Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 185–194

HAMILTON SEQUENCES AND EXTREMALITY FOR CERTAIN TEICHMÜLLER MAPPINGS

Guowu Yao

Peking University, School of Mathematical Sciences Beijing, 100871; wallgreat@lycos.com and Chinese Academy of Sciences, Institute of Mathematics Academy of Mathematics and System Sciences, Beijing, 100080, P. R. China

Abstract. Suppose f is a Teichmüller mapping with complex dilatation

μ(z) = k \frac{\overline{φ(z)}}{|φ(z)|} ,

where φ(z) is holomorphic in the unit disk. If

m(r, φ) = \frac{1}{2π} ∫_{0}^{2π} |φ(re^{iθ})| dθ ≍ log^{s} ( \frac{1}{1 − r} ) /(1 − r), as r → 1,

for some given s > 0, then the putative sequence φ(Rz)/∥φ(Rz)∥, R ↑ 1 is a Hamilton sequence of μ and hence f is extremal.

1. Introduction

Let Δ be the unit disk {|z| < 1} in the complex plane C. Suppose f is a quasiconformal self-mapping of Δ. We denote by Q(f) the class of all quasiconformal self-mappings of Δ which agree with f on the boundary ∂Δ. A quasiconformal mapping f is said to be extremal in Q(f) if it minimizes the maximal dilatations of Q(f), i.e.

K[f] = inf{K[g] : g ∈ Q(f)},

where K[g] is the maximal dilatation of g. The mapping f is uniquely extremal if it is extremal and if there are no other extremal mappings for its boundary values.

Let B(Δ) = {ϕ(z) holomorphic on Δ : ∥ϕ∥ = ∫∫_{Δ} |ϕ(z)| dx dy < ∞}. A necessary and sufficient condition that f is extremal in Q(f) is that [8] its Beltrami differential μ has a so-called Hamilton sequence, namely, a sequence

{ϕ_{n}(z) ∈ B(Δ) : ∥ϕ_{n}∥ = 1, n ∈ N},

2000 Mathematics Subject Classification: Primary 30C75.

The work was partly done during the author’s study at the School of Mathematical Sciences of the Peking University and was supported by the China Postdoctoral Science Foundation.

such that

(1.1)  lim_{n→∞} ∫∫_{Δ} μ(z)ϕ_{n}(z) dx dy = ∥μ∥_{∞}.

In this paper, unless otherwise specified, a Teichmüller mapping f is said to be a quasiconformal mapping of the unit disk onto itself, which has the complex dilatation

(1.2)  μ(z) = \frac{fz̄}{f_{z}} = k \frac{\overline{φ(z)}}{|φ(z)|},  z ∈ Δ,

where φ ≢ 0 is holomorphic on Δ and k ∈ [0, 1) is a constant. It is of interest to know whether f is extremal or, in particular, uniquely extremal among Q(f).

From now on, we call a holomorphic function φ in Δ satisfying the condition of a global Hamilton sequence or call φ GHS if the putative sequence {ϕ_{R}(z) = φ(Rz)/∥φ(Rz)∥, R ↑ 1} is a Hamilton sequence of f; in other words,

(1.3)  lim_{R→1} ∫ ∫_{Δ} \frac{\overline{φ(z)}}{|φ(z)|}|ϕ_{R}(z)| dx dy = 1.

In some papers such as [5], [7] and [8], the following possibility is investigated: If {R_{n}} is a sequence of numbers, R_{n} ∈ (0, 1), lim_{n→∞} R_{n} = 1, does {φ(R_{n}z)/∥φ(R_{n}z)∥} constitute a Hamilton sequence?

In 1974, Reich and Strebel proved

Theorem A ([8]). Suppose φ(z) is holomorphic on Δ and satisfies the growth condition

(1.4)  m(r, φ) = \frac{1}{2π} ∫_{0}^{2π} |φ(re^{iθ})| dθ = O(\frac{1}{1 − r} ),  r → 1.

Then φ is GHS and hence f is extremal. Moreover, the extremality of f is no longer implied if O((1 − r)^{−1}) is replaced by O((1 − r)^{−s}) for any s > 1.

