Univalent Functions in the Möbius Invariant 𝑄_𝐾 Space

Abstract

It is shown that a univalent function 𝑓 belongs to 𝑄_𝐾 if and only if sup_𝑎∈𝔻∫¹₀𝑀²_∞(𝑟,𝑓∘𝜑_𝑎−𝑓(𝑎))𝐾′(log(1/𝑟))𝑑𝑟<∞, where 𝜑_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧), provided 𝐾 satisfies certain regularity conditions. It is also shown that under these conditions 𝑄_𝐾 contains all univalent Bloch functions if and only if ∫¹₀(log((1+𝑟)/(1−𝑟)))²𝐾′(log(1/𝑟))𝑑𝑟<∞.

1. Introduction

The aim of this paper is to characterize univalent functions 𝑓 in the Möbius invariant 𝑄_𝐾 space in terms of the maximum modulus of 𝑓∘𝜑_𝑎−𝑓(𝑎), where 𝜑_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧), provided 𝐾 satisfies certain regularity conditions. We will begin with a brief overview of characterizations of univalent functions in classical function spaces of the unit disc together with necessary definitions. Then, we will state the above-mentioned characterization of univalent functions in 𝑄_𝐾 and its consequences. The proofs are postponed to the end of the paper and will be presented in the sequential order.

2. Notation, Background, and Results

Let ℋ(𝔻) denote the algebra of all analytic functions in the unit disc 𝔻∶={𝑧∶|𝑧|<1}. A function 𝑓∈ℋ(𝔻) is said to be univalent if it is one to one, and the class of all univalent functions is denoted by 𝒰. For the theory of univalent functions, see [1–3].

For 0<𝑝≤∞, the Hardy space 𝐻^𝑝 consists of those 𝑓∈ℋ(𝔻) for which 𝑓_𝐻𝑝∶=lim_𝑟→1−𝑀_𝑝𝑟,𝑓<∞, (2.1) where 𝑀_𝑝1𝑟,𝑓∶=2𝜋₀^2𝜋𝑓𝑟𝑒^𝑖𝜃𝑝𝑑𝜃^1/𝑝,0<𝑝<∞, (2.2) are the standard 𝐿^𝑝-means of the restriction of 𝑓 to the circle of radius 𝑟 centered at the origin, and 𝑀_∞𝑟,𝑓∶=max_{0≤𝜃≤2𝜋}𝑓𝑟𝑒^𝑖𝜃 (2.3) is the maximum modulus function. For the theory of Hardy spaces, see [4, 5].

Hardy and Littlewood [6], Pommerenke [7], and Prawitz [8] proved that if 𝑓∈𝒰, then 𝑓∈𝐻^𝑝, if and only if¹₀𝑀^𝑝_∞𝑟,𝑓𝑑𝑟<∞,0<𝑝<∞. (2.4) For more information on univalent functions in Hardy spaces, see [2, 7–13]. A result due to Baernstein II [14] states that 𝑓∈BMOA, the space of analytic functions in the Hardy space 𝐻¹ with boundary values of bounded mean oscillation, if and only if sup_𝑎∈𝔻‖𝑓∘𝜑_𝑎−𝑓(𝑎)‖_𝐻^𝑝<∞, where 𝜑_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧) is the automorphism of 𝔻 which interchanges the origin and the point 𝑎. This, applied to (2.4), shows that 𝑓∈𝒰 belongs to BMOA if and only ifsup_𝑎∈𝔻¹₀𝑀^𝑝_∞𝑟,𝑓∘𝜑_𝑎−𝑓(𝑎)𝑑𝑟<∞ (2.5) for some (equivalently for all) 0<𝑝<∞. However, Pommerenke [15] has shown that BMOA and the Bloch space ℬ contain the same univalent functions. The Bloch space ℬ consists of those 𝑓∈ℋ(𝔻) for which 𝑓_ℬ∶=sup_𝑧∈𝔻𝑓^′(𝑧)1−|𝑧|²<∞. (2.6) If, in addition, 𝑓′(𝑧)(1−|𝑧|²)→0, as |𝑧|→1⁻, then 𝑓∈ℬ₀. For the theory of Bloch spaces, see [3, 16]. If Ω is a simply connected proper subdomain of the complex plane and 𝑓∈𝒰 such that 𝑓(𝔻)=Ω, then 𝑑_Ω𝑓(𝑧)≤𝑓^′(𝑧)1−|𝑧|²≤4𝑑_Ω𝑓(𝑧),𝑧∈𝔻, (2.7) where 𝑑_Ω(𝑤) stands for the Euclidean distance from 𝑤 to the boundary of Ω, see, for example, [3]. Therefore, univalent functions in the Bloch space can be characterized by the following well-known geometric condition; 𝑓∈ℬ if and only if sup_𝑤∈Ω𝑑_Ω(𝑤)<∞, that is, if the image of 𝔻 under 𝑓 does not contain arbitrarily large discs. For other characterizations of univalent Bloch functions, see [11].

