平面領域の環状幅 (CIRCULAR WIDTH) とその応用

THE CIRCULAR WIDTH OF A PLANE DOMAIN AND ITS APPLICATIONS

須川敏幸 TOSHIYUKI SUGAWA

Abstract. We first give a quantity for non-vanishing analytic functions on a hyperbolic domain to measure their rate of growth in terms of hyperbolic metric. Then we define the circular width of a plane domain not containing the origin. We will see that an estimate of the circular width leads to a criterion for univalence (or quasiconformal extension).

1. The quantity V_{D}(φ)

Conformal invariants play a central role in the modern theory of functions of a complex variable. One of the most important is the hyperbolic metric ρ_{D}(z)|dz| of a hyperbolic plane domain D. Recall that a subdomain D of ℂ is called hyperbolic if D admits an analytic universal covering projection p of the unit disk 𝔻 = {ζ ∈ ℂ : |ζ| < 1} onto D. Then the hyperbolic metric is defined by the equation ρ_{D}(z)|p′(ζ)| = 1/(1 − |ζ|^{2}) for ζ ∈ p^{−1}(z). Note that the density ρ_{D}(z) does not depend on the particular choice of ζ or p. The Poincaré-Koebe uniformization theorem tells us that D ⊂ ℂ is hyperbolic if and only if D is neither the whole plane ℂ nor the punctured plane ℂ \ {a} for any a ∈ ℂ. We denote by d_{D}(z_{0}, z_{1}) the distance between z_{0} and z_{1} measured by the metric ρ_{D}(z)|dz|.

Let φ be a non-vanishing analytic function on a hyperbolic domain D, namely, φ : D → ℂ^{∗} = ℂ \ {0} is holomorphic. Then we set

V_{D}(φ) = sup_{z∈D} ρ_{D}(z)^{−1} |\frac{φ′(z)}{φ(z)}| .

Note also that V_{D}(φ) can be thought of the Bloch semi-norm of the (possibly multi-valued) function log φ. The quantity V_{D}(φ) measures the rate of growth of φ compared with the hyperbolic metric. More precisely, we have the following characterization.

Proposition 1. Let φ be a non-vanishing analytic function on a hyperbolic domain D and let c be a positive constant. Then V_{D}(φ) ≤ c if and only if the double inequality

exp ( − c d_{D}(z_{0}, z_{1})) ≤ \frac{|φ(z_{1})|}{|φ(z_{0})|} ≤ exp (c d_{D}(z_{0}, z_{1})) (1.1)

holds for every pair of points z_{0}, z_{1} in D.

We list properties of this quantity.

Theorem 2. Let D be a hyperbolic domain and let φ and ψ be non-vanishing analytic functions on D. (i) V_{D}(φ · ψ) ≤ V_{D}(φ) + V_{D}(ψ). (ii) V_{D}(φ^{α}) = |α|V_{D}(φ) holds for a complex number α whenever φ^{α} is single-valued in D. (iii) Let p : D_{0} → D be an analytic (unbranched and unlimited) covering projection. Then V_{D_{0}}(φ ∘ p) =

V_{D}(φ). In particular, V_{D}(φ) is conformally invariant in the sense that this does not depend on the source domain.

(iv) V_{D}(L ∘ φ) = V_{D}(φ) holds for any conformal automorphism L of ℂ^{∗}. In particular, V_{D}(1/φ) = V_{D}(φ) = V_{D}(cφ) for any constant c ∈ ℂ^{∗}.

Date: November 11, 2004, at Satellite Plaza of Kanazawa University.

This talk is based on the joint work [8] with Yong Chan Kim.

(v) Let ω : D_{0} → D be a holomorphic map. Then V_{D_{0}}(φ ∘ ω) ≤ V_{D}(φ). (vi) If ψ : D → ℂ^{∗} is univalent and if φ(D) ⊂ ψ(D) then V_{D}(φ) ≤ V_{D}(ψ).

2. Circular width

Let Ω be a hyperbolic plane domain with 0 ∈ ℂ \ Ω. The quantity

W (Ω) = ( inf_{w∈Ω} |w|ρ_{Ω}(w))^{−1}

will be called the circular width of Ω (about the origin). This quantity can be expressed in terms of a covering projection onto Ω.

Lemma 3. Let Ω be a proper subdomain of the punctured plane ℂ^{∗} and let p be an analytic (unbranched) covering projection of a domain D onto Ω. Then W (Ω) = V_{D}(p).

Remark. In general, we can define the circular width W_{a}(Ω) about a point a ∈ ℂ \ Ω in the same way: W_{a}(Ω) = 1/ inf_{w∈Ω} |w − a|ρ_{Ω}(w). It is known that the domain constant C(Ω) = sup_{a∈ℂ\Ω} W_{a}(Ω) is finite if and only if the set ℂ̂ \ Ω is uniformly perfect (see, for example, [10] or [12]). In this context, the constant W_{a}(Ω) appeared essentially in a paper [13] by J.-H. Zheng.

