0}) = 1 whenever ω_{f}(z_{0}) > 0. In the present paper a common course of reasoning “small distortion ⟹ small modulus of continuity” is reversed in an unusual way: “large modulus of continuity ⟹ small distortion”.

In Section 3 we give sharp estimates of the distance between images of two points of maximal stretch. For f a K -quasiregular mapping of the unit disk to itself, Theorem 3.3 gives a sharp upper bound for |f(z)| in terms of |z|, ω_{f}(0), and ω_{f}(z). In contrast to the classical Hersch–Pfluger distortion theorem [11, p. 64], the bound in Theorem 3.3 has a simple closed form.

2. Local behavior at a point of maximal stretch

Our first result describes the behavior of a K -quasiregular mapping f near a point z_{0} with ω_{f}(z_{0}) > 0. Recall that

ω_{f}(z_{0}) = lim sup_{z→z_{0}} |f(z) − f(z_{0})|/|z − z_{0}|^{1/K}.

Theorem 2.1. Let f: Ω → C be a K -quasiregular mapping, z_{0} ∈ Ω. If ω_{f}(z_{0}) > 0, then f is injective in a neighborhood of z_{0} and there exists a continuous function θ: (0, 1) → R such that

(2.1)  lim_{z→z_{0}}\{\frac{f(z) − f(z_{0})}{|z − z_{0}|^{1/K−1}(z − z_{0})} − ω_{f}(z_{0})e^{iθ(|z−z_{0}|)}\} = 0.

According to Theorem 2.1, f is spherically analytic [8], [9] and is asymptotically a rotation on circles [3], [4] at any point of maximal stretch. The proof of Theorem 2.1 is a combination of estimates of conformal capacity and recent results of M. Brakalova and J. A. Jenkins [4]. First, we define the conformal capacity of a compact set E with respect to domain Ω ⊃ E as follows.

cap(Ω, E) = inf{∫_{Ω} |∇u(z)|^{2} dℒ^{2}(z) : u ∈ C_{0}^{∞}(Ω) and u ≥ 1 on E },

where ℒ^{2} is the two-dimensional Lebesgue measure. When E is connected, cap(Ω, E) is equal to the reciprocal of the module of ring domain Ω \ E (see, e.g. [16]).

Proof of Theorem 2.1. Without loss of generality we may assume that z_{0} = f(z_{0}) = 0. The mapping f admits the Stoilow factorization f = ψ ∘ h, where h is K -quasiconformal and ψ is a holomorphic function [11, p. 247]. Now if ψ′ vanishes at h(0), then as z → 0 we have |ψ(h(z))| = O(|h(z)|^{2}) = O(|z|^{2/K}), hence ω_{f}(0) = 0, a contradiction. Thus ψ′(h(0)) ≠ 0, which implies that f is homeomorphic in a neighborhood of 0. We may assume that such a neighborhood contains \overline{D}. Here and in the sequel D(a, r) = \{z : |z − a| < r\}, D(r) = D(0, r), and D = D(0, 1).

Let Ω′ denote the image of D under f. Then Ω′ ⊂ D(R) for some R, because f is continuous in \overline{D}. Choose a sequence r_{j} → 0 such that ω_{f}(0, r_{j}) ≥ ρ_{j} = \frac{1}{2}ω_{f}(0)r_{j}^{1/K} for every j. The set F_{j} = f(\overline{D}(r_{j})) is connected and meets both 0 and ∂D(0, ρ_{j}). By results in [1, Chapter III] we have

cap(Ω′, F_{j}) ≥ cap(D(R), F_{j}) ≥ \frac{2π}{log(4R/ρ_{j})}

= \frac{2πK}{log(1/r_{j})} − \frac{2πK^{2} log(8R/ω_{f}(0))}{(log(1/r_{j}))^{2}} + o(\frac{1}{(log(1/r))^{2}} ).

Therefore,

(2.2)  cap(Ω′, F_{j}) ≥ \frac{2πK}{log(1/r_{j})} − \frac{C}{(log(1/r_{j}))^{2}} ,

where C does not depend on j.

