Georgian Mathematical Journal Volume 8 (2001), Number 3, 513–520

ON A REPRESENTATION OF THE DERIVATIVE OF A CONFORMAL MAPPING

G. KHUSKIVADZE AND V. PAATASHVILI

Abstract. Let ω conformally map the unit circle on a plane singly-connec-ted domain D bounded by a simple rectifiable curve. It is shown that for the function lg ω′ to be represented in the unit circle by a Cauchy type A-integral with density arg ω′, it is necessary and sufficient that D be a Smirnov domain. In particular, for this representation to be done by a Cauchy–Lebesgue type integral with the same density, it is necessary and sufficient that the function lg ω′ belong to the Hardy class H_{1}.

2000 Mathematics Subject Classification: 30C20, 30C35, 30E20, 30E25. Key words and phrases: Conformal mapping, Smirnov domain, extension of Lebesgue integral, A-integral.

Let D be a finite singly-connected domain bounded by a simple rectifiable curve Γ, ω = ω(z) be a function mapping conformally the unit circle U on D, and γ be the boundary of U. Then the derivative ω′ belongs to the Hardy class H_{1}, and almost for all ϑ ∈ [0, 2π] there exists an angular value of the function ω′(z) and

lim_{z⟶^{λ}e^{iϑ}} ω′(z) = ω′(e^{iϑ}) = −ie^{−iϑ} \frac{dω(e^{iϑ})}{dϑ}  (1)

(see, e.g., [1], Ch. III, §1, 1.1,1.6).

Throughout the paper it is assumed that ω′(0) > 0 and arg ω′(0) = 0.

In view of (1), for almost all ϑ we have

lim_{z⟶^{λ}e^{iϑ}} arg ω′(z) = arg ω′(e^{iϑ}) = arg \frac{dω(e^{iϑ})}{dϑ} − ϑ − \frac{π}{2} ,  (2)

where arg \frac{dω(e^{iϑ})}{dϑ} is one of the angles between the tangent to Γ at the point ω(e^{iϑ}) and the abcissa axis.

The present paper is a continuation of [4]; it contains some comments on the well-known formula

lg ω′(z) = lg ω′(0) + \frac{i}{2π} ∫_{0}^{2π}arg ω′(e^{iϑ}) \frac{e^{iσ} + z}{e^{iσ} − z} dσ

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= lg ω′(0) + \frac{1}{π} ∫_{|τ|=1} \frac{arg ω′(τ)}{τ − z} dτ,  (3)

which is valid provided that lg ω′ ∈ H_{1} (this is the Schwarz formula applied to the function i lg ω′). In particular, using a certain extension of the Lebesgue integral, formula (3) is generalized here to Smirnov domains for which

lg ω′(z) = \frac{1}{2π} ∫_{0}^{2π}lg |ω′(e^{iσ})| \frac{e^{iσ} + z}{e^{iσ} − z} dσ = − lg ω′(0) + \frac{1}{πi} ∫_{|τ|=1} \frac{ln |ω′(τ)|}{τ − z} dτ (4)

(see, e.g., [1], Ch. III, §12).

Isolating in (3) the imaginary part, we obtain

arg ω′(z) = \frac{1}{2π} ∫_{0}^{2π}arg ω′(e^{iσ}) \frac{1 − |z|^{2}}{|e^{iσ} − z|^{2}} dσ, |z| < 1.  (5)

Having certain information on the properties of the function arg ω′(e^{iϑ}) = \frac{dω(e^{iϑ})}{dϑ}−ϑ−\frac{π}{2}, we can establish, by formulas (3) and (5), the respective properties of the function ω′ in the circle U and, conversely, knowing the properties of the function ω′ in the circle, it is possible to establish some properties of the function \frac{dω(e^{iϑ})}{dϑ} and, hence, the geometrical properties of the boundary of D, as we do, for instance, in the case of the Lindelöf theorem which states that the smoothness of the boundary of D (the continuity of the inclination angle of the tangent to Γ) is equivalent to the continuity of the function arg ω′ on the closed circle \overline{U}^{1} (see, e.g., [2], pp. 42, 44). However the function arg ω′(z) has been obtained as a boundary function of the harmonic function in U, which does not always give direct information on the properties of arg ω′(e^{iϑ}). Formulas (3) and (5) are useful if the function arg ω′(e^{iϑ}) is constructed using some other arguments as was done, for instance, in [3], [2] (§§3.2, 3.5), [4], [5] (Ch. III).

