Topics around strongly starlike functions

Toshiyuki Sugawa

Department of Mathematics, Graduate School of Science,

Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

sugawa@math.sci.hiroshima-u.ac.jp

May 12, 2003, Osaka City University Strongly starlike functions. A non-constant holomorphic function f on the unit disk 𝔻 = {z ∈ ℂ : |z| < 1} is said to be strongly starlike of order α, if

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

where α ≥ 0 is a constant (J. Stankiewicz, Ann. Univ. Mariae Curie-Skłodowska 1966 and D. A. Brannan and W. E. Kirwan, J. London Math. Soc. 1969). In particular, if 0 ≤ α ≤ 1, the function (f − f(0))/f′(0) is a (normalized univalent) starlike function.

Uniform boundedness. Brannan and Kirwan (J. London Math. 1969) proved that a strongly starlike function f of order α < 1 satisfies the sharp inequality

|f(z) − f(0)| ≤ |zf′(0)|M(α), z ∈ 𝔻.

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)}\}

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

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

Figure 1: The graph of M(α)

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Coefficient estimate. Let f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ be a strongly starlike function of order α ∈ [0, 1). Then it is shown that the order estimate

a_{n} = o (\frac{1}{n}) as n → ∞

is derived and it is sharp (Brannan and Kirwan 1969). Though the sharp estimates

|a_{2}| ≤ 2α and |a_{3}| ≤ max{α, 3α^{2}}

are known (Brannan, Clunie and Kirwan; Canad. J. Math. 1970), general sharp estimates seem to be unknown.

Extremal functions. The function k_{α}(z) determined by the relation

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

and by the normalization k_{α}(0) = 0, k′_{α}(0) = 1 often plays the role of the Koebe function. Let g_{α} be the function k_{α}/M(α) normalized so that g_{α}(1) = 1. The image V_{α} = g_{α}(𝔻) of g_{α} is described as 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}}.

Figure 2: The shape of V_{α} when α = 1/2

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Strongly starlike domains. A domain D in the complex plane is said to be strongly starlike of order α with respect to w_{0} ∈ D if a conformal mapping f : 𝔻 → D of 𝔻 onto D with f(0) = w_{0} is stronlyg starlike of order α. When w_{0} = 0, the domain is simply called strongly starlike of order α. Let U_{α} be the Jordan domain bounded by two congruent circular arcs forming the angle π(1 − α) at the intersection points 0 and 1. For a subset E of ℂ̂ and a point w ∈ ℂ, we write E^{−1} = {1/z : z ∈ E} and w · E = {wz : z ∈ E}. Also set E^{∨} = Ext E^{−1}.

Figure 3: U_{α} ⊂ V_{α} ⊂ Ext V_{α}^{−1} ⊂ Ext U_{α}^{−1}

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Radius function. Let D be a proper subdomain of ℂ containing the origin. We define the periodic function R = R_{D} : ℝ → (0, +∞] by

R(θ) = sup{r > 0; [0, re^{iθ}] ⊂ D}.

Notice that R is lower semi-continuous and reproduces the original domain by the relation D = {re^{iθ}; r < R(θ)} if D is starlike with respect to the origin.

Lemma 1 If R_{D} is continuous, then D is starlike with respect to the origin. If R_{D} is continuous and finite, i.e., R_{D}(ℝ) ⊂ (0, +∞), then D is a bounded starlike Jordan domain and valid is the self-duality relation:

R_{D^{∨}}(θ) = 1/R_{D}(−θ) (θ ∈ ℝ).

Characterizations of strongly starlike domains. Let D be a proper subdomain of ℂ with 0 ∈ D and α be a constant with 0 < α < 1. Then the following seven conditions are equivalent.

(i) D is strongly starlike of order α. (ii) D^{∨} = Ext D^{−1} is strongly starlike of order α. (iii) For each point w ∈ D we have w · U_{α} ⊂ D. (iv) For each point w ∈ ℂ \ \overline{D} we have w · U_{α}^{−1} ⊂ ℂ \ \overline{D}.

