Georgian Mathematical Journal Volume 10 (2003), Number 1, 1–15

ON ONE THEOREM OF S. WARSCHAWSKI

R. ABDULAEV

Abstract. A theorem of S. Warschawski on the derivative of a holomorphic function mapping conformally the circle onto a simply-connected domain bounded by the piecewise-Lyapunov Jordan curve is extended to domains with a non-Jordan boundary having interior cusps of a certain type.

2000 Mathematics Subject Classification: 3OC35.

Key words and phrases: Conformal mapping, corner, non-Jordan boundary.

1. Let a simply-connected domain B be bounded by a closed piecewisesmooth curve γ : z = z(s), 0 ≤ s ≤ S, where s is a natural parameter. Let s_{k}, k = \overline{1, n}, be the points of discontinuity of z′(s). The point z_{k} = z(s_{k}), k = \overline{1, n}, of discontinuity of the function z′(s) will be called the corner of opening ν_{k}π = π − arg(z′(s_{k} + 0) : z′(s_{k} − 0)), where 0 ≤ ν_{k} ≤ 2 and −π < arg · ≤ π.

In [1] S. Warschawski established a result describing the behavior of the derivative of the holomorphic function ω(z) which maps the domain B onto the unit circle 𝔻 in the neighborhood of corners. Namely, it was proved that if the Jordan curve γ is piecewise-Lyapunov and 0 < ν_{k} ≤ 2, k = \overline{1, n}, then

ω′(z) = ω_{0}(z) ∏_{k=1}^{n}(z − z_{k})^{\frac{1}{ν_{k}} −1},  (1)

where ω_{0}(z) is a function holomorphic in B, continuous and non-vanishing in \overline{B}. An analogous representation is valid for (ω^{−1})′ as well. Various aspects of this range of problems were intensively investigated in the subsequent period too. A vast list of works on this topic can be found in the monograph [2]. In [3], using the results of the theory of a discontinuous Riemann problem, the authors showed the validity of representation [1] for a piecewise-smooth Jordan curve γ and 0 < ν_{k} ≤ 2, k = \overline{1, n}. In that case the function ω_{0}(z) belongs to any Smirnov class E_{p}, p > 0.

In this paper a sufficiently simple way is proposed for proving one theorem of Warschawski for a simply-connected domain with a non-Jordan boundary. This proof covers the cases of both a piecewise-Lyapunov and a piecewise-smooth curve γ (0 < ν_{k} ≤ 1, k = \overline{1, n}). In addition to the classical statement, it is proved that for a piecewise-Lyapunov boundary the function ω_{0}(z) satisfies the Hölder condition (condition H(μ))

|ω_{0}(z_{1}) − ω_{0}(z_{2})| < K|z_{1} − z_{2}|^{μ}, 0 < μ ≤ 1,  (2)

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

not only on smooth parts of the curve γ and in the neighborhood of corners with ν_{k} < 2, but also in the neighborhood of cusps (ν_{k} = 2) of a certain type. (In connection with this question see also [3].)

2. Recall that the smoothness of a curve is equivalent to the continuity of an angle formed by the tangent to the curve with a fixed direction. If however this angle as a function of the arc length satisfies the Hölder condition, then the curve is called a Lyapunov curve. Piecewise smoothness imposes on the above-said angle a condition of the existence of one-sided limits at points of discontinuity, while the property of being piecewise-Lyapunov curve implies that the Hölder condition is satisfied on each interval between points of discontinuity, including end-points.

Lemma 1. If z(t) ∈ C^{1,μ}[a, b], 0 < μ ≤ 1 and, z′(a) ≠ 0, z(t) − z(a) ≠ 0 on [a, b], then

arg[z(t) − z(a)] ∈ C^{0,μ}[a, b].

By the condition we have

\begin{aligned}x′(t) & = Re z′(t) = x′(a) + f(t)(t − a)^{μ}, \\ y′(t) & = Im z′(t) = y′(a) + h(t)(t − a)^{μ}, \end{aligned}  (3)

where f(t) and h(t) are bounded on [a, b]. Applying mean value theorem, we obtain

x(t) − x(a) = ∫_{a}^{b} x′(τ)dτ = x′(a)(ξ − a) + x′(t)(t − a)
= (x′(a) − x′(t))(ξ − a) + x′(t)(t − a),

where a ≤ ξ ≤ t. Hence

x(t) − x(a) = \frac{x′(a) − x′(t)}{(t − a)^{μ}} · \frac{ξ − a}{t − a} (t − a)^{μ+1} + x′(t)(t − a)

= x′(a)(t − a) + \frac{x′(t) − x′(a)}{(t − a)^{μ}} (t − a)^{μ+1}

+ \frac{x′(a) − x′(t)}{(t − a)^{μ}} · \frac{ξ − a}{t − a} (t − a)^{μ+1}

= x′(a)(t − a) + φ(t)(t − a)^{μ+1},  (4)

where

φ(t) = f(t)(1 − \frac{ξ − a}{t − a})  (5)

is the function bounded on [a, b].

By a similar reasoning we get

y(t) − y(a) = y′(a)(t − a) + ψ(t)(t − a)^{μ+1},  (6)

where

ψ(t) = h(t)(1 − \frac{η − a}{t − a}), a ≤ η ≤ t,  (7)

is also bounded on [a, b].

Using (3)–(7) we obtain

\frac{d}{dt} arg[z(t) − z(a)] = \frac{d}{dt} arc tg \frac{y(t) − y(a)}{x(t) − x(a)}

= \frac{(x′(a)(t − a) + φ(t)(t − a)^{μ+1})(y′(a) + h(t)(t − a)^{μ})}{(x′(a)(t − a) + φ(t)(t − a)^{μ+1})^{2} + (y′(a)(t − a) + ψ(t)(t − a)^{μ+1})^{2}}

− \frac{(y′(a)(t − a) + ψ(t)(t − a)^{μ+1})(x′(a) + f(t)(t − a)^{μ})}{(x′(a)(t − a) + φ(t)(t − a)^{μ+1})^{2} + (y′(a)(t − a) + ψ(t)(t − a)^{μ+1})^{2}}

= \frac{(t − a)^{μ+1}(x′(a)h(t) + y′(a)φ(t) + h(t)φ(t)(t − a)^{μ})}{(t − a)^{2}[(x′(a) + φ(t)(t − a)^{μ})^{2} + (y′(a) + ψ(t)(t − a)^{μ})^{2}]}

− \frac{(t − a)^{μ+1}[(y′(a)f(t) + x′(a)ψ(t) + f(t)ψ(t)(t − a)^{μ}]}{(t − a)^{2}[(x′(a) + φ(t)(t − a)^{μ})^{2} + (y′(a) + ψ(t)(t − a)^{μ})^{2}]} = \frac{b(t)}{(t − a)^{1−μ}} ,

where the function b(t) is bounded on [a, b] by virtue of the condition |z′(a)| ≠ 0. The latter equality implies

| arg(z(t_{1}) − z(a)) − arg(z(t_{2}) − z(a))| = | ∫_{t_{2}}^{t_{1}} \frac{d}{dt}(arg(z(t) − z(a))dt|

≤ M_{1}|(t_{1} − a)^{μ} − (t_{2} − a)^{μ}| ≤ M_{1}|t_{1} − t_{2}|^{μ}.

