Georgian Mathematical Journal Volume 9 (2002), Number 2, 367–382

CONFORMAL AND QUASICONFORMAL MAPPINGS OF CLOSE MULTIPLY-CONNECTED DOMAINS

Z. SAMSONIA^{†} AND L. ZIVZIVADZE

Abstract. Doubly-connected and triply-connected domains close to each other in a certain sense are considered. Some questions connected with conformal and quasiconformal mappings of such domains are studied using integral equations.

2000 Mathematics Subject Classification: 30C62.

Key words and phrases: Conformal mapping, canonical domain, close domain, quasiconformal mapping, integral equation.

1. Conformal Mapping of Close Triply-Connected Domains

Let G be a finite triply-connected domain of a complex plane Z bounded by the simple Lyapunov curves Γ_{0}, Γ_{1}, Γ_{2}, one of which Γ_{0} envelops the other two and z = 0 ∈ int Γ_{1}.

Assume first that the boundary Γ = ⋃_{i=0}^{2} Γ_{i} belongs to the class C′_{α} (\frac{1}{2} < α ≤

1), while singly-connected domains with boundaries Γ_{1} and Γ_{0} are star-like with respect to z = 0. Let the equations of these curves be given in terms of polar coordinates

t = g_{1}(φ) = ρ_{1}(φ) · e^{iφ}, t = g_{0}(φ) = ρ_{0}(φ) · e^{iφ} (0 ≤ φ ≤ 2π)

and the finite domain with boundary Γ_{2} be star-like with respect to z_{0} ∈ int Γ_{2}. If we assume that the polar axis with a pole in z_{0} is parallel to the abscissa axis, then the parametric equation for Γ_{2} can be written in the form

t = g_{2}(φ) = z_{0} + ρ_{2}(φ)e^{iφ} (0 ≤ φ ≤ 2π).

Let us consider the second triply-connected domain G̃ of type G bounded by the curves Γ̃_{0}, Γ̃_{1}, Γ̃_{2} (with the same properties) whose parametric equations are

t = ρ̃_{1}(φ)e^{iφ}, t = z_{0} + ρ̃_{2}(φ)e^{iφ}, t = ρ̃_{0}(φ)e^{iφ} (0 ≤ φ ≤ 2π).

We introduce the following notation:

d_{1} = ρ(Γ_{1}; Γ_{0}), d_{2} = ρ(Γ_{1}; Γ_{2}), d_{3} = ρ(Γ_{2}; Γ_{0}), d_{0} = min{d_{1}; d_{2}; d_{3}}.

It is assumed that ε ∈ (0; d_{0}/2).

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Definition 1. The domains G and G̃ are called ε-close to each other if the conditions

|ρ_{i}(φ) − ρ̃_{i}(φ)| ≤ ε; ∥ρ′_{i}(φ) − ρ̃_{i}(φ)∥_{C_{α}} ≤ ε (i = 0, 1, 2)  (1)

are fulfilled.

An infinite set of domains ε-close to G are formed for any ε ∈ (0; d/2). We denote it by G_{ε}.

Let us conformally map the domains G and G̃ onto the canonical domains K(ρ; r; 1) and K̃(ρ̃; r̃; 1), respectively, using the assumptions of [1], where K(ρ; r; 1) and K̃(ρ̃; r̃; 1) are annuli with concentric cuts along the arc of the circumferences |W | = r and |W | = r̃, (ρ < r < 1, ρ̃ < r̃ < r < 1), respectively. Then for the definition of radii we have [1]

ln ρ = \frac{1}{π} ∫_{0}^{2π}ν_{1}(φ)\sqrt{ρ_{1}^{2}(φ) + [ρ′_{1}(φ)]^{2}} dφ ,

ln ρ̃ = \frac{1}{π} ∫_{0}^{2π}ν̃_{1}(φ)\sqrt{ρ̃_{1}^{2}(φ) + [ρ̃′_{1}(φ)]^{2}} dφ ,

ln r = \frac{1}{π} ∫_{0}^{2π}ν_{2}(φ)\sqrt{ρ_{2}^{2}(φ) + [ρ′_{2}(φ)]^{2}}dφ ,

ln r̃ = \frac{1}{π} ∫_{0}^{2π}ν̃_{2}(φ)\sqrt{ρ̃_{2}^{2}(φ) + [ρ̃′_{2}(φ)]^{2}} dφ .

(2)

It is assumed here that

ν(t(φ)) = \begin{cases}ν_{1}(φ), & when t ∈ Γ_{1}, \\ ν_{2}(φ), & when t ∈ Γ_{2}, \\ ν_{0}(φ), & when t ∈ Γ_{0}. \end{cases}

The reasoning for ν̃(t̃(φ)) is analogous. It is assumed that ν(t) and ν̃(t) are unique solutions of the integral equations

ν(t_{0}) + \frac{1}{π} ∫_{Γ} K_{0}(t; t_{0})ν(t)dt = − ln |t_{0}|, t_{0} ∈ Γ,  (3)

ν̃(t_{0}) + \frac{1}{π} ∫ ; _{Γ} K̃_{0}(t; t_{0})ν̃(t)dt = − ln |t_{0}|, t_{0} ∈ Γ̃,  (4)

where

K_{0}(t; t_{0}) = \begin{cases}Im (\frac{1}{t − t_{0}} · \frac{dt}{ds}) − 1, & when t, t_{0} ∈ Γ_{j} (j = 1, 2), \\ Im (\frac{1}{t − t_{0}} · \frac{dt}{ds}), & in all other cases. \end{cases}

K̃_{0}(t; t_{0}) is defined analogously.

Let us pose the problem: derive an estimate through ε for a difference of the solutions of equations (3) and (4) (in an appropriate norm), and also for the expressions |ρ − ρ̃|, |r − r̃|.