(v) For each point w ∈ D we have w · V_{α} ⊂ D. (vi) For each point w ∈ ℂ̂ \ \overline{D} we have w · V_{α}^{−1} ⊂ ℂ̂ \ \overline{D}. (vii) The function R = R_{D} is absolutely continuous and satisfies

|R′/R| ≤ tan(πα/2) a.e. in ℝ.

Remarks. The equivalence of conditions (i), (iv) and (vi) was proved by Stankiewicz 1966. The equivalence of conditions (i) and (iii) is due to Ma and Minda (Ann. Univ. Mariae Curie-Skłodowska 1991). The implication (i)⇒(vii) was essentially used by Fait, Krzyż and Zygmunt (Comment. Math. Helv. 1976) though their proof seems to have some gap.

Explicit QC extension of strongly starlike functions. 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 K(α)-quasiconformal reflection λ in ∂D defined by

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

for r > 0 and θ ∈ ℝ, where K(α) = (1 + sin(πα/2))/(1 − sin(πα/2)) (Fait, Krzyż and Zygmunt 1976). By a standard method, we then can give an explicit K(α)-quasiconformal extension of a strongly starlike function. We note the relations

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

at w = re^{iθ}.

Inner radius of univalence. The inner radius of univalence σ(D) of a hyperbolic domain D with the hyperbolic metric ρ_{D}(w)|dw| 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 univalence of f for a nonconstant meromorphic function f on D, where S_{f} is the Schwarzian derivative of f.0

Theorem 2 (S.) A strongly starlike domain D of order α satisfies

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

On the other hand, for 0 < α < 1, the inequality σ(V_{α}) ≤ 2(1 − α)^{2} holds.

Figure 4: Graphs of the lower and upper bounds

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Key ingredients of the proof. The proof uses the characterization of a strongly starlike domain in terms of V_{α}, and the following estimate.

Theorem 3 (Lehto, J. Analyse Math. 1979, S., Fennica 2002) Let D be a quasidisk with a quasiconformal reflection λ in ∂D. Then

σ(D) ≥ 2 ess. inf_{z∈D} \frac{|¯∂λ(z)| − |∂λ(z)|}{|λ(z) − z|^{2}ρ_{D}(z)^{2}} .

Norm estimate of pre-Schwarzian.

Theorem 4 (S., Ann. Univ. Mariae Curie-Skłodowska 1998) For a strongly starlike function f of order 0 < α < 1, the inequality

∥T_{f}∥_{1} = sup_{z∈Δ}(1 − |z|^{2}) |\frac{f′′(z)}{f′(z)}| ≤ N(α) + 2α

holds, where T_{f} = f′′/f′, N(α) is given by

N(α) = \frac{4αc(α)}{(1 − α)c(α)^{2} + 1 + α} = \frac{4αc(α)^{α+1}}{c(α)^{2} + 1} ,

and c(α) is the unique solution of the following equation with respect to x in the interval (1, ∞) :

(1 − α)x^{α+2} + (1 + α)x^{α} − x^{2} − 1 = 0.

Moreover, equality holds above precisely when f is a rotation of k_{α}.

Remark. A lengthy computation yields

\frac{2α(1 + α)^{1+α}(1 − α)^{1−α}}{1 + α^{2}} < N(α)

< 2α(1 − α/2)^{α}(1 + α)^{1+α}(1 − α)^{1−α}

in particular,

N(α) < 2α(1 + α).

Norm estimate of Schwarzian. In view of the result of Fait, Krzyż and Zygmunt, we can easily deduce the inequality

∥S_{f}∥_{2} = sup_{z∈𝔻}(1 − |z|^{2})^{2}|S_{f}(z)| ≤ 6 sin \frac{πα}{2}

for a strongly starlike function f of order α ∈ (0, 1). On the other hand, it is shown that |S_{f}(0)| ≤ 6α for such a function f. Therefore, it may be natural to ask whether the stronger inequality ∥S_{f}∥_{2} ≤ 6α holds or not. Recently, Y.-L. Shen has answered to the above question negatively.