Denote by P_{β} (β > 0) the mapping w = z^{β}, and by E_{α}(q) the angle {z; −πα < arg(z − q) < πα, α < 1, Im q = 0}. For β > 1 the mapping P_{β} is univalent in E_{β^{−1}}(0).

Lemma 2. Let γ_{0} : z = z(s), \overline{s} ≤ s ≤ s^{=} be a piecewise-smooth arc with the corner z(s_{0}) = 0, and let the positive semi-axis be the bisectrix of the interior angle of the opening πν, 0 < ν ≤ 2, at the point z(s_{0}). Then the curve P_{\frac{1}{ν}} ∘ γ is smooth.

If we write the equation of the curve Γ = P_{\frac{1}{ν}} ∘ γ in the form w = w(s) = [z(s)]^{\frac{1}{ν}}, then dw(s) = \frac{1}{ν}[z(s)]^{\frac{1}{ν} −1} · z′(s)ds, dσ(s) = |dw(s)| = \frac{1}{ν}|z(s)|^{\frac{1}{ν} −1} · ds and

\frac{dw(s)}{dσ(s)} = exp [i arg[z(s)]^{\frac{1}{ν} −1} · z′(s)].

Furthermore,

lim_{s→s_{0}+0} arg z(s) = π \frac{ν}{2},  lim_{s→s_{0}+0} arg z′(s) = π \frac{ν}{2} − π,
lim_{s→s_{0}−0} arg z(s) = −π \frac{ν}{2},  lim_{s→s_{0}−0} arg z′(s) = −π \frac{ν}{2}.

Hence we obtain

lim_{s→s_{0}−0} arg[z(s)]^{\frac{1}{ν} −1} · z′(s) = (\frac{1}{ν} − 1) lim_{s→s_{0}−0} arg z(s) + lim_{s→s_{0}−0} arg z′(s)

= (\frac{1}{ν} − 1)π \frac{ν}{2} + π \frac{ν}{2} − π = −\frac{π}{2} ,

lim_{s→s_{0}+0} arg[z(s)]^{\frac{1}{ν} −1} · z′(s) = (\frac{1}{ν} − 1) lim_{s→s_{0}+0} arg z(s) + lim_{s→s_{0}+0} arg z′(s)

= (\frac{1}{ν} − 1)( − π \frac{ν}{2}) − π \frac{ν}{2} = −\frac{π}{2} .

Corollary. If in the conditions of Lemma 2 γ is piecewise-Lyapunov curve, then Γ is Lyapunov curve.

On writing the equation of the curve γ for the parameter σ as γ : z = [w(σ)]^{ν}, \overline{σ} ≤ σ ≤ σ^{=}, we obtain ds = |dz(σ)| = ν|w(σ)|^{ν−1}dσ. Let the points a_{1} and a_{2} lie on the curve γ so that both values s_{1} and s_{2} of the arc variable, which correspond to the points a_{1} and a_{2}, occur either in the interval [\overline{s}, s_{0}] or in [s_{0}, s^{=}], and let A_{j} = P_{\frac{1}{ν}} (a_{j}), j = 1, 2. Then

s(a_{1}, a_{2}) = ∫_{s_{1}}^{s_{2}} ds = ν ∫_{σ(A_{1})}^{σ(A_{2})}|w(σ)|^{ν−1}dσ ≤ M_{2}( ∫_{σ(A_{1})}^{σ(A_{2})}dσ)^{μ}

= M_{2}[σ(A_{1}, A_{2})]^{μ}, 0 < μ ≤ 1.  (8)

By the condition and Lemma 1 we have

| arg \frac{dw}{dσ}(σ(s_{1})) − arg \frac{dw}{dσ}(σ(s_{2}))| ≤ M_{3}|s_{1} − s_{2}|^{μ′}, 0 < μ′ ≤ 1,

for \overline{s} ≤ s_{1}, s_{2} ≤ s_{0} or \overline{s}_{2} ≤ s_{1}, s_{2} ≤ s^{=}. From this, by virtue of (8), we obtain

| arg \frac{dw}{dσ}(σ_{1}) − arg \frac{dw}{dσ}(σ_{2}))| ≤ M_{4}|σ_{1} − σ_{2}|^{μ′′}, 0 < μ′′ ≤ 1,  (9)

where σ_{1} = σ(s_{1}), σ_{2} = σ(s_{2}). But by Lemma 2 the curve Γ is smooth and therefore the fulfilment of condition (9) on the arcs composing Γ implies that this condition is fulfilled on the entire curve ([4], Ch. 1, §5).

Lemma 3. For 0 < β < 1, 0 < α < 1 a > 0 P_{β}(E_{α}(a)) ⊂ E_{α}(a^{β}).

After writing the equation of one of the sides of the angle E_{α}(a) as z = a + t exp(iαπ), 0 ≤ t < ∞, we obtain

arg [(a + t exp(iαπ))^{β}]′ = arg(a + t exp(iαπ))^{β−1} + απ
= (β − 1) arg(a + t exp(iαπ)) + απ ≤ απ.

Next,

( arg(a + t exp(iαπ)))′ = \frac{(a + t cos απ) sin απ − t cos απ sin απ}{(α + t cos απ)^{2} + t^{2} sin^{2} απ}

= \frac{a sin απ}{(a + t cos απ)^{2} + t^{2} sin^{2} απ} > 0,

and therefore

[(arg(a + t exp(iαπ))^{β})′]′ < 0.

Moreover,

[(β − 1) arg(a + t exp(iαπ) + απ]|_{t=0} = απ,
lim_{t→∞} [(β − 1) arg(a + t exp(iαπ) + απ] = βαπ.

Thus arg [(a+t exp(iαπ))^{β}]′ is a decreasing function on [0, ∞) from the value απ to βαπ. Repeating the above arguments for another side of the angle, we ascertain that the lemma is valid.

3. As mentioned above, the boundary γ of the simply-connected domain B is not assumed to be a Jordan curve. We will describe those properties of the boundary curve which are needed for further constructions.

