INNER RADIUS OF UNIVALENCE FOR A STRONGLY STARLIKE DOMAIN

TOSHIYUKI SUGAWA

Abstract. The inner radius of univalence of a domain D with Poincaré density ρ_{D} is the possible largest number σ such that the condition ∥S_{f}∥_{D} = sup_{w∈D} ρ_{D}(w)^{−2}|S_{f}(z)| ≤ σ implies the univalence of f for a nonconstant meromorphic function f on D, where S_{f} is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α.

1. Main result

For a constant 0 ≤ α ≤ 1, a holomorphic function f on the unit disk is called strongly starlike of order α if f′(0) ≠ 0 and if f satisfies the condition

|arg \frac{zf′(z)}{f(z) − f(0)}| ≤ \frac{πα}{2} (z ∈ 𝔻^{∗} = 𝔻 \ \{0\}). (1)

Note that a strongly starlike function f is starlike in the usual sense, namely, f is univalent and the image f(𝔻) is starlike with respect to f(0). Every strongly starlike function f of order α < 1 is bounded. In fact, Brannan and Kirwan [1] showed that

|f(z) − f(0)| ≤ |zf′(0)|M(α) (z ∈ 𝔻). (2)

Here M(α) is defined by

M(α) = exp [∫_{0}^{1} \{(\frac{1 + t}{1 − t})^{α} − 1\} \frac{dt}{t}]

= exp \{2α ∑_{k=0}^{∞} \frac{1}{(2k + 1)(2k + 1 − α)}\} (3)

= \frac{1}{4} exp \{−\frac{Γ′((1 − α)/2)}{Γ((1 − α)/2)} − γ\} ,

where Γ is the Euler gamma function and γ = 0.5772 … is the Euler constant.

A proper subdomain D of the complex plane ℂ is said to be strongly starlike of order α with respect to a point w_{0} ∈ D if D is simply connected and if the conformal map f : 𝔻 → D of D with f(0) = w_{0} is strongly starlike of order α. A strongly starlike domain of order 1 is nothing but a usual starlike domain. In what follows, without any pain, we always assume that w_{0} = 0.

Date: November 22, 2001.

1991 Mathematics Subject Classification. Primary 30C45; Secondary 30C62.

Key words and phrases. strongly starlike, Poincaré metric, inner radius of univalence.

The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS .

We now introduce a standard domain adapted to the strong starlikeness. For a constant α with 0 < α < 1, we denote by V_{α} the bounded domain enclosed by the logarithmic spirals γ_{α} = {exp((− tan(πα/2) + i)θ); 0 ≤ θ ≤ π} and γ̄_{α} = {w; w̄ ∈ γ_{α}}.

Let D be a proper subdomain of ℂ containing the origin. It will be convenient to consider the periodic function R = R_{D} : ℝ → (0, +∞] of period 2π defined by

R(θ) = sup{r > 0; [0, re^{iθ}] ⊂ D},

where [a, b] denotes the closed line segment joining points a and b in ℂ. Note that R is lower semi-continuous.

In the sequel, we will use the convention a · D = {aw; w ∈ D} for a ∈ ℂ and a domain D. Also, set D^{∨} = I(Ext D), where Ext D = ℂ̂ \ \overline{D} and I(z) = 1/z.

The next result will be fundamental for our aim here, whose proof can be found in [7].

Theorem A. Let D be a proper subdomain of ℂ with 0 ∈ D and let α be a constant with 0 < α < 1. Then the following conditions are equivalent.

(a) D is strongly starlike of order α with respect to the origin.

(b) D^{∨} is strongly starlike of order α with respect to the origin.

(c) w · V_{α} ⊂ D holds for each point w ∈ \overline{D}.

(d) The radius function R = R_{D} is bounded, absolutely continuous, and satisfies |R′/R| ≤ tan(πα/2) a.e. in ℝ.

Remark. The implication (a)⇒(d) is essentially due to Fait, Krzyż and Zygmunt [2]. Actually, we will employ their idea which was used to show the quasiconformal extendability of strongly starlike functions.