Hayman and Reich [4] proved that f is also uniquely extremal if φ(z) satisfies the growth condition (1.4).

Sethares solved the unique extremality of Teichmüller mappings for certain holomorphic functions φ:

Theorem B ([9]). Suppose φ(z) is holomorphic on Δ and meromorphic in Δ̄. Then f is uniquely extremal if and only if all poles of φ(z) are of order not exceeding two.

Furthermore, in 1988, Reich considered the relation between the unique extremality and the construction of the Hamilton sequence for certain Teichmüller mappings.

Theorem C ([7]). Suppose φ(z) is holomorphic on Δ and meromorphic in Δ̄. Then the corresponding Teichmüller mapping f is uniquely extremal if and only if φ is GHS.

But, for more general φ, we do not know too much about such a relation. It is interesting for us to consider

Problem 1. If φ is GHS, is the corresponding Teichmüller mapping f uniquely extremal in Q(f)? Conversely, if a Teichmüller mapping f is uniquely extremal in Q(f), is the corresponding φ GHS?

Problem 2. When is φ GHS?

To answer these problems, we have the following two theorems as our main results:

Theorem 1. Given s > 0. Suppose φ(z) is holomorphic on Δ and satisfies the growth condition

(1.5)  m(r, φ) ≍ (\frac{1}{1 − r} log^{s} \frac{1}{1 − r} ),  r → 1.

Then φ is GHS.

As an application of the result in [6] (or see Theorem D in Section 3) and Theorem 1, we answer the first part of Problem 1 negatively. But we notice that the second part of Problem 1 is still open.

Theorem 2. For any given real number s > 1, there exists a holomorphic function φ(z) in Δ satisfying the growth condition (1.5), and hence φ is GHS while the corresponding Teichmüller mapping f is extremal instead of being uniquely extremal.

2. Some lemmas

In order to prove our main results, we need two lemmas. Such results are related to the theories of H^{p} spaces and of univalent functions. We refer the readers to Duren’s book [1].

The following lemma is a counterpart of the results in [2] and [3] by Hardy and Littlewood.

Lemma 1. Suppose φ(z) is holomorphic on Δ. Given s ∈ R, α > 0 and 1 ≤ p < ∞. Then for any positive integer n and with r → 1, the following two conditions are equivalent:

(i) m_{p}(r, φ) = \{\frac{1}{2π} ∫_{0}^{2π} |φ(re^{iθ})|^{p} dθ\}^{1/p} = O(\frac{1}{(1 − r)^{α}} log^{s} \frac{1}{1 − r} ),

(ii) m_{p}(r, φ^{(n)}) = \{\frac{1}{2π} ∫_{0}^{2π} |φ^{(n)}(re^{iθ})|^{p} dθ\}^{1/p} = O(\frac{1}{(1 − r)^{α+n}} log^{s} \frac{1}{1 − r} ).

Proof. By induction, it suffices to show that when n = 1, this lemma holds. Assume that (i) holds. Set R = \frac{1}{2}(1 + r). By the Cauchy formula

φ′(re^{iθ}) = \frac{1}{2πi} ∫_{|z|=R} \frac{φ(z)}{(z − re^{iθ})^{2}} dz = \frac{R}{2π} ∫_{0}^{2π} \frac{φ(Re^{i(t+θ)})e^{i(t−θ)}}{(Re^{it} − r)^{2}} dt.

Minkowski’s inequality then gives

\begin{aligned}m_{p}(r, φ′) & ≤ \frac{1}{2π} ∫_{0}^{2π} \frac{Rm_{p}(R, φ)}{R^{2} − 2Rr cos t + r^{2}} dt \\ & = \frac{R m_{p}(R, φ)}{R^{2} − r^{2}} = O(\frac{1}{(1 − r)^{α+1}} log^{s} \frac{1}{1 − r} ), r → 1. \end{aligned}

Conversely, if (ii) holds, we apply Minkowski’s inequality to the relation

|φ(re^{iθ})| ≤ |φ(0)| + ∫_{0}^{r} |φ′(te^{iθ})| dt

and obtain

m_{p}(r, φ) ≤ |φ(0)| + ∫_{0}^{r} m_{p}(t, φ′) dt = O(\frac{1}{(1 − r)^{α}} log^{s} \frac{1}{1 − r} ), r → 1.