We can obtain such estimates by using the statements proved below. For this, the integral equations (3) and (4) are represented in the complex form:

ν(t_{0}) + \frac{1}{2πi} ∫_{Γ} \frac{1}{t′} K_{∗}(t; t_{0})ν(t)dt = − ln |t_{0}|,  (3_{1})

ν̃(t_{0}) + \frac{1}{2πi} ∫ ; _{Γ} \frac{1}{t′} K̃_{∗}(t; t_{0})ν̃(t)dt = − ln |t_{0}|,  (4_{1})

where

K_{0}(t; t_{0})=\{ \begin{matrix}(t′ − t′_{0}) + (t̄′_{0} − t̄′)\frac{t − t_{0}}{t̄− t̄_{0}} + [t′_{0} \frac{t̄− t̄_{0}}{t − t_{0}} − t̄′_{0}] \frac{t − t_{0}}{t̄− t̄_{0}} \frac{1}{t − t_{0}} − 2i, \\ when t, t_{0} ∈ Γ_{i} (i = 1, 2), \\ (t′ − t′_{0}) + (t̄′_{0} − t̄′)\frac{t − t_{0}}{t̄− t̄_{0}} + [t′_{0} \frac{t̄− t̄_{0}}{t − t_{0}} − t̄′_{0}] \frac{t − t_{0}}{t̄− t̄_{0}} \frac{1}{t − t_{0}} \\ in all other cases. \end{matrix} (5)

Here t′ = g′_{j}(φ), t′_{0} = g′_{j}(φ_{0}) and j take values 0, 1, 2 depending on the fact to which contour Γ_{j} (j = 0, 1, 2) the point t or t_{0} belongs. K̃^{∗}(t; t_{0}), too, is constructed analogously to (5). Clearly, in that case t = g̃_{j}(φ), t_{0} = g̃_{j}(φ_{0}), t′ = g̃′_{j}(φ), t′_{0} = g̃′_{j}(φ_{0}) (j = 0, 1, 2).

Let us represent the integral equations (3_{1}), (4_{1}) in the operator form:

Aν = (I + H)ν = f_{0},  (3′_{1})
Ãν̃ = (I + H̃)ν̃ = f̃_{0}.  (4′_{1})

It is assumed that (analogously to ν[t(φ)]), ν̃(t), f_{0}(t), f̃_{0}(t) are columnmatrices, I is the uni t matrix of third order, and

Hν = (\begin{matrix}H_{11}ν_{1} + H_{12}ν_{2} + H_{10}ν_{0} \\ H_{21}ν_{1} + H_{22}ν_{2} + H_{20}ν_{0} \\ H_{01}ν_{1} + H_{02}ν_{2} + H_{00}ν_{0} \end{matrix}) ,

H̃ν̃ = (\begin{matrix}H̃_{11}ν̃_{1} + H̃_{12}ν̃_{2} + H̃_{10}ν̃_{0} \\ H̃_{21}ν̃_{1} + H̃_{22}ν̃_{2} + H̃_{20}ν̃_{0} \\ H̃_{01}ν̃_{1} + H̃_{02}ν̃_{2} + H̃_{00}ν̃_{0} \end{matrix}) .

Here H_{ij}, H̃_{ij} (i, j = 0, 1, 2) are the concrete integral operators. Before representing them explicitly, let us make some additional observations.

Observe that

\frac{dt}{t − t_{0}} |_{{t∈Γ_{j} \atop t_{0}∈Γ_{k}}} = \frac{g′_{j}(φ)dφ}{g_{j}(φ) − g_{k}(φ_{0})} = M_{jk}(φ; φ_{0})( ctg \frac{φ − φ_{0}}{2} + i)dφ,

where the function

M_{jk}(φ; φ_{0}) = \frac{g′_{j}(φ)}{2ie^{iφ}} · \frac{e^{iφ} − e^{iφ_{0}}}{g_{j}(φ) − g_{k}(φ_{0})}  (6)

satisfies the Hölder condition with index α with respect to its arguments, i.e., belongs to the class C_{α}(Γ_{j} × Γ_{k}).

The function

R_{jk}(t_{0}; t) = \frac{1}{t′} \{(t′ − t′_{0}) + (t̄′_{0} − t̄′)\frac{t − t_{0}}{t̄− t̄_{0}} + [t′_{0} \frac{t̄− t̄_{0}}{t − t_{0}} − t̄′_{0}] \frac{t − t_{0}}{t̄− t̄_{0}} \}

(t ∈ Γ_{j}, t_{0} ∈ Γ_{k}, j, k = 0, 1, 2) figuring in (5) also belongs to the class C_{α}(Γ_{j}×Γ_{k}) and R_{kk}(t_{0}; t_{0}) = 0.

We have in terms of the parameter φ

S_{j k}(φ; φ_{0}) = R_{j k}[g_{j}(φ); g_{k}(φ_{0})] = \frac{1}{g′_{j}(φ)}\{g′_{j}(φ) − g′_{k}(φ_{0})

+ (ḡ′_{k}(φ_{0}) − ḡ′_{j}(φ)) \frac{g_{j}(φ) − g_{k}(φ_{0})}{ḡ_{j}(φ) − ḡ_{k}(φ_{0})}

+ [g′_{k}(φ_{0}) \frac{ḡ_{j}(φ) − ḡ_{k}(φ_{0})}{g_{j}(φ) − g_{k}(φ_{0})} − ḡ′_{k}(φ_{0})]\frac{g_{j}(φ) − g_{k}(φ_{0})}{ḡ_{j}(τ) − ḡ_{k}(τ_{0})}\}, (5′)

S_{jk}(φ; φ_{0}) ∈ C_{α}(Γ_{j} × Γ_{k}), S_{kk}(φ_{0}; φ_{0}) = 0 (0 ≤ φ ≤ 2π).

M̃_{jk}(φ, φ_{0}) and S̃_{jk}(φ; φ_{0}) are defined analogously.

Having introduced the notation, we can write that

[H_{11}ν_{1} + H_{12}ν_{2} + H_{10}ν_{0}](φ_{0})

= \frac{1}{2πi} ∫_{0}^{2π}\{\frac{S_{11}(φ_{0}; φ)M_{11}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β} − 2i\} ν_{1}(φ)dφ

+\frac{1}{2πi} ∫_{0}^{2π} \frac{S_{12}(φ_{0}; φ)M_{12}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}ν_{2}(φ)dφ

+\frac{1}{2πi} ∫_{0}^{2π} \frac{S_{10}(φ_{0}; φ)M_{10}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}ν_{0}(φ)dφ,

[H_{21}ν_{1} + H_{22}ν_{2} + H_{20}ν_{0}](φ_{0})

= \frac{1}{2πi} ∫_{0}^{2π}\{\frac{S_{21}(φ_{0}; φ)M_{21}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}\} ν_{1}(φ)dφ

CONFORMAL AND QUASICONFORMAL MAPPINGS 371

+\frac{1}{2πi} ∫_{0}^{2π}\{\frac{S_{22}(φ_{0}; φ)M_{22}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β} − 2i\} ν_{2}(φ)dφ

+\frac{1}{2πi} ∫_{0}^{2π}\{\frac{S_{20}(φ_{0}; φ)M_{20}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}\} ν_{0}(φ)dφ,

[H_{01}ν_{1} + H_{02}ν_{2} + H_{00}ν_{0}](φ_{0})

= \frac{1}{2πi} ∫_{0}^{2π}\{\frac{S_{01}(φ_{0}; φ)M_{01}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}\} ν_{1}(φ)dφ

+\frac{1}{2πi} ∫_{0}^{2π}\{\frac{S_{02}(φ_{0}; φ)M_{02}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}\} ν_{2}(φ)dφ

+\frac{1}{2πi} ∫_{0}^{2π}\{\frac{S_{00}(φ_{0}; φ)M_{00}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} · ( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}\} ν_{0}(φ)dφ,

where β is any number satisfying the condition \frac{1}{2} < β ≤ α.