We now collect basic properties of the circular width. Before that, we recall the notion of circular symmetrization. For a subdomain Ω of ℂ^{∗} we define the circular symmetrization Ω^{∗} (about the origin) by

Ω^{∗} = {re^{iθ} : θ ∈ I(r, Ω), 0 < r < ∞},

where I(r, Ω) denotes the interval in the form (−t/2, t/2) of the same length as I_{r} = {θ ∈ [−π, π] : re^{iθ} ∈ Ω} if I_{r} ≠ [−π, π] otherwise I(r, Ω) = [−π, π].

Theorem 4. Let Ω and Ω′ be proper subdomains of the punctured plane ℂ^{∗}.

(i) W (Ω) = W (L(Ω)) for any conformal automorphism L of ℂ^{∗}. (ii) If Ω ⊂ Ω′, then W (Ω) ≤ W (Ω′). (iii) Circular symmetrization does not decrease circular width; W (Ω) ≤ W (Ω^{∗}). (iv) If Ω is simply connected, then W (Ω) ≤ 4.

In general, the circular width may not be finite. We give here a characterization of domains with infinite circular width. In particular, if the origin is an isolated boundary point of Ω, then W (Ω) = ∞.

Proposition 5. Let Ω be a proper subdomain of the punctured plane ℂ^{∗}. The circular width W (Ω) is infinite if and only if there is a sequence of annuli A_{n} = {w ∈ ℂ : r_{n} < |w| < R_{n}} with A_{n} ⊂ Ω such that R_{n}/r_{n} → ∞.

The circular width may not behave continuously in Ω. For instance, consider the sequence of domains Ω_{n} = {|w − 1| < 1 + 1/n}. Then Ω_{n} converges to Ω_{∞} = {|w − 1| < 1} in the Hausdorff topology. But W (Ω_{n}) = ∞ by Proposition 5 whereas W (Ω_{∞}) ≤ W (ℍ) = 2 (see Example 3.1). We can, however, show a continuity property of circular width in the following form.

Proposition 6. Let Ω_{n} be a sequence of domains with Ω_{n} ⊂ Ω_{n+1} such that the union Ω = ∪_{n=1}^{∞}Ω_{n} is a proper subdomain of ℂ^{∗}. Then W (Ω_{n}) → W (Ω) as n → ∞.

The circular width W (Ω) dominates the quantity V_{D}(φ) for holomorphic maps φ : D → Ω.

Theorem 7. Let φ : D → Ω be holomorphic. Then V_{D}(φ) ≤ W (Ω).

Proof. By Theorem 2 (v), we have V_{D}(φ) = V_{D}(id_{Ω} ∘ φ) ≤ V_{Ω}(id_{Ω}) = W (Ω).

Combining this with Proposition 1, we have the following. A similar result was obtained by [13].

Corollary 8. Under the same hypotheses in Theorem 7,

exp ( − W (Ω) d_{D}(z_{0}, z_{1})) ≤ \frac{|φ(z_{1})|}{|φ(z_{0})|} ≤ exp (W (Ω) d_{D}(z_{0}, z_{1})), z_{0}, z_{1} ∈ D.

3. Computations of circular widths

In the present section, we give exact values of circular width for several concrete examples. In view of Theorem 4 (iii), we see that circularly symmetric domains are particularly important.

Example 3.1 (sectors). For S(β) = {w : |argw| < πβ/2}, 0 < β ≤ 2, we have W (S(β)) = 2β.

Example 3.2 (half-sectors). Let S(β, r) = {w : |argw| < πβ/2, |w| < r} and S′(β, r) = {w : |argw| < πβ/2, |w| > 1/r} for 0 < β ≤ 2 and 0 < r < ∞. Then W (S(β, r)) = W (S′(β, r)) = 2β.

Example 3.3 (annuli). For the annulus A(r, R) = {w : r < |w| < R}, 0 < r < R < ∞, we have W (A(r, R)) = (2/π) log(R/r).

Example 3.4 (disks). Let 𝔻(a, r) = {w : |w − a| < r} for 0 < r ≤ a. Then

W (𝔻(a, r)) = \frac{2r/a}{1 + \sqrt{1 − (r/a)^{2}}} .

Example 3.5 (parallel strips). Let P (a, b) = {w : a < Re w < b} for 0 ≤ a < b < ∞. Then

W (P (a, b)) = max_{0≤θ≤π/2} \frac{2t cos θ}{1 − tθ} ,

where t is a number with 0 < t ≤ 2/π determined by πt/2 = (b − a)/(b + a).

Example 3.6 (truncated wedges). Let S(β, r, R) = {w : |argw| < πβ/2, r < |w| < R}, 0 < β ≤ 2, 0 < r < R < ∞. Then

W (Ω) = \frac{log(R/r)}{(1 + t)K(t)}, where K(t) = ∫_{0}^{1} \frac{dx}{\sqrt{(1 − x^{2})(1 − t^{2}x^{2})}}

is the complete elliptic integral of the first kind and 0 < t < 1 is a number such that

\frac{K(\sqrt{1 − t^{2}})}{K(t)} = \frac{2πβ}{log(R/r)}.