Let D be a subdomain of ℂ with the Poincaré (or hyperbolic) metric ρ_{D}(w)|dw| of constant curvature −4. The inner radius of univalence of D, which will be denoted by σ(D), is the possible maximal number σ for which the condition ∥S_{f}∥_{D} ≤ σ implies the univalence of the nonconstant meromorphic function f on D, where S_{f} denotes the Schwarzian derivative (f′′/f′)′ − (f′′/f′)^{2}/2 of f and ∥φ∥_{D} = sup_{w∈D} ρ_{D}(w)^{−2}|φ(w)|. Note that σ(D) is Möbius invariant in the sense that σ(L(D)) = σ(D) for a Möbius transformation L. In particular, σ(D^{∨}) = σ(Ext D). The reader may consult the textbook [4] by Lehto as a general reference for the inner radius of univalence and related notions. When D is simply connected, theorems of Ahlfors and Gehring imply that σ(D) > 0 if and only if D is a quasidisk and, furthermore, σ(D) is estimated from below by a positive constant c(K) depending only on K for a K-quasidisk D. However, it is hard to give an explicit lower bound of σ(D) for a concrete quasidisk D in general. Our main result is concerned with the inner radius of univalence of strongly starlike domains.

Theorem 1. A strongly starlike domain D of order α satisfies

σ(D) ≥ \frac{2}{M(α)^{2}} · \frac{cos(πα/2)}{1 + sin(πα/2)}, (4)

where M(α) is defined by (3).

Remarks. 1. When α tends to 0, the right-hand side above tends to 2. On the other hand, it is known that σ(𝔻) = 2. See also the final section.

2. By a result of Fait, Krzyż and Zygmunt [2], we know that a strongly starlike domain D of order α is a K(α)-quasidisk, where (K(α) − 1)/(K(α) + 1) = sin(πα/2). Hence, as a corollary, we have σ(D) ≥ c(K(α)). So, the novelty of this theorem lies in the explicitness of the estimate.

3. From Theorem 1 we observe that D^{∨} is strongly starlike of order α under the assumption of Theorem 1. Hence, we obtain σ(Ext D) ≥ 2 cos(πα/2)/(1+sin(πα/2))M(α)^{2} simultaneously. We also note that the standard domain V_{α} has the property σ(V_{α}) = σ(Ext V_{α}).

2. Mapping function of V_{α}

Let S denote the set of holomorphic univalent functions on the unit disk 𝔻 normalized by f(0) = 0 and f′(0) = 1. For 0 ≤ α ≤ 1, we define the function k_{α} in the class S by the relation

\frac{zk′_{α}(z)}{k_{α}(z)} = (\frac{1 + z}{1 − z} )^{α}

on 𝔻. More explicitly, k_{α} can be expressed by

k_{α}(z) = z exp [∫_{0}^{z} \{(\frac{1 + ζ}{1 − ζ} )^{α} − 1\} \frac{dζ}{ζ} ] .

This function is known to play a role of the usual Koebe function in the class of normalized strongly starlike functions of order α in many cases. Actually k_{1} is nothing but the Koebe function.

Noting k_{α}(1) = M(α), we consider the function

g_{α}(z) = k_{α}(z)/M(α) = exp [∫_{1}^{z} (\frac{1 + ζ}{1 − ζ} )^{α} \frac{dζ}{ζ} ] . (5)

The following fact is useful to note. Although this result was stated in [7], we give a direct proof here for completeness.

Lemma 1. g_{α}(𝔻) = V_{α} for 0 < α < 1.