Even if it is assumed that D is a Smirnov domain, the function lg ω′ may not always belong to the Hardy class H_{1} (see [6]), and the function arg ω′(e^{iϑ}) is not always summable (see equality (13) below). Hence formula (3) cannot be written even for all Smirnov domains^{2} (if the consideration is restricted to the Lebesgue integral). However, if one uses certain generalized Lebesgue integrals in whose sense the conjugate function of the summable function is integrable (for instance, the A-integral, see [7] and [8], Ch. VIII, §18, or the B-integral, see [9], Ch. VII, §4), then formula (3) can be extended to Smirnov domains as well.

1[12], p. 94, gives a wrong statement that the function arg ω′(z) is continuous on the closed circle \overline{U} under an assumption that there only exists a tangent at every point Γ.

2[12], pp. 90–92, gives a wrong statement that formula (3) is valid for all domains bounded by arbitrary rectifiable curves.

1. By the L̃-integral we will mean a minimal extension of the Lebesgue integral in whose sense the conjugate functions

f̃(x) = −\frac{1}{2π} ∫_{0}^{2π}f(t) ctg \frac{t − x}{2} dt

of the summable functions f on [0, 2π] are integrable, and the integral of the conjugate functions is equal to zero (see, for instance, [10], pp. 38, 88, or [5], Ch. I, §6). This is a class of functions of the form f_{1}+f̃_{2}, where f_{1}, f_{2} ∈ L(0, 2π).

Definition. A measurable function on [a, b] is called A-integrable if

|{x ∈ [a, b]; |f(x)| > λ}| = o(λ^{−1})  (6)

and there exists a limit

lim_{λ→∞} ∫_{|f|≤λ} f(x) dx

which is called an A-integral of f with respect to [a, b]. We denote it by

(A) ∫_{a}^{b} f(x) dx.

L̃-integrable functions are A-integrable (and B-integrable) and the integrales coincide (see the above-cited references).

We have equality

(A) ∫_{|t|=1} φ(t)S(f)(t) dt = − ∫_{|t|=1} S(φ)(t)f(t) dt,  (7)

where f is summable, φ satisfies the Lipshitz condition on γ,

S(f)(t) = \frac{1}{πi} ∫_{|τ|=1} \frac{f(τ)}{τ − t} dτ

(see, for instance, [7], [8], Ch. VIII, §18). In equality (7), the A-integral can be replaced by any integral containing the L̃-integral, say, by the B-integral ( see, for instance, [10] or [5]).

Theorem. In order that the formula

lg ω′(z) = lg ω′(0) + \frac{i}{2π} (A) ∫_{0}^{2π}arg ω′(e^{iσ}) \frac{e^{iσ} + z}{e^{iσ} − z} dσ

= lg ω′(0) + \frac{1}{π} (A) ∫_{|τ|=1} \frac{arg ω′(τ)}{τ − z} dτ, |z| < 1,  (8)

be valid, it is necessary and sufficient that D be a Smirnov domain.

Proof. Sufficiency. We will need the following two equalities which are easy to verify:

S(lg |ω′|)(t) = \frac{1}{πi} ∫_{|τ|=1} \frac{lg |ω′(τ)|}{τ − t} dτ = i arg ω′(t) + \frac{1}{2π} ∫_{0}^{2π}lg |ω′(e^{iσ})| dσ, (9)

−(t − z)^{−1} = S((τ − z)^{−1})(t), |z| < 1.  (10)

Let D be a Smirnov domain. Then, taking into account (9), (10) and the equality

\frac{1}{2π} ∫_{0}^{2π}lg |ω′(e^{iσ})| dσ = lg ω′(0)

(the latter equality is valid because D is a Smirnov domain; it follows from (4)), we obtain by virtue of (7)

∫_{|τ|=1} \frac{lg |ω′(τ)|}{τ − z} dτ = − ∫_{|τ|=1} lg |ω′(τ)|S((t − z)^{−1})(τ) dτ

= (A) ∫_{|τ|=1} \frac{S(lg |ω′|)(τ)}{τ − z} dτ = (A)i ∫_{|τ|=1} \frac{arg ω′(τ)}{τ − z} dτ + 2πi lg ω′(0). (11)

Equalities (4) and (11) imply (8).