Lemma 2. Set Δ_{r} = {z ∈ Δ : |z| < r < 1}. Suppose s ≥ −1. If φ satisfies the growth condition

(2.1)  m(r, φ) ≍ (\frac{1}{1 − r} log^{s} \frac{1}{1 − r} ),  r → 1,

then as r → 1

(2.2)

A(r, φ) = ∫∫_{Δ_{r}} |φ(z)| dx dy = ∫_{0}^{r} t dt ∫_{0}^{2π} |φ(te^{iθ})| dθ

≍ \{ \begin{matrix}(log^{s+1} \frac{1}{1 − r} ) & s > −1, \\ (log log \frac{1}{1 − r} ) & s = −1, \end{matrix}

and hence

(2.3)  lim_{r→1} \frac{log^{s} \frac{1}{1 − r}}{A(r, φ)} = 0,

where we prescribe that log^{s}((1/(1 − r)) = 1 as s = 0.

Proof. First let s > −1. By equation (2.1), there exist C_{1} > 0, C_{2} > 0 and r_{0} ≥ 0 such that, when r ≥ r_{0} ,

\frac{C_{2}}{1 − r} log^{s} \frac{1}{1 − r} ≥ m(r, φ) ≥ \frac{C_{1}}{1 − r} log^{s} \frac{1}{1 − r} .

Therefore,

A(r, φ) ≥ ∫_{r_{0}}^{r} t dt m(t, φ) ≥ C_{1} ∫_{r_{0}}^{r} \frac{t}{1 − t} log^{s} \frac{1}{1 − t} dt

≥ C_{1}r_{0} ∫_{r_{0}}^{r} \frac{1}{1 − t} log^{s} \frac{1}{1 − t} dt

= \frac{C_{1}r_{0}}{1 + s} log^{s+1} \frac{1}{1 − t} |_{r_{0}}^{r}

≍ (log^{s+1} \frac{1}{1 − r} ), r → 1.

Further,

A(r, φ) = ∫_{0}^{r} t dt m(t, φ) ≤ ∫_{0}^{r} dt m(t, φ)

≤ C_{2} ∫_{0}^{r} \frac{1}{1 − t} log^{s} \frac{1}{1 − t} dt

= \frac{C_{2}}{1 + s} log^{s+1} \frac{1}{1 − t} |_{0}^{r}

≍ (log^{s+1} \frac{1}{1 − r} ), r → 1.

Thus, we obtain (2.2). When s = −1, we omit the proof, because it is similar to the above. Equation (2.3) is obvious.

190 Guowu Yao

3. Proofs of the main results The proof of Theorem 1 is somewhat similar to that of the “only if” part of

Theorem C. As usual, we write

(3.1)  \frac{∫∫_{Δ}(\overline{φ(z)}/|φ(z)|)φ(Rz) dx dy}{∫∫_{Δ} |φ(Rz)|} = R^{2} + R^{2} \frac{α(R) + β(R)}{A(R, φ)},

where

(3.2)  α(R) = ∫∫_{Δ_{R}} \frac{\overline{φ(z)}}{|φ(z)|}[φ(Rz) − φ(z)] dx dy,
(3.3)  β(R) = ∫∫_{U_{R}} \frac{\overline{φ(z)}}{|φ(z)|}φ(Rz) dx dy,

and U_{R} = {z ∈ Δ : R < |z| < 1}.

It is sufficient to show

(3.4)  lim_{R→1} \frac{α(R)}{A(R, φ)} = 0,
(3.5)  lim_{R→1} \frac{β(R)}{A(R, φ)} = 0.