From the above formulas we obtain

(H_{11} − H̃_{11})ν_{1} = \frac{1}{πi} ∫_{0}^{2π} \frac{S_{11}(φ_{0}; φ)M_{11}(φ_{0}; φ) − S̃_{11}(φ_{0}; φ)M̃_{11}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}}

× \frac{1}{2}( ctg \frac{φ − φ_{0}}{2} + i)|e^{iφ} − e^{iφ_{0}}|^{β}ν_{1}(φ)dφ

= \frac{1}{πi} ∫_{0}^{2π} \frac{S_{11}(φ_{0}; φ)M_{11}(φ_{0}; φ) − S̃_{11}(φ_{0}; φ)M̃_{11}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}}

× \frac{ie^{iφ}}{e^{iφ} − e^{iφ_{0}}} |e^{iφ} − e^{iφ_{0}}|^{β}ν_{1}(φ)dφ

= \frac{1}{πi} ∫_{0}^{2π}[K_{11}(φ_{0}; φ) − K̃_{11}(φ_{0}; φ)]ν_{1}(φ)dφ.  (7)

In (7)

\begin{aligned}K_{11}(φ_{0}; φ) & = K_{11}^{(1)}(φ_{0}; φ)K_{11}^{(2)}(φ_{0}; φ), \\ K̃_{11}(φ_{0}; φ) & = K̃_{11}^{(1)}(φ_{0}; φ)K̃_{11}^{(2)}(φ_{0}; φ), \end{aligned}  (7′)

where

\begin{matrix}K_{11}^{(1)}(φ_{0}; φ) = \frac{M_{11}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} ; K_{11}^{(2)}(φ_{0}; φ) = \frac{iS_{11}(φ_{0}; φ)e^{iφ}}{|e^{iφ} − e^{iφ_{0}}|^{1−β} · e^{i arg(e^{iφ}−e^{iφ_{0}})}} ; \\ K̃_{11}^{(1)}(φ_{0}; φ) = \frac{M̃_{11}(φ_{0}; φ)}{|e^{iφ} − e^{iφ_{0}}|^{β}} ; K̃_{11}^{(2)}(φ_{0}; φ) = \frac{iS̃_{11}(φ_{0}; φ)e^{iφ}}{|e^{iφ} − e^{iφ_{0}}|^{1−β} · e^{i arg(e^{iφ}−e^{iφ_{0}})}} . \end{matrix} (8)

According to [2], if φ(t) ∈ Cμ(Γ), 0 < μ ≤ 1, then the function of two variables (t, t_{0} ∈ Γ)

ψ(t; t_{0}) = \frac{φ(t) − φ(t_{0})}{|t − t_{0}|^{λ}},

0 ≤ λ < μ ≤ 1, satisfies, on Γ, the Hölder condition with index μ−λ. Moreover, the estimate

∥ψ(t_{0}; t)∥_{C_{μ−λ}} ≤ A^{∗}(1 + λ)  (9)

holds true, where A^{∗} ≥ \frac{|φ(t)−φ(t_{0})|}{|t−t_{0}|^{λ}} (see §§5, 6 of [2]). Taking into account the structure of the functions M_{11}(φ; φ_{0}), S_{11}(φ; φ_{0}), M̃_{11}(φ; φ_{0}), S̃_{11}(φ; φ_{0}), this result implies that the functions K_{11}(φ; φ_{0}) and K̃_{11}(φ; φ_{0}) are continuous in the Hölder sense with respect to φ_{0} and φ with index δ, δ = min{α−β; α+β−1} and

∥K_{11}(φ; φ_{0}) − K̃_{11}(φ; φ_{0})∥_{C_{δ}} = ∥K_{11}^{(1)}K_{11}^{(2)} − K̃_{11}^{(1)}K̃_{11}^{(2)}∥_{C_{δ}}
≤ ∥K_{11}^{(2)}∥_{C_{δ}} · ∥K_{11}^{(1)} − K̃_{11}^{(1)}∥_{C_{δ}} + ∥K̃_{11}^{(1)}∥_{C_{δ}} · ∥K_{11}^{(2)} − K̃_{11}^{(2)}∥_{C_{δ}}. (10)

Let us define the order of smallness with respect to ε in (10). Preliminarily, we will prove the validity of the following propositions.

For small values of ε the following inequalities are fulfilled:

I. ∥[g′_{1}(φ) − g′(φ_{0})] − [g̃′_{1}(φ) − g̃′(φ_{0})]∥_{C_{α}} ≤ 2ε,
∥[ḡ′_{1}(φ) − ḡ′(φ_{0})] − [\overline{g̃}′_{1}(φ) − ¯ g̃′(φ_{0})]∥_{C_{α}} ≤ 2ε,
II. ∥\frac{e^{iφ} − e^{iφ_{0}}}{g_{1}(φ) − g_{1}(φ_{0})} · \frac{g′_{1}(φ)}{2ie^{iφ}} − \frac{e^{iφ} − e^{iφ_{0}}}{g̃_{1}(φ) − g̃_{1}(φ_{0})} · \frac{g̃′_{1}(φ)}{2ie^{iφ}} ∥_{C_{α}} ≤ A_{1} · ε,

III. ∥\frac{g_{1}(φ) − g_{1}(φ_{0})}{\overline{g}_{1}(φ) − \overline{g}_{1}(φ_{0})} − \frac{g̃_{1}(φ) − g̃_{1}(φ_{0})}{\overline{g̃}_{1}(φ) − \overline{g̃}_{1}(φ_{0})}∥_{C_{α}} ≤ A_{2} · ε,

IV. |g̃′_{1}(φ)| ≥ \frac{min_{[0,2π]} |g′_{1}(φ)|}{2} for ε ≤ \frac{min_{[0,2π]} |g′_{1}(φ)|}{2} ,

V. ∥(g′_{1}(φ_{0}) · \frac{\overline{g}_{1}(φ) − \overline{g}_{1}(φ_{0})}{g_{1}(φ) − g_{1}(φ_{0})} − g′_{1}(φ_{0}))

(−g̃′_{1}(φ_{0}) · \frac{\overline{g̃}_{1}(φ) − \overline{g̃}_{1}(φ_{0})}{g̃_{1}(φ) − g̃_{1}(φ_{0}} − \overline{g̃}′_{1}(φ_{0}))∥_{C_{α}} ≤ A_{3}ε

and all constants contained in the estimates do not depend on G̃.