Now we fix j and obtain an upper bound for cap(Ω′, F_{j}) as follows. Let g: Ω′ → D be the inverse of f and define u(w) = (log |g(w)|)/ log r_{j} for w ∈ Ω′. Note that u is Hölder continuous in Ω′ \ F_{j}, min{u, 1} ∈ W_{0}^{1,2}(Ω′), and u_{|F_{j}} ≥ 1. Therefore,

(2.3)

cap(Ω′, F_{j}) ≤ ∫_{Ω′\F_{j}} |∇u(w)|^{2} dℒ^{2}(w)
= (log(1/r_{j}))^{−2} ∫_{Ω′\F_{j}} |∇ log |g(w)||^{2} dℒ^{2}(w).

The integrand can be expressed in terms of differential operators ∂ = ∂/∂z and ¯∂ = ∂/∂z̄ (at almost every point of Ω′ \ F_{j}).

|∇ log |g||^{2} = 4|∂ log |g||^{2} = |∂ log g + ∂ log \overline{g}|^{2} = |\frac{∂g}{g} + \overline{(\frac{¯∂g}{g} )}|^{2}.

Now if w = f(z), an easy calculation yields ∂g(w) = \overline{∂f(z)}J_{f}(z)^{−1} and ¯∂g(w) = −¯∂f(z)J_{f}(z)^{−1} . Changing the variable of integration and introducing the notation 214 Leonid V. Kovalev μ = ¯∂f/∂f and φ = arg z, we obtain

∫_{Ω′\F_{j}} |∇ log |g(w)||^{2} dℒ^{2}(w) = ∫_{D\\overline{D}(r_{j})}|\frac{\overline{∂f(z)}}{zJ_{f}(z)} − \overline{(\frac{¯∂f(z)}{zJ_{f}(z)})}|^{2}J_{f}(z) dℒ^{2}(z)

= ∫_{D\\overline{D}(r_{j})} \frac{|∂f(z) − e^{−2iφ} ¯∂f(z)|^{2}}{|∂f(z)|^{2} − |¯∂f(z)|^{2}} |z|^{−2} dℒ^{2}(z)

= ∫_{D\\overline{D}(r_{j})} \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} |z|^{−2} dℒ^{2}(z).

Combining this with (2.2) and (2.3) yields

∫_{D\\overline{D}(r_{j})} \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} |z|^{−2} dℒ^{2}(z) ≥ 2πK log(1/r_{j}) − C,

which is equivalent to

∫_{D\\overline{D}(r_{j})}(K − \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} )|z|^{−2} dℒ^{2}(z) ≤ C.

Since the integrand is non-negative, we conclude that

(2.4)  ∫_{D}|K − \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} ||z|^{−2} dℒ^{2}(z) < ∞.

Furthermore, we have an estimate

0 ≤ \frac{|1 + e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} − \frac{1}{K}

= 2\frac{1 + |μ(z)|^{2}}{1 − |μ(z)|^{2}} − \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} − \frac{1}{K}

= (2\frac{1 + |μ(z)|^{2}}{1 − |μ(z)|^{2}} − \frac{K^{2} + 1}{K}) + (K − \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} )

≤ K − \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} ,

where the last inequality follows from |μ(z)| ≤ (K − 1)/(K + 1). Therefore, (2.4) implies

(2.5)  ∫_{D}|K^{−1} − \frac{|1 + e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} ||z|^{−2} dℒ^{2}(z) < ∞

By (2.4) and (2.5) the mapping f satisfies the conditions of [4, Theorem 1], which says that there exists A > 0 such that |f(z)|/|z|^{1/K} → A as z → 0. Thus f is spherically analytic at 0. It is easy to see that the constant A must be equal to ω_{f}(0). Moreover, by [4, Lemma 3] the mapping f is asymptotically a rotation on circles, that is,

lim_{r→0}(arg f(re^{θ_{2}}) − arg f(re^{θ_{1}}) − (θ_{2} − θ_{1})) = 0

uniformly in θ_{1}, θ_{2} ∈ R. For r ∈ (0, 1) define θ(r) = arg f(r), where the argument is chosen so that θ is continuous. It remains to observe that (2.1) holds with this choice of θ.

Corollary 2.1. Let f: Ω → C be a K -quasiconformal mapping. Suppose that lim inf_{z→z_{0}} |f(z) − f(z_{0})|/|z − z_{0}|^{K} < ∞ for some z_{0} ∈ Ω. Then there exist a constant A and a continuous function θ: (0, 1) → R such that

lim_{z→z_{0}}\{\frac{f(z) − f(z_{0})}{|z − z_{0}|^{K−1}(z − z_{0})} − Ae^{iθ(|z−z_{0}|)}\} = 0.