Necessity. Using the canonical expansion of a function from the class H_{1}, we can write

lg ω′(z) = \frac{1}{2π} ∫_{0}^{2π}lg |ω′(e^{iσ})| \frac{e^{iσ} + z}{e^{iσ} − z} dσ + \frac{1}{2π} ∫_{0}^{2π} \frac{e^{iσ} + z}{e^{iσ} − z} dψ(σ)  (12)

(again keeping in mind the branch for which arg ω′(0) = 0), where ψ is a nondecreasing singular function (see, for instance, [1], p. 220). Isolating, in (12), the imaginary part and passing to the limit, we obtain

lim_{r→1} arg ω′(re^{iϑ}) = arg ω′(ϑ) = { \atop lg |ω′|}(ϑ) + d̃ψ(ϑ),  (13)

where

{ \atop lg |ω′|}(ϑ) = −\frac{1}{2πi} ∫_{0}^{2π}ln |ω′(e^{iσ})| ctg \frac{σ − ϑ}{2} dσ,

d̃ψ(ϑ) = −\frac{1}{2π} ∫_{0}^{2π}ctg \frac{σ − ϑ}{2} dψ(σ)

(the conjugate functions of lg |ω′(e^{iϑ})| and dψ, respectively).

Let (8) be valid, in particular, the function arg ω′(e^{iϑ}) be A-integrable. Then in view of (13) we can write

|{ϑ ∈ [0, 2π]; |d̃ψ(ϑ)| > λ}| = o(λ^{−1})  (14)

and therefore ψ ≡ const (see, for instance, [10], p. 26), which means that D is a Smirnov domain.

Corollary. For (8) to hold with the summable function arg ω′(e^{iϑ}), it is necessary and sufficient that lg ω′ belong to the Hardy class H_{1}.

The sufficiency is obvious. The necessity follows from equality (13), since, when arg ω′(e^{iϑ}) is summable, condition (14) is fulfilled and ψ ≡ const. Then { \atop lg |ω′|}(ϑ) = arg ω′(e^{iϑ}) ∈ L(0, 2π) and therefore lg ω′ ∈ H_{1}.

Remark 1. By equality (3), from condition (14) it follows that the condition |{ϑ ∈ [0, 2π]; | arg ω′(e^{iϑ})| > λ}| = o(λ^{−1}) is necessary and sufficient for D to belong to the Smirnov class.

Remark 2. One can easily verify that formula (8) remains in force if ω′ is assumed to be an arbitrary function of the class H_{1} which is different from zero in U. Hence we have the following assertion:

If f ∈ H_{1} and f(z) ≠ 0 in U, then the parametric representation of f (see [1], pp. 110–111) can be written in the form

f(z) = f(0) exp \{\frac{1}{2π} (A) ∫_{0}^{2π}arg f′(e^{iσ}) \frac{e^{iσ} + z}{e^{iσ} − z} dσ\} exp \{\frac{1}{2π} ∫_{0}^{2π} \frac{e^{iσ} + z}{e^{iσ} − z} dμ\}.

2. As follows from the arguments used in proving the theorem, the A-integral can be replaced by L̃-integral. A further extension of the notion of the integral with an aim to extend formula (8) to non-Smirnov domains leads to a contradiction between the considered formula and the Cauchy and Schwarz integral formulas.

Indeed, let D be a non-Smirnov domain, i.e., the Schwarz formula (4) be invalid. Then we have

lg ω′(z) = lg ω′(0) + \frac{1}{2π} (X) ∫_{0}^{2π}arg ω′(e^{iσ}) \frac{e^{iσ} + z}{e^{iσ} − z} dσ,  (15)

where (X) ∫ ⋯ is some extension of the L̃-integral. From (15) in particular

it follows that (X) ∫_{0}^{2π}arg ω′(e^{iσ}) dσ = 0. By virtue of the latter equality, the

formula of the mean

lg ω′(0) = \frac{1}{2π} ∫_{0}^{2π}lg ω′(e^{iσ}) dσ

and therefore the Cauchy formula

lg ω′(z) = (2πi)^{−1}(X) ∫_{|τ|=1} (τ − z)^{−1} lg ω′(τ) dτ  (16)

give

lg ω′(0) = \frac{1}{2π} ∫_{0}^{2π}lg |ω′(e^{iσ})| dσ

which, on account of equality (12), is valid if and only if D is a Smirnov domain.