For one thing,

|α(R)| = |∫∫_{Δ_{R}} \frac{\overline{φ(z)}}{|φ(z)|}[φ(Rz) − φ(z)] dx dy|

≤ ∫∫_{Δ_{R}} |φ(Rz) − φ(z)| dx dy

≤ ∫_{0}^{R} r dr ∫_{0}^{2π} dθ ∫_{Rr}^{r} |φ′(te^{iθ})| dt = 2π ∫_{0}^{R} r dr ∫_{Rr}^{r} m(t, φ′) dt.

Hamilton sequences and extremality 191

By equation (1.5) and Lemma 1, we have

|α(R)| ≤ 2Cπ ∫_{0}^{R} r dr ∫_{Rr}^{r}(\frac{1}{(1 − r)^{2}} log^{s} \frac{1}{1 − r} ) dt

≤ 2Cπ ∫_{0}^{R} \frac{1}{(1 − r)^{2}} log^{s} \frac{1}{1 − r} (∫_{Rr}^{r} dt)r dr

= 2Cπ(1 − R) ∫_{0}^{R} \frac{r^{2}}{(1 − r)^{2}} log^{s} \frac{1}{1 − r} dr

≤ 2Cπ(1 − R) ∫_{0}^{R} log^{s} \frac{1}{1 − r} d\frac{1}{1 − r}

= 2Cπ(1 − R)[\frac{1}{1 − r} log^{s} \frac{1}{1 − r} |_{0}^{R} −s ∫_{0}^{R} \frac{1}{(1 − r)^{2}} log^{s−1} \frac{1}{1 − r} dr]

≤ 2Cπ(1 − R)(\frac{1}{1 − r} log^{s} \frac{1}{1 − r} |_{0}^{R}).

So we obtain

(3.6)  |α(R)| ≤ 2Cπ log^{s} \frac{1}{1 − R},

where C is a suitable constant.

Next, choose R sufficiently close to 1 such that log(1/(1 − R)) > 1,

|β(R)| = |∫∫_{U_{R}} \frac{\overline{φ(z)}}{|φ(z)|}φ(Rz) dx dy|

≤ ∫_{0}^{2π} dθ ∫_{R}^{1} |φ(Rte^{iθ})|t dt

= \frac{1}{R^{2}} ∫_{0}^{2π} ∫_{R^{2}}^{R} |φ(ue^{iθ})|u du

= \frac{2π}{R^{2}} ∫_{R^{2}}^{R} rm(r, φ) dr

≤ \frac{2π}{R^{2}} ∫_{R^{2}}^{R} m(r, φ) dr

≤ B ∫_{R^{2}}^{R} \frac{1}{1 − r} log^{s} \frac{1}{1 − r} dr

= \frac{B}{1 + s} log^{s+1} \frac{1}{1 − r} |_{R^{2}}^{R}

= \frac{B}{1 + s}[log^{s+1} \frac{1}{1 − R} − log^{s+1} \frac{1}{1 − R^{2}} ]

= \frac{B}{1 + s}[log^{s+1} \frac{1}{1 − R} − (log \frac{1}{1 − R} + log \frac{1}{1 + R})^{s+1}]

≤ \frac{B}{1 + s}[log^{s+1} \frac{1}{1 − R} − (log \frac{1}{1 − R} − 1)^{s+1}],

where B is a constant. Notice that when x > 1, s ≥ 0,

lim_{x→+∞} \frac{x^{s+1} − (x − 1)^{s+1}}{x^{s+1}} = 0  or  lim_{x→+∞} \frac{x^{s+1} − (x − 1)^{s+1}}{x^{s}} = s + 1.

Let x = log(1/(1 − R)). By Lemma 2 we derive

lim_{R→1} \frac{|β(R)|}{A(R, φ)} = lim_{R→1} \frac{log^{s+1} \frac{1}{1 − R} − (log \frac{1}{1 − R} − 1)^{s+1}}{log^{s+1} \frac{1}{1 − R}} = 0.

That is (3.5). Inequality (3.6) and Lemma 2 evidently provide (3.4). This completes the proof of Theorem 1.

In 1995, Lai and Wu obtained the following theorem as an improvement of Theorem 5 in [9] by Sethares.