Proof. Inequality I immediately follows from the definition of domain closeness, i.e., from (1).

Further we have

|g′_{1}(φ)| − |g̃′_{1}(φ)| ≤ ∥g′_{1}(φ) − g̃′_{1}(φ)∥_{C} ≤ ∥g′_{1}(φ) − g̃′_{1}(φ)∥_{C_{α}} < ε

and if it is assumed that ε ≤ \frac{1}{2} min_{[0,2π]} |g′_{1}(φ)|, then the validity of inequality IV is

proved.

Before proving inequality II, note the following: for |s − s_{0}| ≤ \frac{l_{1}}{2} , where l_{1} is the length of Γ_{1}, and s and s_{0} are the arc abscissas of the points t = g_{1}(φ) and t_{0} = g_{1}(φ_{0}), we have

\frac{|t − t_{0}|^{2}}{|φ − φ_{0}|^{2}} = \frac{|g_{1}(φ) − g_{1}(φ_{0})|^{2}}{|s − s_{0}|^{2}} · \frac{|s − s_{0}|^{2}}{|φ − φ_{0}|^{2}} ≥ k_{1}^{2} · ρ_{1}^{2},  (11)

where ρ_{1} = min_{[0,2π]} |g1^{′}(φ)| = ρ(0; Γ_{1}) > 0, and k_{1} (0 < k_{1} < 1) is the constant

defined by giving the contour Γ_{1}.

On the other hand,

[g_{1}(φ) − g_{1}(φ_{0})] − [g̃_{1}(φ) − g̃_{1}(φ_{0})] = [g_{1}(φ) − g̃_{1}(φ)] − [g_{1}(φ_{0}) − g̃_{1}(φ_{0})] = Re[g_{1}(φ) − g̃_{1}(φ)] − Re[g_{1}(φ_{0}) − g̃_{1}(φ_{0})]

+i (Im[g_{1}(φ) − g̃_{1}(φ)] − Im[g_{1}(φ_{0}) − g̃_{1}(φ_{0})])

= (φ − φ_{0}) Re[g1^{′}(ξ) − g̃1^{′}(ξ)] + i (φ − φ_{0}) Im[g1^{′}(η) − g̃1^{′}(η)],

where the numbers ξ and η lie between φ and φ_{0}.

By virtue of (1) we can write that

|[g̃(φ) − g̃(φ_{0})] − [g_{1}(φ) − g_{1}(φ_{0})]| ≤ 4ε|φ − φ_{0}|.

Moreover, since

|g_{1}(φ) − g_{1}(φ_{0})| − |g̃_{1}(φ) − g̃_{1}(φ_{0})|
≤ |[g_{1}(φ) − g_{1}(φ_{0})] − [g̃_{1}(φ) − g̃_{1}(φ_{0})]| ≤ 4ε|φ − φ_{0}|

we have

|g̃_{1}(φ) − g̃_{1}(φ_{0})| ≥ |g_{1}(φ) − g_{1}(φ_{0})| − 4ε|φ − φ_{0}|.

Hence, assuming that ε ≤ \frac{k_{1}ρ_{1}}{8} , by (11) we obtain

\frac{|g̃_{1}(φ) − g̃_{1}(φ_{0})|}{|φ − φ_{0}|} > \frac{k_{1}ρ_{1}}{2} .  (12)

Now we return to proving inequality II. Since

\frac{e^{iφ} − e^{iφ_{0}}}{g_{1}(φ) − g_{1}(φ_{0})} · \frac{g1^{′}(φ)}{2ie^{iφ}} − \frac{e^{iφ} − e^{iφ_{0}}}{g̃_{1}(φ) − g̃_{1}(φ_{0})} · \frac{g̃1^{′}(φ)}{2ie^{iφ}}

= \frac{e^{i φ} − e^{i φ_{0}}}{2 i e^{i φ}} · \frac{g1^{′}(φ)[g̃_{1}(φ) − g̃_{1}(φ_{0})] − g̃1^{′}(φ)[g_{1}(φ) − g_{1}(φ_{0})]}{[g_{1}(φ) − g_{1}(φ_{0})][g̃_{1}(φ) − g̃_{1}(φ_{0})]} = \frac{e^{iφ} − e^{iφ_{0}}}{2ie^{iφ}}

×\frac{(φ − φ_{0})[g1^{′}(φ) ∫_{0}^{1} g̃1^{′}[φ_{0} + u(φ − φ_{0})] du − g̃1^{′}(φ) ∫_{0}^{1} g1^{′}[φ_{0} + u(φ − φ_{0})]du]}{[g_{1}(φ) − g_{1}(φ_{0})][g̃_{1}(φ) − g̃_{1}(φ_{0})]} ,

by virtue of I, IV, (11) and (12) this expression obviously implies that inequality II is valid.

Just in the same way, after carrying out analogous transformations, from

\frac{g_{1}(φ) − g_{1}(φ_{0})}{g̃_{1}(φ) − g̃_{1}(φ_{0})} − \frac{g̃_{1}(φ) − g̃_{1}(φ_{0})}{¯g̃_{1}(φ) − ¯ g̃_{1}(φ_{0})} = \frac{(φ − φ_{0})^{2}}{[ḡ_{1}(φ) − ḡ_{1}(φ_{0})][¯g̃_{1}(φ) − ¯ g̃_{1}(φ_{0})]}

× [∫_{0}^{1} g1^{′}[φ_{0} + u(φ − φ_{0})]du ∫_{0}^{1} ¯ g̃_{1}[φ_{0} + u(φ − φ_{0})]du

− ∫_{0}^{1} g̃1^{′}[φ_{0} + u(φ + φ_{0})]du ∫_{0}^{1} ḡ_{1}[φ_{0} + u(φ − φ_{0})]du]

follows the validity of inequality III.

Inequality V immediately follows from inequalities I, II and IV.

The validity of I–V is proved.