Denote by ⟨γ⟩ the range of values of the mapping γ, by B_{∞}(γ) the component of the set \overline{ℂ}\⟨γ⟩ containing the point at infinity, and by W (γ) the set of points of all other components of the set \overline{ℂ}\⟨γ⟩. The symbol γ[t_{1}, t_{2}] will denote the arc of the parametrized curve corresponding to the variation of t from the value t_{1} to t_{2}, including end-points, while γ(t_{1}, t_{2}) will denote the same arc but without the end-points. If the arcs γ_{1} = γ[a, b] and γ_{2} = γ[c, d] are such that γ(b) = γ(c), then

(γ_{1} · γ_{2})(t) = \begin{cases}γ(t), & a ≤ t ≤ b, \\ γ(t + c − b), & b ≤ t ≤ d + b − c. \end{cases}

Let some point of the curve γ be the impression of two different prime ends a_{1} and a_{2}, and let {d_{k}}_{k=1}^{∞}, d_{k} ⊂ B, k = 1, 2, …, and {d′_{j}}_{j=1}^{∞}, d′_{j} ⊂ B, j = 1, 2, …, be the sequences of domains which determine these prime ends. Denote D^{+}(z_{0}, ρ) = {z ∈ 𝔻, |z −z_{0}| < ρ} and note that for the conformal mapping f of the circle 𝔻 onto B we have f(D^{+}(e^{iθ_{1}}, ρ)) ∩ f(D^{+}(e^{iθ_{2}}, ρ)) = ∅ for sufficiently small ρ and for different θ_{1} and θ_{2}. If now as θ_{1} and θ_{2} we take the values corresponding to the prime ends a_{1} and a_{2} and take into account the fact that for fixed ρ we have f(D^{+}(e^{iθ_{1}}, ρ)) ⊃ d_{k} and f(D^{+}(e^{iθ_{2}}, ρ)) ⊃ d′_{j} for k > N and j > N, then we find that d_{k} ∩ d′_{j} = ∅ holds for sufficiently large values of the indices k and j. From this it immediately follows that the curve γ cannot have points of self-intersection and if γ(s′) = γ(s′′), (s′ ≠ s′′) and γ(s) is differentiable at s′ and s′′, then γ′(s′) = −γ′(s′′).

Lemma 4. Let γ[s′s′′] be a closed Jordan arc of the curve γ, 0 ≤ s′ < s′′ < S, and for some s_{1} ∈ (s′, s′′) there exist a value s_{2} such that γ(s_{1}) = γ(s_{2}). Then s_{2} ∈ (s′, s′′).

Assume that the opposite assumption is true. Let, for definiteness, s_{2} > s′′. Denote γ̃ = γ[s′′, s_{2}] · γ[s_{2}, S] · γ[0, s′]. Since γ has no points of self-intersection, then either γ[s′, s′′] separates W (γ̃) from the point at infinity or γ̃ separates W (γ[s′, s′′]) from the said point. Let us consider the first case. Then B ⊂ W (γ[s′, s′′]) and for any point z_{0} ∈ B we have

κ(z_{0}, γ[s′, s′′]) = \frac{1}{2π} ∫_{γ[s′,s′′]} d Arg(t − z_{0}) = 1.

Hence

κ(z_{0}, γ̃) = \frac{1}{2π} ∫ ; γ̃ d Arg(t − z_{0}) = 0,

i.e., B ⊄ W (γ̃) and therefore B ⊂ W_{∞}(γ̃), from which we obtain B ⊂ W (γ[s′, s′′]) ∩ B_{∞}(γ̃). Let us represent γ as γ = γ[0, s′] · γ[s′, s_{1}] · γ[s_{1}, s′′] · γ[s′′, s_{2}] · γ[s_{2}, S] and consider two closed curves γ_{1} = γ[s′, s_{1}] · γ[s_{2}, S] · γ[0, s′] and γ_{2} = γ[s_{1}, s′′] · γ[s′′, s_{2}]. From the equality

κ(z_{0}, γ) = κ(z_{0}, γ_{1}) + κ(z_{0}, γ_{2}) = 1

we conclude that one of the values κ(z_{0}, γ_{j}), j = 1, 2, is equal to zero, while the second to unity. Let κ(z_{0}, γ_{2}) = 0. This means that W (γ_{2}) ∩ B = ∅. But since W (γ̃) ∩ B = ∅, it follows that γ[s̃′′, s̃_{2}], where [s̃′′, s̃_{2}] ⊂ (s′′, s_{2}) is separated from B, which contradicts the initial assumption that each point of the curve γ is a boundary point.

Arguments for the case with γ̃ separating W (γ[s′, s′′]) from the point at infinity do not differ from those used above.

Two values s′ and s′′ are called twin if in any neighborhoods V (s′) and V (s′′) there are different values s_{1} and s_{2} such that γ(s_{1}) = γ(s_{2}). Let us show that if s′ ≠ s′′, then s′ and s′′ are twin if and only if γ(s′) = γ(s′′).

Indeed, if γ(s′) = γ(s′′), then the values s′ and s′′ themselves can be taken as s_{1} and s_{2}. Assume now that γ(s′) ≠ γ(s′′). Then since γ(s) is continuous, there are neighborhoods V (s′) and V (s′′) such that |γ(s_{1}) − γ(s_{2})| ≥ d > 0 for any s_{1} ∈ V (s′) and s_{2} ∈ V (s′′). An example of the self-twin value is the value s_{0} characterized by the fact that the arcs γ[s_{0} − δ, s_{0}] and γ[s_{0}, s_{0} + δ] coincide up to orientation.

Denote by 𝔐(γ) the set of all segments I = [s′, s′′] (s′ ≤ s′′) whose end-points are twin values. The set 𝔐(γ) is partially ordered with respect to the inclusion. Let r = {I_{α}}, α ∈ A, be a maximal chain (a maximal linearly ordered subset of the set 𝔐(γ))) [5]. As above, the continuity of γ(s) readily implies that \underline{I} = ⋂_{α∈A}I_{α} ∈ r and \overline{I}_{r} = ⋃_{α∈A}I_{α} ∈ r, i.e., any maximal chain contains both the

first and the last element. We will show that \overline{I} = [\overline{s} ′_{r}, \overline{s} ′′_{r}] is the last element of a maximal chain if and only if γ(\overline{s}′) = γ(\overline{s}′′) ∈ ∂B_{∞}(γ). Let

s^{−} = sup s, s ≤ \overline{s} ′_{r}, γ(s) ∈ ∂B_{∞}(γ),
s^{+} = inf s, s ≥ \overline{s} ′′_{r}, γ(s) ∈ ∂B_{∞}(γ).

Let us show that γ(s^{−}) ∈ ∂B_{∞}(γ) and γ(s^{+}) ∈ ∂B_{∞}(γ). Indeed, let γ(s_{k}) ∈

∂B_{∞}(γ), k = 1, 2, …, lim_{k→∞} s_{k} = s^{−} and z_{m}^{(k)} ∈ B_{∞}(γ), lim_{m→∞} z_{m}^{(k)} = γ(s_{k}),

k = 1, 2, . . .. Then it is clear that lim_{m→∞} z_{m}^{(m)} = γ(s^{−}), i.e., γ(s^{−}) ∈ ∂B_{∞}(γ).