Note that the quantity μ(t) = (π/2)K(\sqrt{1 − t^{2}})/K(t) is the modulus of the Grötzsch ring 𝔻 \ [0, t] for 0 < t < 1 and decreasing from +∞ to 0 (see, for example, [1]). Therefore, we can always take such a t satisfying the above relation.

We remark that the essentially same observations for the last example were made by Avhadiev and Aksent’ev [2] though they did not make systematic use of circular width.

4. Applications

In this section, we give a few applications of circular width. More concrete applications can be found in [9] and [11].

Let us introduce some notation. For a locally univalent function f on 𝔻, the quantity T_{f} = f′′/f′ is called the pre-Schwarzian derivative of f and measured by the norm

∥T_{f}∥_{𝔻} = sup_{z∈𝔻}(1 − |z|^{2})|T_{f}(z)|.

Note that this can be described by ∥T_{f}∥_{𝔻} = V_{𝔻}(f′). Let A denote the class of holomorphic functions f on 𝔻 normalized by f(0) = 0, f′(0) = 1.

Theorem 9. Let Ω be a proper subdomain of the punctured plane ℂ^{∗} with W (Ω) < 2. If f ∈ A satisfies f′(𝔻) ⊂ Ω, then |f(z)| < M, z ∈ 𝔻. Here M is a constant depending only on W (Ω).

The constant M can be given by

M = λ[ψ( − \frac{λ}{2}) − ψ(\frac{1−λ}{2})] − 1,

where λ = W (Ω)/2 and ψ(x) = Γ′(x)/Γ(x) is the digamma function (see [7]).

It may be interesting to find a characterization of such subdomains Ω of ℍ that f′(𝔻) ⊂ Ω implies boundedness of f ∈ A. Note that the condition f′(𝔻) ⊂ ℍ implies univalence of f (Noshiro-Warschawski theorem). Recently, Chuaqui and Gevirtz [6] gave a characterization of such subdomains Ω of ℍ that f′(𝔻) ⊂ Ω implies quasiconformal extensibility of f ∈ A.

We next apply Theorem 7 to the problem of quasiconformal extensibility. Our result is based the following theorem due to J. Becker. See, for sharpness, Becker and Pommerenke [5].

Theorem 10 (Becker [4]). Let f ∈ A be locally univalent. If ∥T_{f}∥_{𝔻} ≤ 1, then f is univalent. Furthermore, if ∥T_{f}∥ ≤ k for k ∈ [0, 1), then f has a K-quasiconformal extension to the whole plane, where K = (1 + k)/(1 − k).

Making use of this result, we can show the following.

Theorem 11. Suppose that a proper subdomain Ω of the punctured plane ℂ^{∗} satisfies W (Ω) ≤ k for some k ≤ 1. If f′(𝔻) ⊂ Ω for f ∈ A, then f is univalent and, moreover, f has a K-quasiconformal extension to the whole plane when K = (1 + k)/(1 − k) < ∞.

See [11] for a counterpart of the theorem for meromorphic functions.

Proof. As we noted, the condition f′(𝔻) ⊂ Ω implies that ∥T_{f}∥_{𝔻} ≤ W (Ω) ≤ k. We now apply Theorem 10 to deduce the assertions.

Combining this with examples presented in the previous section, we obtain a series of corollaries. (Remember the fact that circular width is invariant under rotations.) The first corollary was noted by Avhadiev and Aksent’ev [3, pp. 33–34] at least when γ = 0.

Corollary 12. Let 0 < k ≤ 1 and f ∈ A. If |argf′(z) − γ| < πk/4 in |z| < 1 for some real constant γ, then f is univalent and, moreover, it extends to a K-quasiconformal mapping of the whole plane when K = (1 + k)/(1 − k) < ∞.

Note also that the condition |argf′(z)| < M, |z| < 1, implies quasiconformal extensibility of f when M < π/2 (see [6]).

Corollary 13. Let k, r, R be positive numbers with 0 < log(R/r) ≤ πk/2 and f ∈ A. If r < |f′(z)| < M in |z| < 1, then f is univalent and, moreover, it extends to a K-quasiconformal mapping of the whole plane when K = (1 + k)/(1 − k) < ∞.

We omit the other examples.

References

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3. F. G. Avkhadiev and L. A. Aksent’ev, The main results of sufficient conditions for an analytic function to be schlicht, Russian Math. Surveys 30 (1975), 1–64.

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6. M. Chuaqui and J. Gevirtz, Quasidisks and the Noshiro-Warschawski criterion, Complex Variables 48 (2003), 967–985.

7. Y. C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mountain J. Math. 32 (2002), 179–200.

8. , A conformal invariant for non-vanishing analytic functions and its applications, Preprint (2004).

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10. Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. 32 (1979), 192–199.

11. S. Ponnusamy and T. Sugawa, Norm estimates and univalence criteria for meromorphic functions, Preprint (2004).

12. T. Sugawa, Various domain constants related to uniform perfectness, Complex Variables Theory Appl. 36 (1998), 311–345.

13. J.-H. Zheng, Uniformly perfect sets and distortion of holomorphic functions, Nagoya Math J. 164 (2001), 17–34.

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

  E-mail address: sugawa@math.sci.hiroshima-u.ac.jp