Now taking into account formulas (6) and (5′) for M_{11}(φ, φ_{0}), S_{11}(φ, φ_{0}) and analogous formula for M̃_{11}(φ, φ_{0}), S̃_{11}(φ, φ_{0}), we can state by virtue of the above inequalities and S_{11}(φ_{0}, φ_{0}) = S̃_{11}(φ_{0}, φ_{0}) that

\begin{matrix}∥S_{11}(φ, φ_{0}) e^{i arg(e^{i φ}−e^{i φ_{0}})} − S̃_{11}(φ, φ_{0}) e^{i arg(e^{i φ}−e^{i φ_{0}})}∥_{c_{α}} ≤ B_{1} · ε, \\ ∥M_{11}(φ, φ_{0}) − M̃_{11}(φ, φ_{0})∥_{c_{α}} ≤ B_{2} · ε, \end{matrix} (13)

where the constants B_{1} and B_{2} depend only on the domain G.

By virtue of (9) and (13), inequality (10) immediately gives rise to the following estimates of its individual terms:

\begin{matrix}∥K_{11}^{(1)}(φ, φ_{0}) − K̃_{11}^{(1)}(φ, φ_{0})∥_{c_{δ}} ≤ N_{1} · ε, \\ ∥K_{11}^{(2)}(φ, φ_{0}) − K̃_{11}^{(2)}(φ, φ_{0})∥_{c_{δ}} ≤ N_{2} · ε, \\ ∥K̃_{11}^{(1)}(φ, φ_{0})∥_{c_{δ}} ≤ N_{3}, ∥K̃_{11}^{(2)}(φ, φ_{0})∥_{c_{δ}} ≤ N_{4}, \end{matrix}  (14)

(δ = min{α − β; α + β − 1})), where all constants are expressed in terms of the initial domain G, i.e.,for small ε the estimates are uniform with respect to domains G̃ ∈ G_{ε}.

Therefore for the kernels represented by formulas (7′) we have

∥K_{11}(φ; φ_{0}) − K̃_{11}(φ; φ_{0})∥_{c_{δ}} ≤ N_{1}(β) · ε.  (15)

The estimates

\begin{matrix}∥K_{22}(φ; φ_{0}) − K̃_{22}(φ; φ_{0})∥_{c_{δ}} ≤ N_{2}(β) · ε, \\ ∥K_{00}(φ; φ_{0}) − K̃_{00}(φ; φ_{0})∥_{c_{δ}} ≤ N_{0}(β) · ε \end{matrix}  (16)

are established analogously.

If the points t(φ), t(φ_{0}) and the corresponding points t̃(φ), t̃(φ_{0}) lie on different curves Γ_{i}, Γ_{j} and Γ̃_{i}, Γ̃_{j} (i ≠ j), respectively, then estimates for the values ∥K_{ij} − K̃_{ij}∥_{c_{δ}} (i ≠ j, i, j = 0, 1, 2) are established immediately and have order O(ε). This can be verified at once if, for the kernels of equations (3_{1}), (4_{1}) representable by formulas of form (5), the difference

K_{∗}[g_{i}(φ), g_{j}(φ_{0})] − K̃_{∗}[g̃_{i}(φ); g̃_{j}(φ_{0})]

is reduced to the common denominator and the latter is estimated from below by the number d_{0} = min\{d_{1}; d_{2}; d_{3}\} assuming that 0 < ε < \frac{d_{0}}{2} , while the difference of their numerators is estimated using inequality (1).

We have thus shown that the following theorem holds for small values of the parameter ε.

Theorem 1. If the domains G and G̃ ∈ G_{ε} (0 < ε ≤ ε_{0}) belong to the class G′_{α} (\frac{1}{2} < α ≤ 1), then the inequality

∥K_{∗}(φ; φ_{0}) − K̃_{∗}(φ; φ_{0})∥_{c_{δ}} < A_{0}(G; β) · ε  (17)

holds, where K_{∗}(φ; φ_{0}), K̃_{∗}(φ; φ_{0}) are the kernels of the integral equations (3_{1}), (4_{1}), respectively, δ = min\{α − β; α + β − 1\} and \frac{1}{2} < β ≤ α. The constant A_{0}(G; β) and small ε_{0} are completely defined by giving the initial domain G

ε_{0} = \frac{1}{2} min \{d_{0}; kρ/4; g(φ)\},

where k is the constant defined by giving the contour Γ, ρ = min{ρ(0, Γ_{j}) : j =

0, 1, 2}, g(φ) = min{|gj^{′}(φ)| : j = 0, 1, 2}.

The proven theorem makes it possible to obtain an estimate for the difference ν(φ) − ν̃(φ) in an adequate norm. It is of order O(ε), but it can be obtained in a stronger form if we use the result from [3]. This technique implies estimating, through ε, the difference of the corresponding integral operators.

Let us estimate the difference (H − H̃)ν. Having in mind the structure of this difference, it suffices to confine the investigation to the case (H_{11} − H̃_{11})ν_{1}.

From (7) we have

(H_{11} − H̃_{11}) ν_{1}
= \frac{1}{πi} ∫_{0}^{2π}K̃_{11}^{∗}(φ; φ_{0})ie^{i φ} · exp(−i arg(e^{iφ} − e^{iφ_{0}})) \frac{ν_{1}(φ)}{|e^{iφ} − e^{iφ_{0}}|^{1−β}} , (18)

where

K̃_{11}^{∗}(φ, φ_{0}) = \frac{S_{11}(φ; φ_{0})M_{11}(φ; φ_{0}) − S̃_{11}(φ; φ_{0})M̃_{11}(φ; φ_{0})}{|e^{iφ} − e^{iφ_{0}}|^{β}}.  (19)

As has already been noted, the function K̃_{11}^{∗}(φ; φ_{0}) is continuous in the Hölder sense with index α − β. Taking into account (9) and (19) as well as inequalities I–V, we see that the inequality

∥K̃_{11}^{∗}(φ; φ)∥_{c_{α−β}} ≤ C_{1}(β) ε  (20)

is valid for small values of the parameter ε.

Let us use the following result from [2]. A function of form (18)

ω(t(φ_{0})) = \frac{1}{π} ∫_{0}^{2π}K̃_{11}^{∗}(φ; φ_{0}) r(φ; φ_{0}) \frac{ν_{1}(φ) dφ}{|e^{i φ} − e^{i φ_{0}}|^{1−β}} ,

where r(φ; φ_{0}) = ie^{iφ} exp(−i arg(e^{iφ} − e^{iφ_{0}})) belongs to the class C_{δ} for any bounded function ν_{1}(φ) and δ = α − β (see §51 of [2]).