Analogously, γ(s^{+}) ∈ ∂B_{∞}(γ). From the definition of s^{−} and s^{+} it follows that γ(s) ∉ ∂B_{∞}(γ) when s ∈ (s^{−}, s^{+}) and therefore the assumption γ(s^{−}) ≠ γ(s^{+}) would imply the existence of s_{0}, s^{−} < s_{0} < s^{+} such that γ(s_{0}) ∈ ∂B_{∞}(γ). Thus γ(s^{−}) = γ(s^{+}) ∈ ∂B_{∞}(γ), but [s^{−}, s^{+}] ⊇ \overline{I}_{r} and therefore [s^{−}, s^{+}] = \overline{I}_{r}.

Conversely, let [s′, s′′] be some element of the maximal chain and γ(s′) = γ(s′′) ∈ ∂B_{∞}(γ). Assuming that the last element of the chain is another element [s̃′, s̃′′] ⊃ [s′, s′′], by virtue of what has been proved above we would have s̃^{−} ≤ s′ and s̃^{+} ≥ s′′, where s̃^{−} and s̃^{+} are defined for s̃′ and s̃′′. But then γ(s′) = γ(s′′) ∉ ∂B_{∞}(γ). The statement is proved.

Let \underline{I}_{r} = [\underline{s} ′_{r}, \underline{s} ′′_{r}] be the first element of a maximal chain. There are two possible cases:

I. \underline{s} ′_{r} < s′′_{r};

II. \underline{s} ′_{r} = \underline{s} ′′_{r}, i.e., [\underline{s} ′_{r}, \underline{s} ′′_{r}] degenerates into a point.

From Lemma 4 and the definition of \underline{I}_{r} it follows that in case I the curve γ[\underline{s} ′_{r}, \underline{s} ′′_{r}] is a Jordan curve.

Denote by B_{r} the domain bounded by the curve γ[\underline{s} ′_{r}, \underline{s} ′′_{r}] and not containing the point at infinity. Lemma 4 implies that two segments belonging to 𝔐(γ) either have no interior points or one of them is wholly contained within the other. Therefore the number of maximal chains is at most countable and B_{r_{1}}∩B_{r_{2}} = ∅. We will prove that the number of maximal chains of type I is finite, which is equivalent to proving that the number of domains B_{r} is finite. Choose a point a_{r} on each curve γ[\underline{s} ′_{r}, \underline{s} ′′_{r}]. If the set of chosen points is infinite, then it should have at least one limit point which is a boundary point by virtue of the fact that the set of boundary points is closed. Denote it by a_{0} = γ(s_{0}). From the set {a_{r}} choose a sequence {a_{k}}_{k=1}^{∞} tending to a_{0} and assume that γ_{k} are those Jordan curves from the set \{γ[\underline{s} ′_{r}, \underline{s} ′′_{r}]\} on which these points lie. Since lim_{k→∞} diam⟨γ_{k}⟩ = 0, any neighborhood of a_{0} will contain an infinite number of γ_{k}. Since the curve γ is piecewise-smooth, it can be assumed without loss of generality that γ_{k} are smooth and therefore by the property γ′(\underline{s} ′_{k}) = −γ′(\underline{s} ′′_{k}) we have | arg γ′(\underline{s} ′_{k}) − arg γ′(\underline{s} ′′_{k})| = π, k = 1, 2, . . .. But for sufficiently small δ > 0 we have either | arg γ′(s) − arg γ′(s_{0})| < ε when |s − s_{0}| < δ when γ′(s) is continuous at the point s_{0} or | arg γ′(s)−arg γ′(s_{0}−0)| < ε when s_{0}−δ ≤ s ≤ s_{0} and | arg γ′(s) − arg γ′(s_{0} − 0) − πν_{0}| < ε when s_{0} ≤ s < s_{0} + δ at a corner of opening πν_{0}. In both cases the interval of length π cannot be covered by the above-said sets. Hence the set of chains of type I is finite.

Let us consider case II. Again, neglecting the finite number of points of discontinuity of γ′(s), it can be assumed that γ(s) has a derivative at the ends of each interval I = [s′_{α}, s′′_{α}] contained in a chain r of type II and in that case the equality γ′(s′_{α}) = −γ′(s′′_{α}) is fulfilled again. Therefore lim_{s→s_{0}−0} γ′(s) = − lim_{s→s_{0}+0} γ′(s),

where s_{0} = ⋂_{α∈A}I_{α}, i.e., the opening of the angle at the point γ(s_{0}) is equal to

2π. Since, by assumption that γ has a finite number of corners, the number of chains of type II is also finite.

4. Let us fix some maximal chain r′ and consider I = ∪I_{α}, where I_{α} ∈ 𝔐(γ)\r′. Since any union of the form ∪I_{α}, where I_{α} are contained in the same chain, is an element of this chain (i.e., is a segment) and the number of maximal chains is finite, we conclude that the set I is a finite union of segments. Let

Ĩ = m ; ⋃_{k=1}[s′_{k}, s′′_{k}], where \overline{s} ′_{r′} ≤ s′_{1} < s′′_{1} < s′_{2} < s′′_{2} < ⋯ < s′_{m} < s′′_{m} ≤ \overline{s} ′′_{r′}. Choose

in B_{r′} a point z_{r′} and connect it with the point γ(s_{r′}) ∈ γ[\underline{s} ′_{r′}, \underline{s} ′′_{r′}] by means of the simple arc l_{r′} passing through B_{r′}. Let s′_{j} < s_{r′}, j = \overline{1, k}, s′′_{j} < s_{r′}, j = \overline{1, k}, and s′_{k+1} > s_{r′}. Consider the curve

C_{r′} = L_{r′} · γ[\overline{s} ′_{r′}, s′_{1}] · γ[s′′_{1}, s′_{2}] ⋯ γ[s′′_{k}, s_{r′}] · l_{r′},  (10)

where L_{r′} is a simple curve passing through B_{∞}(γ) and connecting the point at infinity with γ(\overline{s} ′_{r′}). The curve C_{r′} is simple by construction. Fix in \overline{ℂ}\⟨C_{r′}⟩ a one-valued branch of the function P (r′) : w = \sqrt{z − z_{r′}}. The function P (r′) conformally maps B onto some domain B(r′) whose boundary contains simple

arcs P (r′)∘γ[\overline{s} ′_{r′}, s′_{1}], P (r′)∘γ[s′′_{1}, s′_{2}], …, P (r′)∘γ[s′′_{m}, \overline{s} ′′_{r′}]. Since P (r′) is analyt-

ically continuable across the both sides of the cut C_{r′}, the images of the corners of the curve γ are the corners of the boundary of the domain B(r′) of the same openings, while new corners do not appear. Since all twin values corresponding to the end-points of segments, contained in the chain r′, cease being twin, the number of maximal chains in 𝔐(P_{r′} ∘ γ) is less by one than in 𝔐(γ), while the points P (r′)(γ(s′_{k})) and P (r′)(γ(s′′_{k})), k = \overline{1, m}, turn out to lie on ∂B_{∞}(P_{r′} ∘ γ) and thus become the last elements of the respective chains.

If now the procedure described above is applied to B(r′), then, without violating the piecewise-smoothness of the boundary, we again decrease the number of maximal chains by one.

Continuing this process, after a finite number of steps we come to the domain B_{0} bounded by the piecewise-smooth curve γ_{0} with the same number of corners and the same angle openings as those of the initial curve γ. But if the set 𝔐(γ) contains maximal chains of type II, the new curve γ_{0} will keep them and it will not be a Jordan curve.

Lemma 5. Let z_{∗} be an accessible from B_{∞}(γ_{∗}) corner of opening ν_{∗}π, ν_{∗} < 1, on ∂B_{∗}. Then there exists a holomorphic and univalent function w = Φ_{∗}(z) in \overline{B}_{∗} such that the mapping ζ = (w − Φ_{∗}(z_{∗}))^{\frac{1}{ν∗}} is univalent in \overline{Φ_{∗}(B_{∗})}.

We will construct the function Φ_{∗} with more specific properties. Namely, Φ_{∗}(γ_{∗}) = 0 and the direction of the bisectrix of the angle E_{∗} with vertex at the point Φ_{∗}(z_{∗}) will coincide with the direction of the positive real semi-axis.

Choose a point a ∈ D(z_{∗}, δ) ∩ C\overline{B}, where D(z_{∗}, δ) = \{z; |z − z_{∗}| < δ\}, and connect it by means of the curve l_{∗}, having no common points with \overline{B}_{∗}, with the point at infinity. This can be done since z_{∗} is accessible from B_{∞}(γ_{∗}). Cut the plane along l_{∗} and map the obtained domain conformally onto the plane cut along the negative real semi-axis. Normalize the mapping w = F (z) by the condition F (z_{∗}) = u_{0}, arg F′(z_{∗}) = η, where u_{0} > 0 and η is chosen so that the direction of the bisectrix of the angle E_{∗} with vertex at z_{∗} be mapped on the direction of the positive real semi-axis.

Take ε > 0 such that \frac{ν_{∗}}{2} + ε < \frac{1}{2} and choose δ > 0 so that w(s) = F (z(s)) ∈ E_{\frac{ν∗}{2} +ε}(u_{0}) for s ∈ (s_{∗} −δ, s_{∗} +δ), where z_{∗} = γ_{∗}(s_{∗}). Next, choose q and 0 < q < u_{0} so that (D(q, ρ)\E_{\frac{ν∗}{2} +ε}(u_{0})) ∩ F (B_{∗}) = ∅. Denote by w_{1} some point of the intersection of the circumference C(q, ρ) = {w : |w − q| = ρ} with the boundary E_{\frac{ν∗}{2} +ε}(u_{0}) and let πβ = | arg w_{1}|. It is obvious that 0 < β < 1. Consider the translation T_{q} : ζ = w −q. For the mapping P_{β} ∘T_{q} the circle D′(q, ρ), cut along the radius directed towards the negative real semi-axis, is mapped into E_{β}(0), while the angle E_{\frac{ν∗}{2} +ε}(u_{0}) is mapped by Lemma 3 onto the domain contained in E_{\frac{ν∗}{2} +ε}((u_{0}−q)^{β}). Thus the domain E_{β}(0)\E_{\frac{ν∗}{2} +ε}((u_{0}−q)^{β}) contains no points of the domain P_{β}(T_{q}(F (B_{∗}))). Make another translation T̃_{(u_{0}−q)^{β}} : ζ̃ = ζ−(u_{0}−q)^{β}. Then T̃_{(u_{0}−q)^{β}}(P_{β}(T_{q}(F (B_{∗})))) ⊂ E_{\frac{ν∗}{2} +ε}(0). But E_{\frac{ν∗}{2} +ε}(0) is a subdomain of the domain where the function P_{\frac{1}{ν∗}} is univalent. Each of the mappings T̃_{(u_{0}−q)^{β}}, P_{β}, T_{q} and F is univalent on the closure of those domains on which they are defined and therefore the mapping Φ_{∗} = T̃_{(u_{0}−q)^{β}} ∘ P_{β} ∘ T_{q} ∘ F satisfies required conditions.

Note that the statement of the lemma also holds for ν_{∗} ≥ 1 since in the case ν_{∗} = 1 the point z_{∗} is not a corner, while for ν_{∗} > 1 we can take as Φ_{∗} the identical mapping. However, for the symmetrical notation of the expressions arising below we will use the common symbols in all cases. In this context, for ν_{∗} > 1 we will take as Φ_{∗}(z) the entire linear function which maps a corner on the origin and the direction of the interior angle bisectrix on the direction of the positive real semi-axis. Our next task is to map the domain B_{0} onto the domain bounded by a Jordan curve without corners.

Let z_{1} = γ_{0}(s_{∗}) be the first element of the maximal chain r_{1} ∈ 𝔐(γ_{0}) of type II. The natural parameter on the curve γ_{0} is again denoted by s and it is assumed that s′_{j} and s′′_{j} are the same notations for r_{1} as in the case of a chain of type I. An auxiliary curve C_{r_{1}} has the same form as (10) but with the only difference that the curve l_{r_{1}} is absent and the last cofactor in the expression for C_{r_{1}} is γ_{0}[s′′_{k}, s_{∗}], where s′′_{k} < s_{∗}, and s′_{k+1} > s_{∗}. Let us make a mapping P_{\frac{1}{2}} ∘ Φ_{1}, where Φ_{1} is the holomorphic function from Lemma 5 (the entire linear function in the considered case). By Lemma 2 the point P_{\frac{1}{2}} ∘ Φ_{1}(z_{1}) is not a corner of the curve γ_{1} = P_{\frac{1}{2}} ∘ Φ_{1} ∘ γ_{0}. Moreover, like in the case of a chain of type I, the number of maximal chains in 𝔐(γ_{1}) is less by one than the number of chains in 𝔐(γ_{0}). All corners of the curve γ_{1}, except the point P_{\frac{1}{2}} ∘ Φ_{1}(z_{1}), have the same opening as their preimages.

Continue this process until after performing a finite number of steps the obtained curve γ_{n_{0}} becomes a Jordan curve. Number the remaining corners in an arbitrary manner starting from n_{0} + 1. The following notation will be used below: Φ_{j} (j ≥ 2) will denote the function from Lemma 5 for the domain

(P_{\frac{1}{ν_{j−1}}} ∘ Φ_{j−1} ∘ ⋯ ∘ P_{\frac{1}{ν_{1}}} ∘ Φ_{1})(B_{0})

and the point (P_{\frac{1}{ν_{j−1}}} ∘ Φ_{j−1} ∘ ⋯ ∘ P_{\frac{1}{ν_{1}}} ∘ Φ_{1})(z_{j}).

Recall that ν_{1} = ν_{2} = ⋯ = ν_{n_{0}} = 2 and z_{1}, z_{2}, …, z_{n_{0}} are the points corresponding to the first elements of type II in 𝔐(γ_{0}).

Denote ω̃ = P_{\frac{1}{ν_{k}}} ∘ Φ_{n} ∘ ⋯ ∘ P_{\frac{1}{ν_{1}}} ∘ Φ_{1}. From Lemma 2 and the above constructions it follows that the curve γ_{n} = ω̃ ∘ γ_{0} is a Jordan smooth curve.

5. Fix k, 1 ≤ k ≤ n, and write ω̃ in the form

ω̃ = X_{k} ∘ P_{\frac{1}{ν_{k}}} ∘ X̃_{k},

where

X̃_{k} = Φ_{k} ∘ P_{\frac{1}{ν_{k}}} ∘ ⋯ ∘ P_{\frac{1}{ν_{1}}} ∘ Φ_{1},
X_{k} = Φ_{\frac{1}{νn}} ∘ Φ_{n} ∘ P_{\frac{1}{ν_{n−1}}} ∘ ⋯ ∘ Φ_{k+1}.

Since the univalent function \overline{B}_{0} in X̃_{k} is holomorphic at the point z_{k} and X̃_{k}(z_{k}) = 0, for z sufficiently close to z_{k} we have

X̃_{k}(z) = ∑_{m=1}^{∞} ã_{m}(z − z_{k})^{m}, ã_{1} ≠ 0.

Hence we obtain

X̃(z) = (z − z_{k})R_{k}(z),

where R_{k}(z) ≠ 0 for |z − z_{k}| < δ_{k}. Therefore

\frac{d}{dz} (P_{\frac{1}{ν_{k}}} (X̃_{k}(z))) = \frac{1}{ν_{k}} (z − z_{k})^{\frac{1}{ν_{k}} −1}(R_{k}(z))^{\frac{1}{ν_{k}} −1}[R_{k}(z) + (z − z_{k})R′_{k}(z)]
= (z − z_{k})^{\frac{1}{ν_{k}} −1}g_{k}(z),

where g_{k}(z) = \frac{1}{ν_{k}} (R_{k}(z))^{\frac{1}{ν_{k}} −1}[R_{k}(z) + (z − z_{k})R′_{k}(z)] is a non-vanishing holomorphic function in D(z_{k}, δ_{k}).

Since X_{k}(w) is holomorphic in the neighborhood w = 0, for w = P_{\frac{1}{ν_{k}}} (X̃(z)), where z ∈ D^{+}(z_{k}, δ_{k}), we have

\frac{dω̃(z)}{dz} = \frac{dX_{k}(w)}{dw} · \frac{dw}{dz} = \frac{dX_{k}(P_{\frac{1}{ν_{k}}} (X̃(z)))}{dw} · \frac{d}{dz} (P_{\frac{1}{ν_{k}}} (X̃_{k}(z)))

= \frac{dX_{k}(P_{\frac{1}{ν_{k}}} (X̃(z)))}{dw} · (z − z_{k})^{\frac{1}{ν_{k}} −1} · g_{k}(z).

Denoting

\frac{dX_{k}(P_{\frac{1}{ν_{k}}} (X̃(z)))}{dw} · g_{k}(z) = ω̃_{k}(z),

we obtain the local representation

ω̃′(z) = (z − z_{k})^{\frac{1}{ν_{k}} −1} · ω̃_{k}(z),  (11)

where ω̃_{k}(z) is holomorphic in \overline{D}^{+}(z_{k}, δ_{k})\\{z_{k}\}, continuous in \overline{D}(z_{k}, δ_{k}) because P_{\frac{1}{ν_{k}}} (X̃_{k}(z)) is continuous and non-vanishing because X_{k}(w) is univalent in the neighborhood of zero.

Consider the function

ω̃_{0}(z) = ω̃′(z) ∏_{k=1}^{n}(z − z_{k})^{1− \frac{1}{ν_{k}}} .  (12)

By virtue of (11) the function ω̃_{0}(z) is holomorphic in \overline{B}_{0}\ n ; ⋃_{k=1}\{z_{k}\}, continuous

in \overline{B}_{0} and non-vanishing. From (12) we obtain the representation

ω̃′ = ω̃_{0} (z) ∏_{k=1}^{n}(z − z_{k})^{\frac{1}{ν_{k}} −1}.

To obtain a similar representation of the function Ω̃′(ζ) = (ω̃^{−1}(ζ))′, where ζ ∈ B_{n} = ω̃(B_{0}), we write Ω̃ in the form

Ω̃ = X̃ _{k}^{−1} ∘ P_{ν_{k}} ∘ X_{k}^{−1},

where

X_{k}^{−1} = Φ_{k+1}^{−1} ∘ P_{ν_{k+1}} ∘ Φ_{k+2}^{−1} ∘ ⋯ ∘ P_{ν_{n}},
X̃ _{k}^{−1} = Φ_{1}^{−1} ∘ P_{ν_{1}} ∘ Φ_{2}^{−1} ∘ ⋯ ∘ Φ_{k}^{−1}

and investigate the behavior of its derivative in the neighborhood of the point ζ_{k} = ω̃(z_{k}). Repeating the previous arguments for the function Ω̃, we obtain the representation

Ω̃′(ζ) = Ω̃_{0}(ζ) ∏_{k=1}^{n}(ζ − ζ_{k})^{ν_{k}−1},  (13)

where the function Ω̃_{0}(z) is holomorphic in \overline{B}_{n}\ n ; ⋃_{k=1}\{ζ_{k}\}, continuous and non-

vanishing in \overline{B}_{n}.

So far we have been investigating the behavior of the derivative of the function mapping B_{0} onto 𝔻. But B_{0} is obtained from the domain B by means of the conformal mapping violating neither the piecewise-smoothness of the boundary nor the openings of corners and therefore the reasoning used above for ω̃ and Ω̃ can be applied both to the function mapping B onto 𝔻 and to the inverse function.

6. Before we proceed to investigating the nature of the continuity of the considered functions it is appropriate to make the following remark: if the domain B (or B_{0}) is bounded by a non-Jordan curve, then an inequality of form (2) cannot be satisfied globally all over the boundary. Again, since the mapping B → B_{0}, as mentioned above, is analytically continuable across the boundary, it inherits all the local boundary properties of the function ω̃ which we are interested in.