Proof. If we set g_{α}(e^{it}) = r(t)e^{iΘ(t)} = R(θ)e^{iθ}, then we have e^{it}g′_{α}(e^{it})/g_{α}(e^{it}) = Θ′(t) − ir′(t)/r(t). Since arg(zg′_{α}(z)/g_{α}(z)) = πα/2 for z = e^{it} with t ∈ (0, π), we obtain

\frac{R′(θ)}{R(θ)} = \frac{r′(t)}{r(t)Θ′(t)} = − tan \frac{πα}{2},

which yields log R(θ) = −θ tan(πα/2) for θ = Θ(t) ∈ (0, π). In the same way, we have log R(−θ) = −θ tan(πα/2) for θ ∈ (0, π). These imply that the radius function R of g_{α}(𝔻) agrees with that of V_{α}, and hence g_{α}(𝔻) = V_{α}.

3. Proof of Main Theorem

First, we recall the construction of a quasiconformal reflection in the boundary of a strongly starlike domain given by [2]. Let D be a strongly starlike domain of order α ∈ (0, 1) with respect to the origin and let R be its radius function. Then we can take the quasiconformal reflection λ in ∂D defined by

λ(re^{iθ}) = \frac{R(θ)^{2}}{r} e^{iθ}

for all r > 0 and θ ∈ ℝ. We then calculate

∂λ = i\frac{RR′}{r^{2}} and ¯∂λ = \frac{e^{2iθ}}{r^{2}} (iRR′ − R^{2}) (6)

at w = re^{iθ}.

Now we use a general estimate on the radius of univalence in terms of quasiconformal reflections to prove our main result. The following estimate is first shown by Lehto [3] in a restricted situation and is generalized to the present form by [6].

Theorem B. Let D be a quasidisk with quasiconformal reflection λ in ∂D. Then the following inequality holds:

σ(D) ≥ ε(λ, D) := 2 ess. inf_{w∈D} \frac{|¯∂λ(w)| − |∂λ(w)|}{|λ(w) − w|^{2}ρ_{D}(w)^{2}} . (7)

Let us return to our case. By (6) and Theorem A (d), we obtain the estimates

|¯∂λ(w)| − |∂λ(w)| = \frac{R^{2}}{r^{2}} (\sqrt{1 + |R′/R|^{2}} − |R′/R|)

≥ \frac{R^{2}}{r^{2}} (\sqrt{1 + tan^{2}(πα/2)} − tan(πα/2))

= \frac{R^{2}}{r^{2}} · \frac{cos(πα/2)}{1 + sin(πα/2)}

and

|λ(w) − w| = \frac{R^{2}}{r} − r = \frac{R^{2} − r^{2}}{r}

for almost all w = re^{iθ} ∈ D.

Secondly, we estimate ρ_{D} from above. Fix w = re^{iθ} and set R = R(θ). If we think of the domain W = w_{0} · V_{α}, where w_{0} = Re^{iθ} ∈ ∂D, from Theorem A (c), we have W ⊂ D. The monotoneity property of the Poincaré metric then implies ρ_{D}(w) ≤ ρ_{W} (w). Now we write τ_{α} = ρ_{V_{α}}. Then ρ_{W} (w) = τ_{α}(w/w_{0})/|w_{0}| = τ_{α}(r/R)/R. Consequently, we have ρ_{D}(w) ≤ τ_{α}(r/R)/R.

Summarizing the above, we have the estimate

\frac{|¯∂λ(w)| − |∂λ(w)|}{|λ(w) − w|^{2}ρ_{D}(w)^{2}} ≥ \frac{R^{2}}{r^{2}} · (\frac{r}{R^{2} − r^{2}} )^{2} · \frac{R^{2}}{τ_{α}(r/R)^{2}} · \frac{cos(πα/2)}{1 + sin(πα/2)}

= \frac{1}{(1 − (r/R)^{2})^{2}τ_{α}(r/R)^{2}} · \frac{cos(πα/2)}{1 + sin(πα/2)}.

Hence,

ε(λ, D) ≥ \frac{2}{sup_{0<u<1}(1 − u^{2})^{2}τ_{α}(u)^{2}} · \frac{cos(πα/2)}{1 + sin(πα/2)}.

Therefore, if we can show the following lemma, the proof of our main theorem will be finished.

Lemma 2. The Poincaré density τ_{α} of V_{α} satisfies

sup_{0<u<1}(1 − u^{2})τ_{α}(u) = M(α).

Proof. Since g_{α} : 𝔻 → V_{α} is biholomorphic by Lemma 1, we have (1 − |z|^{2})^{−1} = τ_{α}(g_{α}(z))|g′_{α}(z)| for z ∈ 𝔻. Note here that u = u(x) = g_{α}(x) > 0 and g′_{α}(x) > 0 for positive x. If we set

Q(x) = (1 − u(x)^{2})τ_{α}(u(x)) = \frac{1 − u(x)^{2}}{(1 − x^{2})u′(x)}

for x ∈ (0, 1), we have only to show that Q is non-increasing in the interval (0, 1) because lim_{x→0} Q(x) = τ_{α}(0) = 1/|g′_{α}(0)| = M(α).

Since xu′/u = {(1 + x)/(1 − x)}^{α}, we have the expression

Q = \frac{1 − u^{2}}{1 − x^{2}} · \frac{x}{u} · (\frac{1 − x}{1 + x})^{α} = \frac{x}{u} · \frac{1 − u^{2}}{(1 + x)^{1+α}(1 − x)^{1−α}} .

Taking the logarithmic derivative, we obtain

x\frac{Q′}{Q} = 1 − \frac{xu′}{u} · \frac{1 + u^{2}}{1 − u^{2}} + \frac{2x(x − α)}{1 − x^{2}} = \frac{1 − 2αx + x^{2}}{1 − x^{2}} − \frac{1 + u^{2}}{1 − u^{2}} (\frac{1 + x}{1 − x})^{α} .

Therefore, Q′ ≤ 0 if and only if

\frac{1 − u^{2}}{1 + u^{2}} ≤ (\frac{1 + x}{1 − x})^{α} · \frac{1 − x^{2}}{1 − 2αx + x^{2}} = \frac{(1 + x)^{1+α}(1 − x)^{1−α}}{1 − 2αx + x^{2}} =: P (x).

By representation (5), we see

\frac{1 − u^{2}}{1 + u^{2}} = tanh [− ∫_{1}^{x} (\frac{1 + t}{1 − t})^{α} \frac{dt}{t}] .

Hence, the assertion Q′ ≥ 0 on the interval (0, 1) is further equivalent to the validity of the statement that

∫_{x}^{1} (\frac{1 + t}{1 − t})^{α} \frac{dt}{t} ≤ arctanhP (x)

holds whenever P (x) < 1.

We now investigate the behaviour of the function P on (0, 1). Since

\frac{P′(x)}{P (x)} = \frac{4(x − α)(αx − 1)}{(1 − x^{2})(1 − 2αx + x^{2})},

P is increasing in (0, α) and decreasing in (α, 1). Noting P (0) = 1 and P (1) = 0, we observe that P (x) > 1 for x ∈ (0, β) and that 0 < P (x) < 1 for x ∈ (β, 1) for some number β between α and 1. Here, we use the following elementary fact.

Lemma 3. Let S and T be continuous functions on the interval (β, 1] that are positive, have continuous integrable derivatives on (β, 1) and satisfy S(1) = T (1) = 0 and S′(x) ≤ T′(x) for x ∈ (β, 1). Then S(x) ≥ T (x) for x ∈ (β, 1].

Thus it is enough to show the inequality

−\frac{1}{x} (\frac{1 + x}{1 − x})^{α} ≥ \frac{d}{dx}arctanhP (x) = \frac{P′(x)}{1 − P (x)^{2}}

= \frac{−4(x − α)(1 − αx)}{(1 − 2αx + x^{2})^{2} − (1 + x)^{2+2α}(1 − x)^{2−2α}} (\frac{1 + x}{1 − x})^{α}

for x ∈ (β, 1). This inequality is equivalent to

(1 − 2αx + x^{2})^{2} − (1 + x)^{2+2α}(1 − x)^{2−2α} ≤ 4x(x − α)(1 − αx)
⇔ (1 + x)^{2+2α}(1 − x)^{2−2α} ≥ (1 − 2αx + x^{2})^{2} − 4x(x − α)(1 − αx) = (1 − x^{2})^{2}
⇔ (\frac{1 + x}{1 − x})^{2α} ≥ 1.

The last inequality is certainly valid for x ∈ (0, 1). So, now the proof is complete.

Remark. We can see from the proof that ε(λ, V_{α}) = 2 cos(πα/2)/(1+sin(πα/2))M(α)^{2} holds, where λ is the quasiconformal reflection constructed for V_{α} as above.

4. Upper estimate of the radius of univalence

Set ξ(α) = 2 cos(πα/2)/(1+sin(πα/2))M(α)^{2} and let η(α) be the possible largest value for which η(α) ≤ σ(D) holds for any strongly starlike domain D of order α. Theorem 1 says that ξ(α) ≤ η(α) for 0 < α < 1. In this section, we give an upper bound for η(α) in order to examine how close the bound ξ(α) is to η(α). To this end, we give a rough upper estimate of σ(V_{α}).

Theorem 2. For 0 < α < 1, the inequality σ(V_{α}) ≤ 2(1 − α)^{2} holds.

Proof. We consider the holomorphic function f(w) = log(1−w) on the domain ℂ\[1, +∞). Although f is univalent, f(V_{α}) has an outward pointing cusp. So, f(V_{α}) is not a quasidisk. On the other hand, for a quasidisk D, if ∥S_{f}∥_{D} < σ(D), we know that f(D) is also a quasidisk (see [4, p. 120]). Hence, we conclude σ(V_{α}) ≤ ∥S_{f}∥_{V_{α}} for the above f.

Now we estimate ∥S_{f}∥_{V_{α}}. First note that V_{α} ⊂ W := {w; | arg(1 − w)| < (1 − α)π/2}. By the monotoneity of the Poincaré metric, we have ∥S_{f}∥_{V_{α}} ≤ ∥S_{f}∥_{W} . Since (1−w)^{1/(1−α)} maps W conformally onto the right half plane, we compute

ρ_{W} (w) = \frac{|1 − w|^{α/(1−α)}}{2(1 − α)Re [(1 − w)^{1/(1−α)}]}.

On the other hand, S_{f}(w) = 1/2(1 − w)^{2}. Thus, we calculate

∥S_{f}∥_{W} = sup_{w∈W} 2(1 − α)^{2}Re [\frac{(1 − w)^{1/(1−α)}}{|1 − w|^{1/(1−α)}} ] = 2(1 − α)^{2}.

Now the proof is completed.

Remark. Since V_{α} has a corner of opening (1 − α)π, the above estimate follows also from Lemma 3.3 in the paper [5] by Miller-Van Wieren.

Note that ξ(α) ≤ η(α) ≤ σ(V_{α}) ≤ 2(1 − α)^{2}. We exhibit the graphs of the functions ξ(α) and 2(1 − α)^{2} below.

References

1. D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431–443.

2. M. Fait, J. G. Krzyż, and J. Zygmunt, Explicit quasiconformal extensions for some classes of univalent functions, Comment. Math. Helv. 51 (1976), 279–285.

3. O. Lehto, Remarks on Nehari’s theorem about the Schwarzian derivative and schlicht functions, J. Analyse Math. 36 (1979), 184–190.

4. , Univalent Functions and Teichmüller Spaces, Springer-Verlag, 1987.

5. L. Miller-Van Wieren, On Nehari disks and the inner radius, Comment. Math. Helv. 76 (2001), 183– 199.

6. T. Sugawa, A remark on the Ahlfors-Lehto univalence criterion, to appear in Ann. Acad. Sci. Fenn. A I Math.

7. , A self-duality of strong starlikeness, preprint.

Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

Current address: Department of Mathematics, University of Helsinki, Yliopistonkatu 5, 00014, Helsinki, Finland

  E-mail address: sugawa@kusm.kyoto-u.ac.jp