Now, by virtue of this result, (18) and (20) give rise to the estimate

∥(H_{11} − H̃_{11}) ν_{1}∥_{c_{α−β}} ≤ C′(β) ∥ν_{1}∥_{C_{α−β}} · ε,  (21)

where C′(β) and C_{1}(β) from (20) are the absolute constants, while ν(φ) is any function of the class C_{α−β}.

The proof that the values ∥(H_{ii} − H̃_{ii})ν_{i}∥_{C_{α−β}} for i = 0, 2 have order O(ε) repeats the proof of inequality (21).

The estimate for the values ∥(H_{ij} − H̃_{ij})ν_{j}∥_{C_{α−β}} when i ≠ j, i.e., when the points t(φ) and t_{0}(φ_{0}) belong to different curves Γ_{1} and Γ_{2}, while the corresponding points t̃(φ) and t̃_{0}(φ_{0}) belong to the contours Γ̃_{1} and Γ̃_{2}, is obtained immediately if it is assumed, for instance, ε < \frac{d_{0}}{2} . The constants in all such estimates are expressed only in terms of the curves Γ_{1} and Γ_{2}.

Hence the estimate

∥(H − H̃) ν∥_{c_{α−β}} ≤ C_{0}(β) · ∥ν∥_{C_{α−β}} · ε,  (22)

where C_{0}(β) is the absolute constant, is valid.

Assume now that ν(φ) and ν̃(φ) are solutions of equations (3_{1}) and (4_{1}), respectively. From (3′_{1}) and (4′′_{1}) we have

ν − ν̃ = Ã^{−1}(Ã − A) ν − Ã^{−1} (f̄_{0} − f_{0})
= Ã^{−1}(H̃ − H) ν − Ã^{−1}(f̃_{0} − f_{0}).  (23)

But by (22)

∥A − Ã∥_{c_{α−β}} = sup_{∥ν∥_{C_{α−β}} =1} ∥(Ã − A) ν∥_{c_{α−β}}
≤ sup_{∥ν∥_{C_{α−β}} =1} C_{0}(β)∥ν∥_{c_{α−β}} · ε = C_{0}(β) · ε  (24)

and

∥(Ã − A) A^{−1}∥_{c_{α−β}} ≤ ∥A − Ã∥_{c_{α−β}} · ∥A^{−1}∥_{c_{α−β}} ≤ C_{0}(β) ∥A^{−1}∥_{c_{α−β}} · ε

and if it is assumed that ε < 1/(C_{0}(β)·∥A^{−1}∥_{C_{α−β}}), then the norm of the inverse operator Ã^{−1} [4] is uniformly (with respect to Γ̃) bounded in the space C_{α−β},

∥Ã^{−1}∥_{C_{α−β}} ≤ \frac{∥A^{−1}∥_{C_{α−β}}}{1 − ∥(H̃ − H)A^{−1}∥_{C_{α−β}}} .

Using further the obvious estimate ∥f_{0} − f̃_{0}∥ and (24), from (23) it follows that the inequality

∥ν(φ) − ν̃(φ)∥_{C_{α−β}} < B_{0}(β) · ∥ν∥_{Cα−β} · ε

is valid for all ε ≤ ε_{1}, where

ε_{1} = \{\frac{d_{0}}{2} ; \frac{g(φ)}{2}; \frac{1}{C_{0}(β)∥A^{−1}∥_{C_{α−β}}} \} ,

B_{0}(β) is the constant depending only on G and β (β is any positive number smaller than α). Thus we have proved

Theorem 2. If the boundaries of the domains G and G̃ ∈ G_{ε}, 0 < ε ≤ ε_{1}, belong to the class C′_{α} (0 < α < 1), then the inequality

∥ν(φ) − ν̃(φ)∥_{C_{α−β}} < B_{0}(β) · ∥ν∥_{C_{α−β}} · ε  (25)

is valid, where ν(φ) and ν̃(φ) are unique solutions of the integral equations (3_{1}), (4_{1}), respectively. The constant B_{0}(β) and small ε_{1} are defined by giving the initial domain G; β is any positive number smaller than α.

The proven theorem solves the problem we have posed. As for estimating the differences ∥r − r̃∥ and ∥ρ − ρ̃}, such an estimate is immediately implied by Theorem 2 and formulas (2).

Corollary 1. If the triply-connected domains G and G̃ are ε-close to each other (0 < ε ≤ ε_{1}), then the inequalities

∥ρ − ρ̃∥ < Q_{1} · ε; ∥r − r̃∥ < Q_{2} · ε  (26)

are valid, where ρ, r, ρ̃, r̃ are the radii defining the canonical domains K(ρ; r; 1) and K̃(ρ̃; r̃; 1), respectively, while the constant Q_{1} and Q_{2} depend only on the domain G.

2. Conformal Mapping of Close Doubly-Connected Domains

In the complex plane Z let us consider the finite doubly-connected domain G whose boundary Γ consists of the simple closed Lyapunov curves Γ_{0} and Γ_{1}, one of which Γ_{0} envelops the other, and z = 0 ∈ int Γ_{1}.

Assume that the boundary Γ = Γ_{0} ∪ Γ_{1} of the given domain belongs to the class C′_{α} (0 < α < 1) and is given parametrically by the equations

t = g_{1}(τ), t = g_{0}(τ) (0 ≤ τ ≤ 2π; g_{i}(0) = g_{i}(2π), i = 0, 1).

Let d_{0} = ρ(Γ_{1}; Γ_{0}) and assume that 0 < ε < d_{0}/2.

Consider another doubly-connected domain G̃ of type G.

Definition 2. The domains G and G̃ whose boundary Γ̃ consists of Γ̃_{0} and Γ̃_{1} whose parametric equations are

t = g̃_{1}(τ), t = g̃_{0}(τ) (0 ≤ τ ≤ 2π; g̃_{i}(0) = g̃_{i}(2π), i = 0, 1)

are called ε-close to each other if the conditions

|g_{i}(τ) − g̃_{i}(τ)| ≤ ε, ∥g′_{i}(τ) − g̃′_{i}(τ)∥_{C_{α}} ≤ ε (i = 0, 1)  (1′)

are fulfilled.

As has already been noted, G_{ε} denotes a set of domains ε-close to G for any 0 < ε < d_{0}/2.

Let us map conformally (under the assumptions of [5]) the close domains G and G̃ onto the canonical domains K(ρ; 1) and K̃(ρ̃; 1), where K(ρ; 1) and K̃(ρ̃; 1) are respectively annuli with ρ < |w| < 1 and ρ̃ < |w| < 1, while the radii ρ and ρ̃ are defined by the formulas

ln ρ = \frac{1}{π} ∫_{0}^{2π}ν_{1}(τ) |g′_{1}(τ)|dτ,

ln ρ̃ = \frac{1}{π} ∫_{0}^{2π}ν̃_{1}(τ) |g̃′_{1}(τ)|dτ,

where

ν(t) = \begin{cases}ν_{1}(t), & when t ∈ Γ_{1}, \\ ν_{0}(t), & when t ∈ Γ_{0}. \end{cases}

Apply an analogous treatment to ν̃(t) too. Note that ν(t) and ν̃(t) are solutions of integral equations of form (3_{1}) and (4_{1}), respectively, derived for the doubly-connected domains G and G̃.

Using the methods from Section 1 one can similarly obtain an estimate for the norm ∥ν − ν̃∥_{C_{α−β}} of difference of solutions of integral equations. Namely, we have

Theorem 3. If the boundaries of doubly-connected domains G and G̃ ∈ G_{ε}, 0 < ε ≤ ε_{0}, belong to the class C′_{α} (\frac{1}{2} < α < 1), then the inequalities

∥ν(τ) − ν̃(τ)∥_{Cα−β} < B′_{0}(β) · ∥ν∥_{Cα−β} · ε,
| ln ρ − ln ρ̃| < Q_{0} · ε

are valid, where 0 < β < α and the constants B′_{0}(β), Q_{0} and ε_{0} are defined by giving the initial domain G.

These estimates allow us to construct, with the aid of the function ν(t) defined by giving the initial domain G, an approximation to the function w = f̃_{ν}(z) (f̃_{ν}(z̃_{1}) = 1, z̃_{1} > 0, z̃_{1} ∈ Γ̃_{0}) which maps conformally an arbitrary doublyconnected domain G̃ ∈ G_{ε} (0 < ε ≤ ε_{0}) onto the canonical domain K(ρ̃; 1).

By virtue of inequalities (25) and (26), we can regard the function (see [3])

w = f̃_{ν}(z) = z · exp (\frac{1}{πi} ∫ ; _{Γ} \frac{ν(t)dt}{t − z} + ic)  (27)

(f̃_{ν}(z_{1}) = 1, z_{1} > 0, z_{1} ∈ Γ̃_{0}), where ν(t) is a solution of equation (3_{1}), as an approximation to the function w = f̃_{ν}(z). By the proof of the Plemelj–Privalov theorem [2] (§18) the use of (25) and (26) leads to

Theorem 4. The function w = f̃_{ν}(z) (f̃_{ν}(z_{1}) = 1, z_{1} > 0, z_{1} ∈ Γ̃_{0}) given in the doubly-connected domain G̃ by formula (27), where G̃ ∈ G_{ε} (0 < ε ≤ ε_{1}^{∗}), admits, in G̃, an estimate

|f̃_{ν}(z) − f̃_{ν}(z)| < P · ε,

where P depends only on the domain G.

Note that it is assumed here that the given boundary points z_{1} and z̃_{1} correspond to one and the same value of the parameter τ (say, τ = 0).

3. To the Quasiconformal Mapping of Close Doubly-Connected Domains

Let us consider the problem of quasiconformal mapping of close domains. As a construction tool we take the method of integral equations [5], which stipulates the knowledge of the concrete global homeomorhism of the Beltrami equation

Wz̄ = q(z) · W_{z},  (28)
|q(z)| ≤ q_{0} < 1,

constructed by I. N. Vekua’s scheme [6]. This homeomorphism figures in the kernels of integral equations whose solutions are used to construct the wanted functions.

Assume that the coefficient q(z) of the Beltrami equation is given in some doubly-connected domain G_{0} containing the initial domain G and all domains G̃ ∈ G_{ε} which are ε-close to G (in the sense of (1′)). As G_{0} we can take, for instance, an annulus \frac{ρ_{0}}{2} < |z| < R_{0}, where ρ_{0} = ρ(0; Γ_{1}), R_{0} = max_{t_{0}∈Γ_{0}} ρ(0; t_{0}).

Assume further that the boundary of the domain G belongs to the class C′_{α} (0 < α < 1), and q(z) ∈ C′_{γ}(Ḡ_{0}), (0 < γ < 1), and the so-called Vekua basic homeomorphism W̃(z) of equation (28) is constructed with the coefficient

q̃(z) = \begin{cases}q(z), & when z ∈ Ḡ_{0}, \\ 0, & when z lies outside G_{0}. \end{cases}

In that case q̃(z) belongs to any Lebesgue class L_{p}(E) (where E is the entire complex plane) and, according to [7], W̃z̄ and W̃_{z} satisfy the Hölder condition with index γ_{0} and 0 < γ_{0} < min{α; γ}. In what follows this global homeomorphism of equation (28) is denoted by W̃_{G_{0}}(z).

We use the technique of exit in the plane of this homeomorphism. Then the integral equations (3_{1}) and (4_{1}) take the form

μ(ξ_{0}) + \frac{1}{2πi} ∫_{_{W_{G_{0}}}(Γ)} \frac{1}{ξ′} K_{∗}(ξ; ξ_{0})μ(ξ)dξ
= − ln |ξ_{0} − W̃_{G_{0}}(0)|, ξ_{0} ∈ W̃_{G_{0}}(Γ),  (29)
μ̃(ξ_{0}) + \frac{1}{2πi} ∫_{_{W_{G_{0}}}(_{Γ})} \frac{1}{ξ′} K̃_{∗}(ξ; ξ_{0})μ̃(ξ)dξ
= − ln |ξ_{0} − W̃_{G_{0}}(0)|, ξ_{0} ∈ W̃_{G_{0}}(Γ̃),  (30)

where ξ = W̃_{G_{0}}[g_{k}(τ)], ξ = W̃_{G_{0}}[g̃_{k}(τ)] (k = 0, 1) are the parametric equations of the curves W̃_{G_{0}}(Γ_{k}), W̃_{G_{0}}(Γ̃_{k}), respectively. K_{∗}(ξ; ξ_{0}) = K_{∗}[W̃_{G_{0}}[g_{k}(τ); g_{l}(τ_{0})]] (k, l = 0, 1). In an analogous manner we define K̃_{∗}(ξ; ξ_{0}). Moreover, [W̃_{G_{0}}(z)]z̄ [W̃_{G_{0}}(z)]_{z} ∈ C_{γ_{0}}(Ḡ_{0}) [7] and μ[ξ(τ)], μ̃[ξ(τ)] ∈ C_{γ_{0}}[0; 2π]. We can also write that μ(ξ_{0}) = ν[W̃_{G_{0}}^{−1}(g(τ_{0}))].