Proof. Apply Theorem 2.1 to f^{−1} .

Even though the statement of Corollary 2.1 almost mirrors that of Theorem 2.1, the similarity is not complete. Corollary 2.1 does not apply to general K -quasiregular mappings, as follows from considering the mapping f(z) = z^{2} at z_{0} = 0.

Since 1-quasiregular mappings are holomorphic, for K = 1 the conclusion of Theorem 2.1 can be strengthened as follows:

lim_{z→z_{0}} \frac{f(z) − f(z_{0})}{z − z_{0}} = f′(z_{0})

exists. One might ask if the same is true for K > 1. The answer is given by the following theorem.

Theorem 2.2. For any K > 1 there exists a K -quasiconformal mapping f: C → C such that the assumptions of Theorem 2.1 are satisfied, but the limit

lim_{z→z_{0}} \frac{f(z) − f(z_{0})}{|z − z_{0}|^{1/K−1}(z − z_{0})}

does not exist.

Proof. Let γ: (0, 1] → (0, 1] be an increasing continuous function such that

(2.6)  ∫_{0}^{1} γ(r) \frac{dr}{r} = ∞ and ∫_{0}^{1} γ(r)^{2} \frac{dr}{r} < ∞.

For instance, γ(r) = (log(e + 1/r))^{−1} would work. Fix K > 1 and let k = (K − 1)/(K + 1). Let f: C → C be the normal solution [1, p. 91] of the Beltrami equation ¯∂f = μ∂f, where

μ(re^{iφ}) = \{ \begin{matrix}−ke^{i(2φ+γ(r))}, & r ≤ 1; \\ 0, & r > 1. \end{matrix}

Then

\begin{aligned}∫_{D}|K − \frac{|1 − e^{−2iφ}μ(z)|^{2}}{1 − |μ(z)|^{2}} | \frac{dℒ^{2}(z)}{|z|^{2}} & = 2π ∫_{0}^{1}|\frac{2k(1 − cos γ(r))}{1 − k^{2}} | \frac{dr}{r} \\ & = \frac{8πk}{1 − k^{2}} ∫_{0}^{1} sin^{2}(\frac{1}{2}γ(r)) \frac{dr}{r} < ∞, \end{aligned}

so (2.4) holds. It follows that f satisfies the assumptions of Theorem 2.1 with z_{0} = 0.

Since e^{−2iφ}μ(re^{iφ}) does not depend on φ, it follows from the uniqueness of normal solutions that f is a rotation on circles. That is, there exist continuous functions g: [0, ∞) → [0, ∞) and δ: (0, ∞) → R such that f(re^{iφ}) = g(r)e^{i(φ+δ(r))} . Actually, g and δ are absolutely continuous on every compact subinterval of (0, ∞), which can be seen as follows. Fix r_{0} > 0 and choose ε so that 0 < ε < r_{0} and |δ(r) − δ(r_{0})| < \frac{1}{4}π whenever |r − r_{0}| < ε. Now consider the mapping

h(x + iy) = log f(e^{x+iy}) = log g(e^{x}) + i(y + δ(e^{x}))

defined on the rectangle

R = \{(x, y) : log(r_{0} − ε) < x < log(r_{0} + ε), |y| < \frac{1}{4}π\}.

Since h is a quasiconformal mapping, it is absolutely continuous on almost every horizontal segment lying in R [11, p. 162]. Thus log g(e^{x}) and δ(e^{x}) are absolutely continuous in a neighborhood of log r_{0} , which implies the absolute continuity of g and δ in a neighborhood of r_{0} .

It remains to prove that δ(r) does not have a finite limit as r ↓ 0. For almost every r ∈ (0, 1) we can compute ∂f and ¯∂f to find that

μ(re^{iφ}) = e^{2iφ} \frac{g′(r) − g(r)/r + ig(r)δ′(r)}{g′(r) + g(r)/r + ig(r)δ′(r)}.

Rearranging the terms, obtain

ke^{iγ(r)} = \frac{1 − (rg′(r)/g(r) + irδ′(r))}{1 + (rg′(r)/g(r) + irδ′(r))}.

