Xue MENG Sihui ZHANG

Abstract An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given.

Keywords Quasiconformal mapping， Uniquely extremal quasiconformal mapping，Reich sequence

1 Introduction

Let Ω be a plane domain with at least two boundary points.The Teichm¨uller space T(Ω)is the space of equivalence classes of quasiconformal maps f from Ω to a variable domain f(Ω).Two quasiconformal maps f from Ω to f(Ω) and g from Ω to g(Ω) are equivalent if there is a conformal map c from f(Ω) onto g(Ω) and a homotopy through quasiconformal maps htmapping Ω onto g(Ω) such that h0= c ◦f， h1= g and ht(p) = c ◦f(p) for every t ∈[0，1]and every p in the boundary of Ω.Denote by [f] the Teichm¨uller equivalence class of f; also sometimes denote the equivalence class by [μ] where μ is the Beltrami coefficient of f.

Denote by M(Ω) the open unit ball in L∞(Ω).For μ ∈M(Ω)， define

We say that μ is extremal in [μ] if ‖μ‖∞=k0([μ])， and uniquely extremal if ‖ν‖∞＞k0(μ) for any other ν ∈[μ].The corresponding f is also called extremal or uniquely extremal.

A quasiconformal mapping f will be said to be of Teichm¨uller type if its Beltrami cofficient μ is of Teichm¨uller type， i.e.，

where k ∈(0，1)is a constant and φ0(z)/=0 a.e.is a measurable function in Ω.In particular，if φ0is holomorphic in Ω，we call f a Teichm¨uller mapping and the corresponding μ a Teichm¨uller differential.

Let B(Ω)be the Banach space consisting of holomorphic functions φ(z)belonging to L1(Ω)，with norm

For μ ∈M(Ω) and φ ∈B(Ω)， set

In [4]， Reich proved the following theorem.

Theorem 1.1(see [4])Letμ ∈M(Ω)be given by(1.1).Suppose that there exists a sequence of functionsφn∈B(Ω)，n=1，2， ··· ，such that

pointwise a.e.inΩ，

Thenfis uniquely extremal.

In [5]，Reich showed that(1.3)can be replaced by the weaker assumption of boundedness of{δ[φn]}，provided that(1.2)is strengthened appropriately.This was done in his theorem which is stated below.

Theorem 1.2(see [5])Letμ ∈M(Ω)be given by(1.1).Suppose that there exists a sequence of functionsφn∈B(Ω)， n=1，2，··· ，such that

pointwise a.e.inΩ，φ0∈(Ω)，

uniformly with respect ton， whereΩ(n，A)={z ∈Ω:A|φ0(z)|＜|φn(z)|}.Thenfis uniquely extremal.

Generally， if μ satisfies Reich’s condition above， following [3] we call {φn} a Reich sequence for μ on Ω.Note that a Reich sequence {φn} for μ is not necessarily convergent pointwise on Ω (see [1–2]).Therefore， if in addition φnconverges to some φ0pointwise a.e.on Ω， we call{φn} a normal Reich sequence for μ on Ω.

Let μ ∈M(Ω) be given by (1.1).We say that μ satisfies weak Reich’s condition on Ω if there exists a sequence {φn} in B(Ω) such that (1.4)–(1.6) hold.If μ satisfies weak Reich’s condition， we call {φn} a weak Reich sequence for μ on Ω.

In order to show the unique extremality of f，when we cannot find the normal Reich sequence by Theorem 1.1， we can find the weak Reich sequence by Theorem 1.2 instead.For example，define

Ever since [8]， the family (1.7) has been known to possess a precise transition point a =3.It is known that the horizontal stretch of Ωais uniquely extremal if and only if 3 ≤a ≤∞.It is easy to check that

provides a normal Reich sequence when 3 ＜a ≤∞， but it fails to do so for the critical case a = 3.For a long time， in [7]， Reich gave an explicit example of a normal Reich sequence for the critical case a=3.

In this paper， we consider another uniquely extremal quasiconformal mapping， in a borderline case between uniqueness and non-uniqueness， a normal Reich sequence is given explicitly in this case for the first time.

Define ωa={z =x+iy :y ＞|x|a， z /=ib， b ＞0}， T(z)the Teichm¨uller mapping generated by the quadratic differential φ0(z)=The complex dilatation is1.

In [6]， it was proved that if a ≥， then T(z) is uniquely extremal， and in the proof， the normal Reich sequence is given when a ＞， but when a =， the normal Reich sequence is failed to work.In this paper， we construct the normal Reich sequence for this case.

Theorem 1.3The functions

wheredenotes the branch in the right half-plane that is real on the positive y-axis， provide a normal Reich sequence for.

2 Proof of Theorem 1.3

We have

and

We mainly focus on the left half part ofto the upper of the line Im z =N， N ＞max{2b，1}first， and denote it by Ω， i.e.， Ω=Re z ＜0， Im z ＞N}.We also letRe z ＞0， Im z ＞N}.Obviously，

equals

Consider

From

and

when n is sufficiently large， we obtain that

Since

when Re w ＞0， from (2.1) we have

Now we estimate the terms on the right of (2.2) when n is sufficiently large.From

we get

From

we obtain that

Since N ＞2b， it is easy to seewhen z ∈Ω， hence

From above we obtain

Now we estimate another part of

Let w =eiθ=r cos θ+ir sin θ.Then

Let

From above we get

From above we get

Hence from (2.3)–(2.4)， we obtain

which approaches 0 when n →∞.The proof of Theorem 1.3 is complete.

AcknowledgementThe authors sincerely thank the referees for their valueable suggestions.