3. In this subsection we give the proof of one well-known statement on ω′(z) (see [11]), based on representation (3).

Statement ([11]). Let Γ be a closed smooth rectifiable curve with the equation ζ = ζ(s), 0 ≤ s ≤ ℓ, (s is an arc abscissa), and δ = δ(s) be a slope angle of the tangent to the point ζ(s) with the abscissa axis, which changes continuously on [0, ℓ]. If the continuity modulus ρ(δ, t), t ∈ (0, ℓ) of the function δ satisfies the Dini condition

∫_{0}^{ℓ} \frac{ρ(δ, t)}{t} dt < ∞,  (17)

then the derivative of the conformal mapping of the unit circle on the finite domain bounded by Γ is continuous in the closed circle.

Proof. Let δ(s) = δ(s(ζ)) and ζ = ω(e^{iϑ}). Then the function ζ = ζ(ϑ) is uniquely defined on [0, 2π] and therefore, by the Lindelöf theorem, we have arg ω′(e^{iϑ}) = δ(ζ(ϑ)) − ϑ − \frac{π}{2}. Since Γ is a smooth curve, we can rewrite (3) as

ω′(z) = ω′(0) exp \{\frac{1}{π} ∫_{|τ|=1} \frac{δ(ζ(ϑ)) − ϑ − \frac{π}{2}}{τ − z} dτ \}, τ = e^{iϑ}.  (18)

Let us set ν(ϑ) = δ(ζ(ϑ)) − ϑ − \frac{π}{2} and show that the continuity modulus ρ(ν, t), t ∈ (0, 2π), satisfies the Dini condition. We obtain

|ν(ϑ + h) − ν(ϑ)| = |δ(ζ(ϑ + h)) − δ(ζ(ϑ)) − h|
≤ ρ(δ, sup_{0<σ≤h} |ζ(ϑ + σ) − ζ(ϑ)| + h) ≤ ρ(δ, sup_{0<σ≤h} ∫_{ϑ}^{ϑ+h}|ω′(e^{iu})| du + h). (19)

Since Γ is a smooth curve, we have ω′ ∈ ∩_{p>1}H_{p} (see, for instance, [2]). Therefore

ω′(e^{iu}) ∈ ∩_{p>1}L_{p}(0, 2π) and hence (19) implies that for any α ∈ (0, 1) there exists M_{α} such that

sup_{ρ≤h} |ν(ϑ + σ) − ν(ϑ)| ≤ ρ(δ, M_{α}h^{α}) + h.

But then

∫_{0}^{2π} \frac{ρ(ν, t)}{t} dt ≤ ∫_{0}^{ℓ} \frac{ρ(ν, Mt^{α})}{t} dt + 2π

≤ k ∫_{0}^{ℓ} \frac{ρ(δ, u)}{u^{1/α}} u^{1/α−1} du + 2π, k = \frac{1}{αM_{α}^{1/α}} .  (20)

Hence, by virtue of (20), we conclude that ∫_{0}^{2π} \frac{ρ(ν,t)}{t} dt < ∞ and thus

ω′(z) = ω′(0) exp \{\frac{1}{π} ∫_{|τ|=1} \frac{ν(ϑ)}{τ − z} dτ \}, τ = e^{iϑ},  (21)

where condition (17), i.e., the Dini condition is fulfilled for the continuity modulus of the function ν. As is known, in that case a Cauchy type integral with density ν is a continuous function in the closed circle (see, for instance, [2]). Hence, by virtue of (21), it follows that the function ω′(z) is continuous too.

Other applications of representation (3) can be found in [2]–[5], where the function arg ω′(e^{iϑ}) is assumed to be given.

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(Received 26.03.2001)

Authors’ address:

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia