A conformal invariant for

non-vanishing analytic functions

Toshiyuki Sugawa

Department of Mathematics, Graduate School of Science,

Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

sugawa@math.sci.hiroshima-u.ac.jp
http://sugawa@cajpn.org/

Joint work with Yong Chan Kim

August 9, 2004, Yeungnam Univ., Korea Hyperbolic metric.

A plane domain D ⊂ ℂ is called hyperbolic if ∃ an analytic universal covering projection p : 𝔻 → D. Here 𝔻 = {z ∈ ℂ : |z| < 1}. The hyperbolic metric ρ_{D}(w)|dw| of D is determined by the equation

\frac{1}{|p′(z)|(1 − |z|^{2})} = ρ_{D}(w), w = p(z).

The hyperbolic metric is conformally invariant in the sense that the pull-back

f^{∗}ρ_{D′}(z) := ρ_{D′}(f(z))|f′(z)|

of ρ_{D′}(w)|dw| under a conformal map f : D → D′ is equal to ρ_{D}(z).

The quantity V_{D}(φ).

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 that V_{D}(φ) can be thought of the Bloch semi-norm of the (possibly multi-valued) function log φ.

Subordination.

A holomorphic function f on 𝔻 is said to be subordinate to another holomorphic function g if there exists a holormorphic function ω : 𝔻 → 𝔻 such that f = g ∘ ω and ω(0) = 0. When we do not require the condition ω(0) = 0 the function f is said to be weakly subordinate to g.

When using these notions, the Schwarz-Pick lemma:

\frac{|ω′(z)|}{1 − |ω(z)|^{2}} ≤ \frac{1}{1 − |z|^{2}} , ω : 𝔻 → 𝔻,

plays an important role. For instance, this leads to the inequality

f^{∗}ρ_{D′}(z) ≤ ρ_{D}(z)

for a holomorphic map f : D → D′, where equality holds at some (and thus every) point z iff f : D → D′ is a(n unbranched) covering.

Basic properties of V_{D}(φ).

(1) V_{D}(φ · ψ) ≤ V_{D}(φ) + V_{D}(ψ).

(2) V_{D}(φ^{α}) = |α|V_{D}(φ) for α ∈ ℂ as long as the power φ^{α} is defined as a single-valued analytic function on D.

(3) V_{D_{0}}(φ∘p) = V_{D}(φ) for analytic (unbranched) covering p : D_{0} → D.

(4) 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 ∀c ∈ ℂ^{∗}.

(5) V_{𝔻}(φ) ≤ V_{𝔻}(ψ) if φ is weakly subordinate to ψ.

(6) V_{D}(φ) ≤ V_{D}(ψ) if ψ : D → ℂ^{∗} is univalent and if φ(D) ⊂ ψ(D).

Circular width.

Let Ω be a proper subdomain of ℂ^{∗}. Then Ω admits an analytic universal covering projection p of a simply connected proper subdomain D of ℂ onto it. Then the quantity W (Ω) = V_{D}(p) is independent of the particular choice of p : D → Ω and will be called the circular width of Ω (around the origin). Usually, the domain D is chosen to be the unit disk 𝔻 but sometimes another domain is more appropriate to compute the value of W (Ω).

When Ω is simply connected, we can choose the identity map as p, and thus obtain the relation

\frac{1}{W (Ω)} = inf_{w∈Ω} |w|ρ_{Ω} (w).

Basic properties of W (Ω).

(1) W (L(Ω)) = W (Ω) for any conformal automorphism L of ℂ^{∗}.

(2) W (Ω) ≤ 4 for a simply connected domain Ω ⊂ ℂ^{∗}.

(3) W (Ω) ≤ W (Ω_{1}) if Ω ⊂ Ω_{1} ⊂ ℂ^{∗}.

(4) Let Ω^{∗} denote the circular symmetrization of a domain Ω ⊂ ℂ^{∗}. Then W (Ω) ≤ W (Ω^{∗}).

Applications.

Theorem 1 (Y. C. Kim -S.) Let Ω be a proper subdomain of ℂ^{∗}. Suppose that a holomorphic function f on 𝔻 satisfies f′(z) ∈ Ω for any z ∈ 𝔻. If W (Ω) ≤ 1, then f is univalent. Moreover, if W (Ω) ≤ k, for some 0 ≤ k < 1, then f extends to a k-quasiconformal map of the Riemann sphere.

Theorem 2 (S. Ponnusamy -S.) Let Ω be a proper subdomain of ℂ^{∗}. Suppose that a meromorphic function

F (z) = z + b_{0} + b_{1}/z + b_{2}/z^{2} + ⋯

on |z| > 1 satisfies F′(z) ∈ Ω. If W (Ω) ≤ 1/2, then F is univalent. Moreover, if W (Ω) ≤ k/2, for some 0 ≤ k < 1, then F extends to a kquasiconformal map of the Riemann sphere.

Some computations.

(1) Sector. Ω = {w ∈ ℂ : |arg w| < πα/2}

⇒ W (Ω) = 2α.

(2) Annulus. Ω = {w : r < |w| < R}, 0 ≤ r < R ≤ ∞

⇒ W (Ω) = \frac{2}{π} log \frac{R}{r}.

(3) Parallel strip. Ω = {w : a < Re w < b}, 0 < a < b < ∞

⇒ W (Ω) = max_{0≤θ≤π/2} \frac{k cos θ}{1 − kθ/2} ,

where k is a positive number such that

\frac{b}{a} = \frac{4 + kπ}{4 − kπ} .

(4) Disk. Ω = {w : |w − a| < r}, 0 < r ≤ a

⇒ W (Ω) = \frac{2r/a}{1 + √1 − (r/a)^{2}} .

(5) Wedge. Ω = {w : r < |w| < R, |arg w| < πα/2}

⇒ W (Ω) = \frac{log (R/r)}{(1 + t)K(t)} ,

where

K(t) = ∫_{0}^{1} \frac{dx}{√(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(√1 − t^{2} )}{K(t)} = \frac{2πα}{log (R/r)} .