Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 819–825

Estimates for Quasiconformal Mappings onto Canonical Domains

Vo Dang Thao

Abstract. In this paper we establish estimates for K-quasiconformal mappings z = g(w) of a domain bounded by two circles |w| = 1, |w| = q and n continua situated in q < |w| < 1 onto a circular ring Q(g) < |z| < 1 that has been slit along n arcs on the circles |z| = R_{j}(g) (j = 1, …, n) such that |z| = 1 and |z| = Q correspond to |w| = 1 and |w| = q, respectively. The bounds in the estimates for Q, R_{j} and |g(w)| are explicitly given, most of them are optimal. They are deduced mainly from [17].

Keywords: K-quasiconformal mappings, Riemann moduli of a multiply-connected domain,

monotony of the modulus of a doubly-connected domain

AMS subject classification: 30 C 62, 30 C 75, 30 C 80, 30 C 30

1. Introduction and notations

The generalization of Carleman’s (see [1: p. 212], [2: p. 177], [12: p. 15]) area inequality for doubly-connected domains to multiply-connected domains in [15] improves many Grötzsch’s [4, 6, 8] and Rengel’s [13] significant circular slits theorems for conformal mappings. In [16] we establish further area inequalities for K-quasiconformal mappings (see the definition in [10: p. 16]). Combining this with Grötzch’s [4, 5, 7] inequalities yields in [17] sharp or asymptotic sharp estimates for K-quasiconformal mappings of the circular ring Q < |z| < 1 with concentric circular slits onto domains lying in q < |w| < 1. In this paper we establish estimates for the inverse mappings of those studied in [17]. Here the consideration is partly similar to the case of conformal mappings (K = 1, see [14: pp. 121 - 124]) using two auxiliary functions introduced in Section 4.

Let now B be any domain given in the w-plane, bounded by two circles |w| = 1, |w| = q and pn (p, n ∈ ℕ) boundary components σ_{1}, …, σ_{pn} lying in (0 <) q < |w| < 1, and transformed into itself by the rotation t = e^{i \frac{2π}{p}} w. We shall write B = B_{0} when all σ_{j} are circular arcs concentric with the circular ring. Let G be the family of all Kquasiconformal mappings z = g(w) each of which maps B onto a circular ring Q(g) < |z| < 1 that has been slit along pn circular arcs L_{1}(g), …, L_{pn}(g) concentric with the circular ring such that |z| = 1, |z| = Q and L_{j} correspond to |w| = 1, |w| = q and σ_{j} (j = 1, …, pn), respectively, g(1) = 1 and satisfies the p times rotation symmetry

Vo Dang Thao: Univ. Ho Chi Minh City, Dept. Math., 227 Nguyen Van Cu, Q 5, Ho Chi Minh City, Vietnam e-mail: thaovd@usa.net ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

g(e^{i \frac{2π}{p}} w) = e^{i \frac{2π}{p}} g(w)  (w ∈ B).  (1.1)

It is clear that the symmetry condition is trivial for p = 1. Therefore we see that the inverse mapping f of g, g ∈ G, belongs to the family F studied in [17], because from (1.1) for all z ∈ A, A = g(B), and f = g^{−1}, g ∈ G, the relation

e^{i \frac{2π}{p}} f(z) = f(e^{i \frac{2π}{p}} z)

follows. Put R_{j}(g) = |z| with z ∈ L_{j}(g) and g ∈ G

c_{j} = min |w| and d_{j} = max |w| with w ∈ σ_{j} (j = 1, …, pn)
c = min{c_{1}, …, c_{pn}} and d = max{d_{1}, …, d_{pn}}
μ = sup ∏_{j=1}^{J} \frac{r′_{j}}{r_{j}}, where r_{j} < |w| < r′_{j} are pairwise disjoint sets in B.

Furthermore, denote by S the inner area of B by s_{j} the external area of the compact set bounded by σ_{j}.

Clearly,

\frac{c}{qd} ≤ μ ≤ \frac{1}{q}  (1.2)

and

S + s = π(1 − q^{2})  with s = ∑_{j=1}^{pn} s_{j}.  (1.3)

Our task is to estimate the radii Q(g) and R_{j}(g) (g ∈ G; j = 1, …, pn) that are nothing but the Riemann moduli of the domain B in the case K = 1 (see [11: p. 334]); as well as |g(w)| (g ∈ G, w ∈ B), by at most the quantities K, p, q, c_{j}, d_{j}, s_{j}, μ, S and |w|. The bounds in the estimates will be explicitly calculated by simple functions or the auxiliary function R(p, t, s) introduced in Section 4. Most of them are the best among the bounds that depend on the same quantities.

2. Estimates of Q

The estimate of Q plays an important role in establishing estimates for other quantities. Therefore we begin with this estimate.