Theorem D ([6]). Let z_{1}, …, z_{m} be points of ∂Δ such that removing an arbitrary neighborhood D_{i} of z_{i} from Δ results in a region of a finite φ-area. Let α_{1}, …, α_{m} be non-zero complex numbers and let t_{1}, …, t_{m} be real numbers such that φ satisfies, for each i = 1, …, m, the growth condition

(3.7)  |\frac{\sqrt{φ(z)}}{log^{t_{i}}(z_{i} − z)} − \frac{α_{i}}{z_{i} − z} | = O(1),  z → z_{i}.

Then f is uniquely extremal if and only if t_{i} ≤ \frac{1}{2} , i = 1, …, m.

It is easy to see that φ with the condition (3.7) satisfies the growth estimate

(3.8)  m(r, φ) = \frac{1}{2π} ∫_{0}^{2π} |φ(re^{iθ})| dθ < C \frac{1}{1 − r} log \frac{1}{1 − r} ,

where C is a positive constant.

We continue to show that

φ(z) = \frac{log^{s}(1 − z)}{(1 − z)^{2}}, s > 0,

satisfies the growth condition (1.5).

On the one hand, by virtue of

(3.9)  lim_{r→1} \frac{|1 − re^{i(1−r)}|}{1 − r} = \sqrt{2} ,

we have

(3.10)  \begin{aligned}∫_{0}^{2π} |φ(re^{iθ})| dθ & ≥ ∫_{0}^{1−r} \frac{|log \frac{1}{1 − re^{iθ}} |^{s}}{|1 − re^{iθ}|^{2}} dθ \\ & ≥ C_{1} \frac{1}{1 − r} log^{s} \frac{1}{1 − r} ,  r → 1. \end{aligned}

On the other hand,

m(r, φ) = \frac{1}{2π} ∫_{0}^{2π} |φ(re^{iθ})| dθ

= \frac{1}{2π} ∫_{0}^{2π} \frac{|log \frac{1}{1 − re^{iθ}} |^{s}}{|1 − re^{iθ}|^{2}} dθ

≤ C_{2} \frac{1}{1 − r} log^{s} \frac{1}{1 − r} , r → 1,

where C_{1} and C_{2} are two positive constants. Therefore, φ assumes the growth condition (1.5).

Thus, combining Theorem 1 and Theorem D, we have

Corollary 3.1. Suppose s > 1 is a real number and

φ(z) = \frac{log^{s}(1 − z)}{(1 − z)^{2}}.

Then φ(z) is GHS and f is extremal instead of being uniquely extremal. Here some suitable univalent branch is chosen for φ in Δ.

Now, Theorem 2 is evident.

Acknowledgement. I would like to thank Professor Reich for referring me to the paper [5] of Huang and for his useful suggestions concerning this paper.

194 Guowu Yao

References

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[3] Hardy, G. H., and J. E. Littlewood: Some properties of conjugate functions. - J. Reine Angew. Math. 167, 1931, 405–423.

[4] Hayman, W. K., and E. Reich: On the Teichmüller mappings of the disk. - Complex Variables Theory Appl. 1, 1982, 1–12.

[5] Huang, X. Z.: On the extremality for Teichmüller mappings. - J. Math. Kyoto Univ. 35, 1995, 115–132.

[6] Lai, W. C., and Z. M. Wu: On extremality and unique extremality of Teichmüller mappings. - Chinese Ann. Math. Ser. B 16, 1995, 399–406.

[7] Reich, E.: Construction of Hamilton sequences for certain Teichmüller mappings. - Proc. Amer. Math. Soc. 103, 1988, 789–796.

[8] Reich, E., and K. Strebel: Extremal quasiconformal mappings with given boundary values. In: Contributions to Analysis, A Collection of Papers Dedicated to Lipman Bers, Academic Press, New York, 1974, pp. 375–392.

[9] Sethares, G. C.: The extremal property of certain Teichmüller mappings. - Comment. Math. Helv. 43, 1968, 98–119.

Received 13 May 2003