By virtue of (1′) we have

|W̃_{G_{0}}[g_{j}(τ)] − W̃_{G_{0}}[g̃_{j}(τ)]| ≤ C_{j}(W̃_{G_{0}}; G_{0})|g_{j}(τ) − g̃_{j}(τ)|
< C_{j}(W̃_{G_{2}}; G_{0}) · ε (j = 0, 1).  (31)

For convenience, denote W̃_{G_{0}}(z) = W̃(z).

We have

(W̃[g_{j}(τ)])′_{τ} = W̃_{t}[g(τ)](g′_{j}(τ))_{τ} + W̃t̄[g_{j}(τ)](ḡ_{j}(τ))′_{τ},
(W̃[g̃_{j}(τ)])′_{τ} = W̃_{t}[g̃(τ)](g̃′_{j}(τ))_{τ} + W̃t̄[g̃_{j}(τ)](¯g̃_{j}(τ))′_{τ}

and

(W̃[g_{j}])′_{τ} ∈ C_{γ_{0}}[0, 2π], (W̃[g̃_{j}(τ)])′_{τ} ∈ C_{γ_{0}}[0; 2π]; (j = 0, 1).

Compose the difference

(W̃[g_{j}(τ)])′_{τ} − (W̃[g̃_{j}(τ)])′_{τ} = W̃_{t}(g_{j})(g′_{j})_{τ} − W̃_{t}(g_{j})(g̃′_{j})_{τ} − W̃_{t}(g̃_{j})(g̃_{j})′_{τ}

+W̃_{t}(g_{j})(g̃′_{j})_{τ} + W̃t̄(g_{j})(g̃_{j})′_{τ} − W̃t̄(g_{j})( ¯g̃_{j})′_{τ}
−W̃t̄(g̃_{j})( ¯g̃_{j})′_{τ} + W̃t̄(g_{j})( ¯g̃_{j})′_{τ} (j = 0, 1).

Using (1′) in these expressions and taking into account the inequalities

∥g′_{j}(τ) − g̃′(τ)∥_{C_{γ_{0}/2}} ≤ const_{j1} · ∥g′_{j}(τ) − g̃_{j} ′(τ)∥_{C_{α}},
∥ḡ′_{j}(τ) − ¯ g̃_{j}′(τ)∥_{C_{γ_{0}/2}} ≤ const_{j2} · ∥ḡ′_{j}(τ) − ¯ g̃_{j}′(τ)∥_{C_{α}},

we obtain

∥W̃_{t}(g_{j}) − W̃_{t}(g̃_{j})∥_{C_{γ_{0}/2}} ≤ C_{2j}^{∗}(W̃_{G_{0}}; G_{0})|g_{j}(τ) − g̃_{j}(τ)|^{γ_{0}/2}
≤ C_{3j}^{∗} ε^{γ_{0}/2} (j = 0, 1).  (32)

We see that the closeness conditions (1′) of the contours Γ_{j} and Γ̃_{j} in the plane of the homeomorphism W̃_{G_{0}}(z) for the corresponding curves W̃_{G_{0}}(Γ_{j}) and W̃_{G_{0}}(Γ̃_{j}) are replaced by conditions (31) and (32).

Now, by virtue of the estimate established in Theorem 3, we come to the validity of the following proposition.

Theorem 5. If the doubly-connected domains G ⊂ G_{0} and G̃ ⊂ G_{0} whose boundaries belong to the class C′_{α} (0 < α < 1) are ε-close to each other in the sense of (1′), then the estimate

∥μ(ξ) − μ̃(ξ)∥_{C_{\frac{γ_{0}}{2} −β}} < Q^{∗}(W̃_{G_{0}}; G; β) · ∥μ∥_{C_{\frac{γ_{0}}{2} −β}} · ε^{\frac{γ_{0}}{2}}  (33)

holds for all ε ∈ [0; ε^{∗}]; here μ(ξ) and μ̃(ξ) are unique solutions of the integral equations (29) and (30), respectively, β is any positive number smaller than γ_{0}/2, the constant Q^{∗} and small ε^{∗} are completely defined by giving the initial domain G and the homeomorphism W̃_{G_{0}}(z).

Note that in Theorem 5 the order of smallness for ε can be obtained arbitrarily close to γ_{0} [3], for instance, O(ε^{γ_{0}−η}), where η < γ_{0} is any positive number, but in that case we can estimate only “small” norms of the value ∥μ(ξ) − μ̃(ξ)∥_{C_{η}} (the Hölder index η decreases). In the considered situation the choice η = \frac{γ_{0}}{2} seems optimal to us.

In conclusion, also note that, analogously to the conformal case, the estimate in terms of ε for the difference of modules ln ρ(q̃) − ln ρ̃(q̃) calculated by [8] can be established with the aid of estimate (33).

References

1. L. Zivzivadze, On conformal and quasiconformal transformations of triply connected fields. Bull. Georgian Acad. Sci. 163(2001), No. 2, 226–229.

2. N. I. Muskhelishvili, Singular integral equations. Boundary value problems of the theory of functions and some of their applications in mathematical physics. (Russian) 3rd ed. Nauka, Moscow, 1968; English translation from 1st Russian ed. (1946): P. Noordhoff, Groningen, 1953, corrected reprint Dover Publications, Inc. , N. Y., 1992.

3. Z. Samsonia and I. Samkharadze, On quasiconformal mappings corresponding to the Beltrami equation. (Russian) Ukrain. Mat. Zh. 51(1999), No. 10, 1391–1397.

4. V. Trenogin, Functional analysis. (Russian) Nauka, Moscow, 1980.

5. D. Kveselava, On the application of integral equations in the theory of conformal mappings. (Russian) Trudy Vychisl. Centra Akad. Nauk Gruzin. SSR 2(1961), 3–15.

6. I. Vekua, Generalized analytic functions. (Russian) Nauka, Moscow, 1988; English translation from the 1st Russian edition: Pergamon Press, Oxford-London-New York-Paris, 1962; Addison-Wesley, Reading, Mass.–London.

7. G. Mandzhavidze, A boundary value problem of linear conjugation with displacements and its connection to the theory of generalized analytic functions. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 33(1967), 82–87.

8. D. Kveselava and Z. Samsonia, On the quasiconformal mapping of domains. Metric Problems of the Function Theory (Russian), 53–65, Naukova Dumka, Kiev,1980.

(Received 13.09.2001)

Author’s address:

L. Zivzivadze A. Tsereteli Kutaisi State University 55, Queen Tamar St., Kutaisi 384000 Georgia