Theorem 1. Under the above hypothesis and notations we have, for every g ∈ G, the estimate

(1 + \frac{S}{πq^{2}} )^{− \frac{K}{2}} ≤ Q(g) ≤ μ^{− \frac{1}{K}} ,  (2.1)

where the equality on the left-hand side holds if and only if B = B_{0} and g(w) = w|w|^{K−1} (w ∈ B), and the equality on the right-hand side holds if and only if B = B_{0} and g(w) = w|w|^{\frac{1}{K} −1} (w ∈ B).

Proof. Applying [17: Theorem 3.1] to the mapping f = g^{−1}, g ∈ G, we have

S ≥ πq^{2}(Q^{− \frac{2}{K}} − 1),

hence the lower bound of Q in (2.1) follows. Here the equality holds if and only if f(z) = z|z|^{\frac{1}{K} −1} (z ∈ A), i.e. B = B_{0} and g(w) = w|w|^{K−1} (w ∈ B). On the other hand, applying [16: Theorem 1] to the mapping g ∈ G, we obtain

π ≥ πQ^{2}μ^{\frac{2}{K}} ,

hence the upper bound of Q in (2.1) follows. Here with the help of [17: Formula (2.5)] we have the assertion on the occurence of the equality

Corollary 1. By (1.3), the lower bound of Q in (2.1) may be written in the form

Q(g) ≥ q^{K} (1 − \frac{s}{π} )^{− \frac{K}{2}}  (g ∈ G)  (2.2)

hence

Q(g) ≥ q^{K}  (g ∈ G).  (2.3)

Equality in (2.2) and (2.3) can only occur if B = B_{0} and g(w) = w|w|^{K−1} (w ∈ B).

Corollary 2. From (2.1) and (1.2) we obtain the estimate

Q(g) ≤ (\frac{dq}{c} )^{\frac{1}{K}} (< 1) (g ∈ G)

where the equality can only occur if c = d and g(w) = w|w|^{\frac{1}{K} −1} (w ∈ B).

3. Lower bounds of R_{j}

Since R_{j}(g) > Q(g) (g ∈ G; j = 1, …, pn), with the help of (2.2) or (2.3) we can get lower bounds of R_{j}. However, we want to establish other relations that, in certain situations, may give sharper estimates.

Theorem 2. Under the hypothesis and notations given in Section 1, for every g ∈ G with Q(g) = Q > q^{K} we have the estimates

R_{j}(g) > Q[\frac{ps_{j}}{π(Q^{\frac{2}{K}} − q^{2})}]^{\frac{K}{2}}  (j = 1, …, pn)  (3.1)

and

max_{1≤j≤pn} R_{j}(g) > Q[\frac{s}{π(Q^{\frac{2}{K}} − q^{2})}]^{\frac{K}{2}} .  (3.2)

Proof. First, we notice that by (1.1), each circle |z| = R_{j} contains at least p slits belonging to the boundary of A = g(B), g ∈ G. By [17: Formula (3.1)], we therefore have

ps_{j} ≤ πRj^{\frac{2}{K}} (1 − q^{2}Q^{− \frac{2}{K}}),

hence for every g ∈ G with q < Q^{\frac{1}{K}}, i.e., Q(g) > q^{K}, estimate (3.1) follows. Similarly we obtain, with the help of [17: formula (3.2)], estimate (3.2)

4. The auxiliary functions R(p, t, s) and T (p, r, s)

In order to establish other estimates of R_{j}, Q and |g(w)| we will introduce the following two real functions.

Definition. The real functions

\begin{matrix}r & = R(p, t, s) (0 ≤ s < t < 1) \\ t & = T (p, r, s) (0 ≤ s < r < 1) \end{matrix} \} (p ∈ ℕ)

are defined in such a way that the circular ring s < |w| < 1 with p radial slits

P_{j} = \{w| s ≤ |w| ≤ t and arg w = j \frac{2π}{p} \} (j = 0, …, p − 1)

and the circular ring r < |z| < 1 can be schlicht conformal mapped onto each other.

Because of the monotony of the modulus of a doubly-connected domain (see [2: p. 176]) we have the following monotonies of the auxiliary function R(p, t, s) with p ∈ ℕ:

s < R(p, t, s) < t (0 ≤ s < t < 1)  (4.1)
R(p, t_{1}, s) < R(p, t_{2}, s) (0 ≤ s < t_{1} < t_{2} < 1)  (4.2)
R(p, t, s_{1}) < R(p, t, s_{2}) (0 ≤ s_{1} < s_{2} < t < 1)  (4.3)
R(p, t, s) > R(1, t, s) (0 ≤ s < t < 1; p ≥ 2)

With the help of Hersch’s [9: p. 316] and Nehari’s [11: p. 295] formulae, I myself found in [14: pp. 101 - 104] the following expression for R(p, t, s):