Let 0 < ν_{k} < 2. Then, as is known ([4], §6 and Appendix 1), the function t^{\frac{1}{ν_{k}}} satisfies on the curve t = X̃_{k} ∘ γ_{0}(s) the condition H(μ), where μ = min(1, \frac{1}{ν_{k}}).

Therefore for z_{1}, z_{2} ∈ ⟨γ⟩ ∩ D(z_{k}, δ_{k}) we have

|\frac{dX_{k}(P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{1})))}{dw} − \frac{dX_{k}(P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{2})))}{dw}| ≤ M_{5}|P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{1}) − P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{2}))|

≤ M_{6}|X̃_{k}(z_{1}) − X̃_{k}(z_{2})|^{μ} ≤ M_{7}|z_{1} − z_{2}|^{μ}.  (14)

It obviously follows that on ∂D(z_{k}, δ_{k})∩\overline{B}_{0} the function ^{\frac{dX_{k}(P_{\frac{1}{ν_{k}}} (X̃_{k}(z)))}{dw}} satisfies the condition H(1) and thus it satisfies the condition H(μ) all over \overline{D}^{+}(z_{k}, δ_{k}) = \overline{D(z_{k}, δ_{k}) ∩ B_{0}} ([4], § 15 and Appendix II).

Let us consider the case ν_{k} = 2 (cusp). Let L_{1} and L_{2} be the arcs of the curve X̃_{k} ∘ γ which are adjacent to the corner z̃_{k} = 0, and let in a sufficiently small neighborhood of zero these arcs be represented in the form

L_{1} : ỹ = χ_{1}(x̃), L_{2} : ỹ = −χ_{2}(x̃),

where χ_{j}(x̃) = |x̃|^{p_{j}}φ_{j}(x̃), 0 < k ≤ φ_{j}(x̃) ≤ K, j = 1, 2; 0 ≤ −x̃ ≤ ε, 1 < p_{j} < ∞. In that case the point z_{k} is called a cusp of finite order. It is obvious that the numbers p_{j}, j = 1, 2, are invariant with respect to diffeomorphisms of the domain enclosing \overline{B}_{0}. Let z̃_{j} = x̃_{j} + iỹ_{j} ∈ L_{j}, j = 1, 2. Then

|z̃_{j}| = \sqrt{x̃ _{j}^{2} + χ2_{j}(x̃_{j})} = \sqrt{[\frac{χ_{j}(x̃_{j})}{φ_{j}(x̃_{j})}]^{\frac{2}{pj}} + χ2_{j}(x̃_{j})}

= |χ_{j}(x̃_{j})|^{\frac{1}{pj}} \sqrt{[φ_{j}(x̃_{j})]^{− \frac{2}{pj}} + [χ_{j}(x̃_{j})]^{2(1− \frac{1}{pj})}}
≤ M_{8}|χ_{j}(x̃_{j})|^{\frac{1}{pj}} ≤ M_{8}|χ_{j}(x_{j})|^{\frac{1}{p}} ,  (15)

where p = max(p_{1}, p_{2}).

On the other hand, we have

|z̃_{1} − z̃_{2}| ≥ |χ_{1}(x̃_{1})| + |χ_{2}(x̃_{2})|.  (16)

From (15) and (16) we obtain

|z̃_{1} − z̃_{2}| ≥ M_{9}(|z̃_{1}|^{p} + |z̃_{2}|^{p}).  (17)

Now using the inequality

a^{p} + b^{p} ≥ 2^{1−p}(a + b)^{p},  (18)

where a ≥ 0, b ≥ 0 and p > 1 ([6], Section 3.5), from (17) we get

|z̃_{1} − z̃_{2}| ≥ M_{10}(|z̃_{1}| + |z̃_{2}|)^{p}.

Let w_{j} = P_{\frac{1}{2}} (z̃_{j}), j = 1, 2. Then

|z̃_{1} − z̃_{2}| ≥ M_{10}(|w_{1}|^{2} + |w_{2}|^{2})^{p}

and again using inequality (18) we obtain

|z̃_{1} − z̃_{2}| ≥ M_{11}(|w_{1}| + |w_{2}|)^{2p} ≥ M_{11}|w_{1} − w_{2}|^{2p},

i.e.,

|w_{1} − w_{2}| ≤ M_{12}|z_{1} − z_{2}|^{\frac{1}{2p}} .  (19)

Inequality (19) means that for a cusp of finite order the function w = z̃^{\frac{1}{2}} satisfies, on X̃_{k} ∘ γ in a neighborhood of zero, the condition H(\frac{1}{2p}) in the socalled strong form ([4], Appendix II). Hence, repeating the arguments we used for (14), we conclude that ω̃_{0}(z) also satisfies, on ⟨γ⟩ ∩ D(z_{k}, δ_{k})), the condition H(\frac{1}{2p}) in the strong form and therefore satisfies this condition in \overline{D}^{+}(z_{k}, δ_{k}).

Since ω̃_{0} is holomorphic in \overline{B}_{0}\ n ; ⋃_{k=1}\{z_{k}\}, the latter fact immediately implies

that if γ_{0} is a Jordan curve and all cusps are of finite order, then ω̃_{0} satisfies the Hölder condition all over \overline{B}_{o}.

The proof that \overline{Ω}_{0} is a Hölder continuous function in \overline{B}_{n} is simpler since γ_{n} = ∂B_{n} is smooth and an estimate of form (14) for the function

\frac{dX_{k}^{−1}(P_{ν_{k}}(X_{k}^{−1}(ζ)))}{dz̃},

where z̃ = P_{ν_{k}}(X_{k}^{−1}(ζ)), ζ ∈ \overline{B}_{n}, implies that it is a Hölder continuous function in \overline{B}_{n}.

7. Let us perform the last mapping Φ : B_{n} → 𝔻. Note that if z_{k} is a corner of opening πν_{k}, ν_{k} < 2, or a cusp of finite order, then for z_{1}, z_{2} ∈ \overline{D}^{+}(z_{k}, δ_{k}) we have

|ω̃(z_{1}) − ω̃(z_{2})| = |X_{k}(P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{1}))) − X_{k}(P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{2})))|
≤ M_{13}|P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{1})) − P_{\frac{1}{ν_{k}}} (X̃_{k}(z_{2}))| ≤ M_{14}|X̃_{k}(z_{1}) − X̃_{k}(z_{2})|^{μ}
≤ M_{15}|z_{1} − z_{2}|^{μ}, μ > 0.