Therefore,

\frac{rg′(r)}{g(r)} + irδ′(r) = \frac{1 − ke^{iγ(r)}}{1 + ke^{iγ(r)}} = \frac{1 − k^{2} − 2ik sin γ(r)}{|1 + ke^{iγ(r)}|^{2}}.

Comparing the imaginary parts, we conclude that

−δ′(r) = \frac{2k sin γ(r)}{r|1 + ke^{iγ(r)}|^{2}} ≥ \frac{2k}{(1 + k)^{2}} \frac{sin γ(r)}{r}.

By (2.6) the right-hand side is not integrable, so lim_{r↓0} δ(r) = +∞.

3. Distortion of distance between points of maximal stretch

The main tool used in the present section is the generalized reduced module of a domain Ω ⊆ C which was introduced in a series of papers by V. N. Dubinin [5], [6]. Besides Ω itself, the module depends on parameters Z = (z_{1}, …, z_{n}), Δ = (δ_{1}, …, δ_{n}), and Ψ = (ψ_{1}, …, ψ_{n}). Here Z is an n-tuple of distinct points in Ω, δ_{j} ∈ R \ {0}, and ψ_{j}: [0, ∞) → [0, ∞) is a function of the form ψ_{j}(r) = μ_{j}r^{ν_{j}} with μ_{j}, ν_{j} > 0 for each j.

The conformal capacity of Ω with respect to Z, Δ, Ψ, and r > 0 is defined as follows.

cap C(r; Ω, Z, Δ, Ψ) = inf{∫_{Ω} |∇u(z)|^{2} dℒ^{2}(z) : u ∈ C_{0}^{∞}(Ω),
u = δ_{j} on \overline{D}(z_{j}, ψ_{j}(r))\}.

Let ν = (∑_{j=1}^{n} δ_{j}^{2}/ν_{j})^{−1} . The generalized reduced module of (Ω, Z, Δ, Ψ) is equal to

M(Ω, Z, Δ, Ψ) = lim_{r↓0}\{(cap C(r; Ω, Z, Δ, Ψ))^{−1} + \frac{ν}{2π} log r\},

provided that the limit exists. It is evident from the definition that M(Ω, Z, Δ, Ψ) is monotone with respect to Ω.

Before stating the basic result concerning the generalized reduced module we need a few definitions. Given a domain Ω ⊂ C, its Green’s function g_{Ω}: Ω × Ω → (0, +∞] is determined by the conditions that g_{Ω}( · , ζ) is harmonic in Ω \ {ζ}, vanishes at the regular boundary points of ∂Ω, and

g_{Ω}(z, ζ) = − log |z − ζ| + γ(Ω, ζ) + o(1) as z → ζ.

The number γ(Ω, ζ) is called the Robin constant of Ω at ζ. The domain Ω is called Greenian if g_{Ω} exists, or, equivalently, if its complement contains a compact set of positive logarithmic capacity [15]. We will use the following theorem, which is a combination of results found in [6], [7].

Theorem 3.1 ([6], [7]). If Ω is a Greenian domain, then

(3.1) M(Ω, Z, Δ, Ψ) = \frac{ν^{2}}{2π} \{∑_{j=1}^{n} \frac{δ_{j}^{2}}{ν_{j}^{2}} (γ(Ω, z_{j}) − log μ_{j}) + ∑_{j≠k} \frac{δ_{j}δ_{k}}{ν_{j}ν_{k}} g_{Ω}(z_{j}, z_{k})\}.

If Ω is not Greenian and ∑_{j=1}^{n} δ_{j}/ν_{j} = 0, then

(3.2)  M(Ω, Z, Δ, Ψ) = −\frac{ν^{2}}{2π} \{∑_{j=1}^{n} \frac{δ_{j}^{2}}{ν_{j}^{2}} log μ_{j} + ∑_{j≠k} \frac{δ_{j}δ_{k}}{ν_{j}ν_{k}} log |z_{j} − z_{k}|\}.

Since M(Ω, Z, Δ, Ψ) is defined in terms of conformal capacity, it is not surprising that it exhibits a similar quasiinvariance property. The special case n = 1 of the following proposition appeared in [13].