R(p, t, 0) = exp \{\frac{−πK′(t^{p})}{2pK(t^{p})}\}  (0 < t < 1; p ∈ ℕ)  (4.4)

with

\begin{aligned}K(k) & = ∫_{0}^{1} \frac{dx}{\sqrt{(1 − x^{2})(1 − k^{2}x^{2})}} \\ K′(k) & = K(\sqrt{1 − k^{2}}) \end{aligned}

and for 0 < s < t < 1

R(p, t, s) = exp \{\frac{−πK′(u)}{2pK(u)}\}

with

u = 1 + h − \sqrt{h(2 + h)}

where

h = \frac{(1 − k)(1 − ak)}{k(1 + a)}, k = 4s^{p} ∏_{n=1}^{∞} [\frac{1 + s^{4pn}}{1 + s^{2p(2n−1)}} ]^{4}

a = sn (b + \frac{i2pb}{π} log \frac{t}{s}, k) , b = K(k).

Here sn(z, k) means the Jacobian elliptic sinus with the parameter k. Another expression for R(p, t, s) was shown by Graeser in [3: pp. 77 - 78].

In view of [14: Formula (1.21)] we have the estimate

4^{− \frac{1}{p}} t < R(p, t, s) < t  (0 ≤ s < t < 1; p ∈ ℕ),  (4.5)

hence R(p, t, s) ≈ t when p → ∞.

The evaluation of K(t^{p}) and K′(t^{p}) (see [18: p. 177]) yields the asymptotic behaviour of R(p, t, 0):

R(p, t, 0) ≈ 4^{− \frac{1}{p}} t  when t → 0  (4.6)

and

1 − R(p, t, 0) ≈ \frac{π^{2}}{2p log \frac{8}{p(1−t)}} when t → 1.

Successive approximations for R(1, t, 0) are given by Lehto [10: p. 64]. The expression for T (p, r, s), that is not needed here, was shown by Thao ([14: pp. 102 - 105] or [17: p. 61]).

5. Other estimates of R_{j}, Q and |g(w)|

Using the auxiliary functions studied in Section 4, other estimates for R_{j}, Q and |g(w)| will be given. In particular, when s_{j} = 0 or s = 0, they may be sharper than ones of (3.1) and (2.2).

Theorem 3. Under the hypothesis and notations given in Sections 1 and 4, for every g ∈ G, w ∈ B and j = 1, …, pn, we have the estimates

R^{K}(p, d_{j}, q) < R_{j}(g) < Q(g)R^{−K}(p, \frac{q}{c_{j}} , q)  (5.1)
Q(g) > R^{K}(p, d_{j}, q) R^{K}(p, \frac{q}{c_{j}} , q)  (5.2)
R^{K}(p, |w|, q) < |g(w)| < Q(g)R^{−K}(p, \frac{q}{|w|}, q).  (5.3)

Proof. Considering the mapping f = g^{−1}, g ∈ G, with the help of [17: Theorem 6.1] we obtain

d_{j} < T (p, Rj^{\frac{1}{K}} , q) = t  and  \frac{q}{c_{j}} < T [p, (\frac{Q}{R_{j}} )^{\frac{1}{K}} , q] = t′.

Hence the definition of the auxiliary functions and the monotony (4.2) yield the relations

Rj^{\frac{1}{K}} = R(p, t, q) > R(p, d_{j}, q)  and  (\frac{Q}{R_{j}} )^{\frac{1}{K}} = R(p, t′, q) > R(p, \frac{q}{c_{j}} , q).

Thus we have estimate (5.1). Estimate (5.2) is just a consequence of (5.1). Using [17: Formula (6.10)], we obtain similary estimate (5.3)

Corollary 3. From Theorem 3 and (4.5), for every g ∈ G, w ∈ B and j = 1, …, pn we obtain the simple estimates

4^{− \frac{K}{p}} d_{j}^{K} < R_{j}(g) < 4^{\frac{K}{p}} Q(g)(\frac{c_{j}}{q} )^{K}  (5.4)
Q(g) > 4^{−2 \frac{K}{p}} (\frac{qd_{j}}{c_{j}} )^{K}  (5.5)

4^{− \frac{K}{p}} |w|^{K} < |g(w)| < 4^{\frac{K}{p}} Q(g)(\frac{|w|}{q})^{K}.  (5.6)

In view of (4.3) and (4.6) we see that the coefficients in (5.4) - (5.6) are the best possible.

Acknowledgement. This paper has been prepared during the research visit of the author at the TU Berlin in July and August 1998 under the patronage of DAAD. I would like to thank the DAAD, Professor Pommerenke and his group for their kind support.

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Received 21.09.1998; in revised form 12.07.1999