Let γ be a piecewise-Lyapunov curve. Then, by the corollary of Lemma 2, ∂B_{n} is a Lyapunov curve and Φ′_{ζ}(ζ) satisfies in \overline{B}_{n} the Hölder condition and is different from zero [7]. Denote ω = Φ ∘ ω̃. Then

\frac{dω}{dz} = \frac{dΦ(ζ)}{dζ} · \frac{dω̃(z)}{dz}.

If z_{k} is a corner with ν_{k} < 2 or a cusp of finite order, then for z_{1}, z_{2} ∈ \overline{D}^{+}(z_{k}, δ_{k}) we have

|Φ′_{ζ}(ω̃(z_{1})) − Φ′_{ζ}(ω̃(z_{2}))| ≤ M_{16}|ω̃(z_{1}) − ω̃(z_{2})|^{μ_{1}} ≤ M_{17}|z_{1} − z_{2}|^{μ_{2}}, μ_{2} > 0.

Therefore in the representation

ω′(z) = Φ′_{ζ}(ω̃(z)) ω̃_{k}(z)(z − z_{k})^{\frac{1}{ν_{k}} −1}

given by equality (11) the function Φ′_{ζ}(ω̃(z)) ω̃_{k}(z) satisfies in \overline{D}^{+}(z_{k}, δ_{k}) the Hölder condition and is different from zero. This fact allows us to conclude that in the representation

ω′(z) = ω_{0}(z) ∏_{k=1}^{n}(z − z_{k})^{\frac{1}{ν_{k}} −1}

the holomorphic function ω_{0}(z) in B_{0} is continuous in \overline{B}_{0} and satisfies the Hölder condition on each smooth simple arc of the curve γ_{0}. If however γ_{0} is a Jordan curve and all cusps are of finite order, then ω_{0} is a Höder continuous function in \overline{B}_{0}.

Let us denote Φ^{−1} = Ψ and consider the behavior of the function Ω = Ω̃∘Ψ in the neighborhood of the point τ_{k} = Φ(ω̃(z_{k})) = ω(z_{k}). Since ∂B_{n} is a Lyapunov curve, the function Ψ′(τ) satisfies in \overline{𝔻} the Hölder condition and is different from zero [7]. Using (13), we obtain

Ω′(τ) = Ψ′(τ) · Ω̃ ′_{ζ}(Ψ(τ)) = Ψ′(τ) · Ω̃_{0}(Ψ(τ)) · ∏_{k=1}^{n} (Ψ(τ) − Ψ(τ_{k}))^{ν_{k}−1}

= Ψ′(τ) · Ω̃_{0}(Ψ(τ)) ∏_{k=1}^{n} (\frac{Ψ(τ) − Ψ(τ_{k})}{τ − τ_{k}})^{ν_{k}−1} ∏_{k=1}^{n}(τ − τ_{k})^{ν_{k}−1}.

Since Ω̃_{0}(ζ) and Ψ(τ) are Hölder continuous functions, the composition Ω̃_{0} ∘ Ψ is a Hölder continuous function in \overline{𝔻}. Consider the continuous function

u(τ, τ_{k}) = Arg \frac{Ψ(τ) − Ψ(τ_{k})}{τ − τ_{k}}  (20)

in 𝔻. By Lemma 1 this function satisfies the Hölder condition on each of the arcs l_{k}^{−} = {e^{iθ}, θ_{k} −ε ≤ θ ≤ θ_{k}} and l_{k}^{+} = {e^{iθ}, θ_{k} ≤ θ ≤ θ_{k} +ε}, where τ_{k} = e^{iθ_{k}}. Therefore u(τ, τ_{k}) satisfies the Hölder condition on l^{−} ∪ l^{+}. On the remaining part of the unit circumference the Hölder continuity of the function u(τ, τ_{k}) is obvious. But in that case the function

W (τ, τ_{k}) = ln \frac{Ψ(τ) − Ψ(τ_{k})}{τ − τ_{k}}

satisfies the Hölder condition in \overline{𝔻} [4]. Therefore

|\frac{Ψ(τ′) − Ψ(τ_{k})}{τ′ − τ_{k}} − \frac{Ψ(τ′′) − Ψ(τ_{k})}{τ′′ − τ_{k}} | = | exp W (τ′, τ_{k}) − exp W (τ′′, τ_{k})|

= | ∑_{n=1}^{∞} \frac{W^{n}(τ′, τ_{k}) − W^{n}(τ′′, τ_{k})}{n!}| ≤ |W (τ′, τ_{k}) − W (τ′′, τ_{k})| · ∑_{n=1}^{∞} \frac{nM_{18}^{n}}{n!}

≤ M_{19}|τ′ − τ′′|^{α}, 0 < α ≤ 1.

In conclusion let us consider the case with the piecewise-smooth curve γ. In that case ∂B_{n} is a smooth curve and therefore Ψ′(τ) ∈ H_{p}(𝔻) for all p > 0([7], Ch. IX). The function Ω̃_{0}(Ψ(τ)) is bounded in 𝔻 since Ω̃_{0}(ζ) is continuous in \overline{B}_{n}. The continuity of the function u(τ, τ_{k}) in \overline{𝔻} implies that

[\frac{Ψ(τ) − Ψ(τ_{k})}{τ − τ_{k}}]^{±1} ∈ H_{p}

for all p > 0 ([7], Ch. IX), from which it follows that

Ψ′(τ)Ω̃_{0}(Ψ(τ)) ∏_{k=1}^{n} [\frac{Ψ(τ) − Ψ(τ_{k})}{τ − τ_{k}}]^{ν_{k}−1} ∈ H_{p}

for all p > 0.

Finally, the identity

|ω(Ω(τ))\sqrt[p]{Ω′(τ)}|^{p} = |τ|^{p} |Ω′(τ)|, τ ∈ 𝔻,

immediately implies that ω(z) belongs to ⋂_{p>0} E_{p}(B_{0}).

References

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2. Ch. Pommerenke, Boundary behavior of conformal maps. Grundlehren der Mathematischen Wissenschaften 299. Springer-Verlag, Berlin, 1992.

3. G. Khuskivadze and V. Paatashvili, On the conformal mapping of a circle onto the domain with a piecewise-smooth boundary. Proc. A. Razmadze Math. Inst. 116(1998), 123–132.

4. N. I. Muskhelishvili, Singular integral equations. (Russian) Nauka, Moscow, 1968; English translation: P. Noordhoff, Groningen, 1953.

5. A. N. Kolmogorov and S. V. Fomin, Elements of the function theory and functional analysis. (Russian) Nauka, Moscow, 1968.

6. G. H. Hardy, J. E. Littlewood, and G. Polia, Inequalities. Cambridge Univ. Press, Cambridge, 1934.

7. G. M. Goluzin, Geometric theory of functions of a complex variable. (Russian) Nauka, Moscow, 1966.

(Received 18.09.2002)

Author’s address:

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: ramaz@rmi.acnet.ge