Proposition 3.1. Let f: Ω → C be a K -quasiregular mapping and suppose that z_{1}, …, z_{n} ∈ Ω are points of maximal stretch. Let Δ, Ψ, and ν be as above. For 1 ≤ j ≤ n let w_{j} = f(z_{j}) and ψ̃_{j}(r) = ω_{f}(z_{j})ψ_{j}(r)^{1/K}. Suppose that either (i) f is quasiconformal, or (ii) w_{1}, …, w_{n} are distinct and δ_{1}, …, δ_{n} are of the same sign. Then

(3.3)  M(f(Ω), W, Δ, Ψ̃) ≥ K^{−1}M(Ω, Z, Δ, Ψ).

Proof. It follows from Theorem 2.1 that for small r > 0

\overline{D}(w_{j}, (1 + o(1))ψ̃_{j}(r)) ⊆ f(\overline{D}(z_{j}, ψ_{j}(r))) ⊆ \overline{D}(w_{j}, (1 + o(1))ψ̃_{j}(r)).

If f is quasiconformal, these inclusions together with [6, Lemma 1] and quasiinvariance of capacity imply

(3.4)  (cap C(r; f(Ω), W, Δ, Ψ̃))^{−1} ≥ (K cap C(r; Ω, Z, Δ, Ψ))^{−1} + o(1).

Adding (ν/2πK) log r to both sides and passing to the limit r ↓ 0, we obtain (3.3).

Now suppose that the assumption (ii) holds. Without loss of generality we may assume that δ_{1}, …, δ_{n} are positive. If δ_{j} = 1 for every j, then the inequality (3.4) follows from [14, Theorem 7.1] and [6, Lemma 1]; the rest of the proof is as above. In the general case let δ_{j}^{∗} = 1, ψ_{j}^{∗}(r) = ψ_{j}(r^{1/δ_{j}}), and ν^{∗} = (∑_{j=1}^{n} δ_{j}/ν_{j})^{−1} (recall that ψ_{j}(r) = μ_{j}r^{ν_{j}}). Then

M(Ω, Z, Δ, Ψ) = \frac{ν^{2}}{(ν^{∗})^{2}} M(Ω, Z, Δ^{∗}, Ψ^{∗})

≤ \frac{Kν^{2}}{(ν^{∗})^{2}} M(f(Ω), W, Δ^{∗}, Ψ̃^{∗})

= KM(f(Ω), W, Δ, Ψ̃).

Here the first step is based on Theorem 3.1; the inequality for reduced modules is true because δ_{j}^{∗} = 1 for every j; and in the last step we have used the fact that transformations ψ ↦ ψ^{∗} and ψ ↦ ψ̃ commute.

With an appropriate choice of Z, Δ and Ψ, Theorem 3.1 and Proposition 3.1 yield sharp distortion estimates for quasiregular mappings. Our first result is that the points of maximal stretch are mapped as far from each other as the Hölder continuity allows.

Theorem 3.2. Let f: Ω → C be a K -quasiconformal mapping. For any domain Ω_{1} ⋐ Ω there exists a constant C = C(Ω, Ω_{1}) such that

(3.5)  |f(z_{1}) − f(z_{2})| ≥ C\sqrt{ω_{f}(z_{1})ω_{f}(z_{2})} |z_{1} − z_{2}|^{1/K}

for any z_{1}, z_{2} ∈ Ω_{1} . When Ω = C one can take C = 1, and this value is best possible for every K ≥ 1.

Proof. Assume that ω_{f}(z_{1}), ω_{f}(z_{2}) > 0 and z_{1} ≠ z_{2} ; otherwise the statement is trivial. Set parameters Z = (z_{1}, z_{2}), Δ = (1, −1), ψ_{1}(r) = r = ψ_{2}(r), W = (f(z_{1}), f(z_{2})), ψ̃_{1}(r) = ω_{f}(z_{1})r^{1/K}, and ψ̃_{2}(r) = ω_{f}(z_{2})r^{1/K}. By Proposition 3.1 and monotonicity of the generalized reduced modulus we have

(3.6)  M(C, W, Δ, Ψ̃) ≥ M(f(Ω), W, Δ, Ψ̃) ≥ K^{−1}M(Ω, Z, Δ, Ψ).

If C\Ω is not Greenian, one can plug (3.2) into the inequality (3.6) to obtain (3.5) with C = 1.