Aulaskari et al. [17] improved the result by Pommerenke by showing that ℬ∩𝒰=𝑄_𝑝∩𝒰 for any 𝑝>0. Recall that 𝑄_𝑝 is a Möbius invariant subspace of ℬ and consists of those 𝑓∈ℋ(𝔻) for which 𝑓²_𝑄𝑝∶=sup_{𝑎∈𝔻𝔻}𝑓^′(𝑧)²𝑔^𝑝(𝑧,𝑎)𝑑𝐴(𝑧)<∞, (2.8) where 𝑑𝐴(𝑧) is the Euclidean area element on 𝔻. In particular, 𝑄₁=BMOA and 𝑄_𝑝=ℬ for all 𝑝>1. Moreover, 𝑄₀ is the classical Dirichlet space 𝒟 which consists of all 𝑓∈ℋ(𝔻) with finite area of image counting multiplicities. The Dirichlet space 𝒟 is a special case of classical Besov spaces. A geometric characterization of univalent functions in Besov spaces was found by Walsh [18], see also the related results by Donaire et al. [19]. For the theory of 𝑄_𝑝 spaces, see [20, 21].

For 𝐾∶[0,∞)→[0,∞), the Möbius invariant space 𝑄_𝐾 consists of those 𝑓∈ℋ(𝔻) for which 𝑓²_𝑄𝐾∶=sup_{𝑎∈𝔻𝔻}𝑓^′(𝑧)²𝐾𝑔(𝑧,𝑎)𝑑𝐴(𝑧)<∞. (2.9) Moreover, if the integral above tends to zero as 𝑎 approaches to the boundary of 𝔻, then 𝑓∈𝑄_𝐾,0. If 𝐾(𝑡)=𝑡^𝑝, then 𝑄_𝐾=𝑄_𝑝, and therefore 𝑄_𝐾 can be viewed as a generalized 𝑄_𝑝 space. For results on 𝑄_𝐾, see [22–24] and the references therein. From now on, the weight 𝐾 is assumed to admit the following basic properties: (1)𝐾(0)=0;(2)∫₀^1/𝑒𝐾(log(1/𝑟))𝑟𝑑𝑟<∞;(3)𝐾′(𝑡)≥0 for all 𝑡∈(0,1);(4)𝐾^′′(𝑡)≤0 for all 𝑡∈(0,1).

Requirements (1) and (2) are standard; the first one ensures that 𝐾 indeed plays an essential role in the definition, and the second one guarantees the nontriviality of 𝑄_𝐾 as well as the inclusions 𝑄_𝐾⊂ℬ and 𝑄_𝐾,0⊂ℬ₀. Conditions (3) and (4) are, of course, restrictions, yet, for example, 𝐾(𝑡)=𝑡^𝑝 satisfies both of them for all 0<𝑝≤1.

Before proceeding further, we give an example related to 𝑄_𝐾 spaces in order to illustrate the variety of spaces induced by different choices of 𝐾.

Example 2.1. For 0<𝑝<∞ and 𝑞∈ℝ, consider the weight 𝐾(𝑡)(𝑡)𝑡=𝐾𝑝,𝑞∶=^𝑝1log𝑡^𝑞,0<𝑡<𝑒^−𝑞/𝑝,𝑒^−𝑞𝑞𝑝^𝑞,𝑒^−𝑞/𝑝𝑡≤𝑡<1,0≤𝑞<∞,∶=^𝑝1log𝑡^𝑞1,0<𝑡<2,2^−𝑝log2^𝑞,12≤𝑡<1,−∞<𝑞<0. (2.10) Obviously, 𝑄_𝐾(𝑝,0)=𝑄_𝑝 for all 0<𝑝<∞. Since 𝐾′(𝑡)=𝑡^𝑝−11log𝑡^𝑞−11𝑝log𝑡−𝑞 (2.11) for either 0<𝑡<𝑒^−𝑞/𝑝 (the case 𝑞≥0) or 0<𝑡<1/2 (the case 𝑞<0), the weight 𝐾 is increasing. If 0<𝑝≤1, then the integral ¹₀𝐾1log𝑟1−𝑟²⁻²𝑟𝑑𝑟 (2.12) diverges for all 𝑞∈ℝ, and therefore 𝑄_𝐾⊊ℬ by [22, Theorem 2.3]. Moreover, [22, Theorem 2.6] shows that 𝑄_𝑝1⊊𝑄_{𝐾(𝑝1,−𝑞1)}⊊𝑄_{𝐾(𝑝1,−𝑞2)}⊊𝑄_{𝐾(𝑝2,𝑞2)}⊊𝑄_{𝐾(𝑝2,𝑞1)}⊊𝑄_𝑝2 (2.13) for all 0<𝑝₁<𝑝₂≤1 and 0<𝑞₁<𝑞₂<∞.