Now suppose that G is Greenian. Consider the regular part of Green’s function h_{Ω}: Ω × Ω → R, defined by h_{Ω}(ζ_{1}, ζ_{2}) = g_{Ω}(ζ_{1}, ζ_{2}) + log |ζ_{1} − ζ_{2}| if ζ_{1} ≠ ζ_{2} , and h_{Ω}(ζ_{1}, ζ_{1}) = γ(Ω, ζ_{1}) otherwise. Since h_{Ω} is continuous, the function |h_{Ω}| attains a finite maximum H on the compact set \overline{Ω_{1}} × \overline{Ω_{1}} . By (3.1) we have for any z_{1}, z_{2} ∈ Ω_{1}

M(Ω, Z, Δ, Ψ_{1}) = \frac{1}{8π} \{γ(Ω, z_{1}) + γ(Ω, z_{2}) − 2g_{Ω}(z_{1}, z_{2})\}

≥ \frac{1}{8π} \{2 log |z_{1} − z_{2}| − 4H\}.

This inequality together with (3.6) and (3.2) yields (3.5) with C = e^{−2H}.

It remains to show that the constant C in (3.5) cannot be greater than 1. Fix K ≥ 1 and introduce functions h_{1}(z) = z/(1 − z), h_{2}(z) = |z|^{1/K−1}z, and h_{3}(z) = z/(1 + z), which are understood as bijective self-maps of the Riemann sphere \overline{C}. The composition h = h_{3} ∘h_{2} ∘h_{1} is a K -quasiconformal automorphism of C. Straightforward computations show that h(0) = 0, h(1) = 1, and ω_{h}(0) = 1 = ω_{h}(1). Thus the value C = 1 is best possible in the case Ω = C.

Theorem 3.2 is not valid for K -quasiregular mappings, even when K = 1. Indeed, for a holomorphic function f every point z with f′(z) ≠ 0 is a point of maximal stretch. Obviously, two non-critical points of f may be mapped by f into the same point.

Next we consider a quasiregular mapping f: D → D that has at least two points of maximal stretch. By composing f with appropriate Möbius transformations, we may assume that 0 is both a fixed point and a point of maximal stretch for f. Note that under these assumptions ω_{f}(0) ≤ 4^{1−1/K}, as follows from the Hersch–Pfluger distortion theorem and calculations in [11, p. 65].

Theorem 3.3. Let f: D → D be a K -quasiregular mapping such that f(0) = 0. Then for any z ∈ D

(3.7)  |f(z)| ≤ (1 + ω_{f}(0)ω_{f}(z)|z|^{−2/K}(1 − |z|^{2})^{1/K})^{−1/2}.

For any K ≥ 1 and any z ∈ D there exists a K -quasiconformal mapping f for which equality is attained in (3.7).

Proof. Assume that f(z) ≠ 0, ω_{f}(0) > 0, and ω_{f}(z) > 0; otherwise there is nothing to prove. Let Z = (0, z), Δ = (1, 1), ψ_{1}(r) = r = ψ_{2}(r), W = (0, f(z)), ψ̃_{1}(r) = ω_{f}(0)r^{1/K}, and ψ̃_{2}(r) = ω_{f}(z)r^{1/K}. By Proposition 3.1 and the monotonicity of M we have M(D, W, Δ, Ψ̃) ≥ K^{−1}M(D, Z, Δ, Ψ). Using (3.1) and the identities γ(D, z) = log(1 − |z|^{2}) and g_{D}(z, 0) = − log |z|, obtain

log \frac{1 − |f(z)|^{2}}{ω_{f}(0)ω_{f}(z)} + 2 log \frac{1}{|f(z)|} ≥ \frac{1}{K} \{log(1 − |z|^{2}) + 2 log \frac{1}{|z|}\}.

This inequality readily implies (3.7).

To prove that (3.7) is sharp, fix z_{0} ∈ (0, 1). There is t ∈ (0, 1) such that z_{0} = 2t/(1 + t^{2}). The holomorphic function h_{1}(z) = (z^{2} − t^{2})/(1 − t^{2}z^{2}) maps the half-disk D ∩ {Re z > 0} conformally onto D \ (−1, −t^{2}). Composing h_{1} with the radial stretch map f_{K}(z) = |z|^{1/K−1}z and an appropriate branch of h_{2}(z) = \sqrt{(z + t^{2/K})/(1 + t^{2/K}z)} , we find that h_{2}∘f_{K} ∘h_{1} is a K -quasiconformal automorphism of D ∩ {Re z > 0} which maps D ∩ {Re z = 0} onto itself. It can be extended by reflection to a K -quasiconformal map F_{1}: D → D. Note that F_{1}(±t) = ±t^{1/K}. Finally, let F = h_{3} ∘ F_{1} ∘ h_{0} , where h_{0}(z) = (z − t)/(1 − tz) and h_{3}(z) = (z + t^{1/K})/(1 + t^{1/K}z).

We have F (0) = 0 and F (z_{0}) = 2t^{1/K}/(1 + t^{2/K}). Furthermore,

\begin{aligned}ω_{F} (0) & = |h′_{0}(0)h′_{1}(−t)|^{1/K}|h′_{2}(0)h′_{3}(−t^{1/K})| = (\frac{2}{1 + t^{2}} )^{1/K} \frac{1 + t^{2/K}}{2} ; \\ ω_{F} (z_{0}) & = |h′_{0}(z_{0})h′_{1}(t)|^{1/K}|h′_{2}(0)h′_{3}(t^{1/K})| = (\frac{2(1 + t^{2})}{(1 − t^{2})^{2}})^{1/K} \frac{(1 − t^{2/K})^{2}}{2(1 + t^{2/K})}. \end{aligned}

Hence

1 + ω_{F} (0)ω_{F} (z_{0})(\frac{1 − |z_{0}|^{2}}{|z_{0}|^{2}} )^{1/K}= 1 + (\frac{4}{(1 − t^{2})^{2}} )^{1/K}\frac{(1 − t^{2/K})^{2}}{4} (\frac{(1 − t^{2})^{2}}{4t^{2}} )^{1/K}

= 1 + \frac{(1 − t^{2/K})^{2}}{4t^{2/K}} = \frac{(1 + t^{2/K})^{2}}{4t^{2/K}}

= |F (z_{0})|^{−2} as required.

4. Concluding remarks

One can generalize some of the above results by taking into account the local degree of a mapping. For example, let f be a K -quasiregular mapping of degree m at a point z. If lim sup_{δ→0} δ^{−m/K}ω_{f}(z, δ) > 0, then f is spherically analytic at z, as follows from Theorem 2.1 and the Stoilow factorization. The following problems indicate other possible directions for further research.

Problem 4.1. Extend Theorem 2.1 to dimension n ≥ 3.

It is clear that in higher-dimensional setting the definition of ω_{f}(z) and the statement of Theorem 2.1 will have to be modified. In particular, the constant K should be replaced with K_{I}^{1/(n−1)} , where K_{I} is the inner distortion of f. See [10] for comparison of different distortion functions.

It is known [12] that the linear distortion H_{f}(z) of a K -quasiconformal mapping in the plane is bounded by K whenever Df(z) exists and |Df(z)| > 0. By Theorem 2.1 we have H_{f}(z) = 1 at the points where lim sup_{δ→0} δ^{−1/K}ω_{f}(z, δ) > 0. It may be possible to interpolate between the two estimates as follows.

Problem 4.2. Prove that for any α ∈ [1/K, 1] and any planar K -quasiregu-lar mapping f the condition lim sup_{δ→0} δ^{−α}ω_{f}(z, δ) > 0 implies H_{f}(z) ≤ αK.

As an example, consider h_{1}(z) = |z|^{α−1}z and h_{2}(x + iy) = αKx + iy. The mapping f = h_{2} ∘ h_{1} satisfies the above conditions at the origin, where its linear distortion is H_{f}(0) = αK. There could be a higher-dimensional analogue of Problem 4.2, but the situation is complicated by the fact that despite recent progress (e.g. [9]), the local topological behavior of spatial quasiregular mappings is not yet fully understood.

Acknowledgements. The author is thankful to Albert Baernstein and Matti Vuorinen for reading the manuscript and making valuable suggestions. The helpful discussions with Melkana Brakalova and David Opěla are also acknowledged. The referee’s constructive comments led to several improvements.

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222 Leonid V. Kovalev

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Received 1 July 2003