Let us turn back to univalent functions. A special case of [11, Theorem 4] shows that a univalent function 𝑓 belongs to 𝑄_𝑝 if and only ifsup_𝑎∈𝔻¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓log𝑟^𝑝−1𝑑𝑟<∞,0<𝑝≤1. (2.14) Of course, this is not a natural way to state the result because 𝒰∩𝑄_𝑝=𝒰∩ℬ for all 𝑝, but it appears to be useful for our purposes. The case 𝑝=1 of (2.14), which corresponds toBMOA, reduces to (2.5) with 𝑝=2. An appropriate interpretation of (2.14) allows us to conclude what happens with 𝑄_𝐾. Namely, since 𝑝(log(1/𝑟))^𝑝−1=𝐾′(log(1/𝑟)) for 𝐾(𝑡)=𝑡^𝑝, condition (2.14), with (log(1/𝑟))^𝑝−1 being replaced by 𝐾′(log(1/𝑟)), gives a candidate for a characterization of univalent functions in 𝑄_𝐾. The main result of this paper is Theorem 2.2 which shows that this is indeed the case, provided−𝐾^′′(𝑡)+𝐾′(𝑡)𝐾(𝑡)𝐾^′(𝑡)²≤𝐶,0<𝑡<∞, (2.15) for some positive constant 𝐶.

Theorem 2.2. Let 𝑓∈𝒰 and assume that 𝐾 satisfies condition (2.15). Then, 𝑓∈𝑄_𝐾 if and only if sup_𝑎∈𝔻¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝐾′log𝑟𝑑𝑟<∞. (2.16) Moreover, 𝑓∈𝑄_𝐾,0 if and only if lim_{|𝑎|→1−}¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎−𝑓(𝑎)𝐾^′1log𝑟𝑑𝑟=0. (2.17)

It is an immediate consequence of the proof that the assertions in Theorem 2.2 remain valid for areally mean 𝑞-valent functions. In addition to standard techniques of univalent functions, the proof of Theorem 2.2 uses a result by Pavlović and Peláez [25] on weighted integrals of analytic functions and their derivatives. An application of this result yields the requirement (2.15). We will analyze the importance of (2.15) after discussing consequences of Theorem 2.2 and its proof.

The first part of the proof of Theorem 2.2 shows that 𝑓∈ℋ(𝔻) belongs to 𝑄_𝐾 if and only if sup_𝑎∈𝔻¹₀𝐴𝑓(Δ(1𝑎,𝑟))𝐾′log𝑟𝑑𝑟<∞, (2.18) where 𝐴(𝑓(Δ(𝑎,𝑟))) denotes the area of image of the pseudohyperbolic disc Δ(𝑎,𝑟)∶={𝑧∶|𝜑_𝑎(𝑧)|<𝑟} under 𝑓 counting multiplicities. This, together with Theorem 2.2, shows that for 𝑓∈𝑄_𝐾∩𝒰, the quantities 𝐴(𝑓(Δ(𝑎,𝑟))) and 𝑀²_∞(𝑟,𝑓∘𝜑_𝑎−𝑓(𝑎)) are of the same growth (uniformly in 𝑎) when measured in terms of 𝐿¹(𝐾′(log(1/𝑟))𝑑𝑟).

Since ℬ∩𝒰=𝑄_𝑝∩𝒰 for any 𝑝>0 by a result due to Aulaskari et al. [17], it is natural to ask when does the identity ℬ∩𝒰=𝑄_𝐾∩𝒰 hold. An application of Theorem 2.2 yields Corollary 2.3 which answers this question provided 𝐾 satisfies (2.15).

Corollary 2.3. Assume that 𝐾 satisfies condition (2.15). Then, 𝑄_𝐾∩𝒰=ℬ∩𝒰 if and only if ¹₀log1+𝑟1−𝑟²1𝐾′log𝑟𝑑𝑟<∞. (2.19) Moreover, if (2.19) is satisfied, then 𝑄_𝐾,0∩𝒰=ℬ₀∩𝒰.

Wulan [26] showed that every areally mean 𝑞-valent Bloch function belongs to 𝑄_𝐾 whenever 𝐾 satisfies¹₀1log1−𝑟²𝐾log(1/𝑟)1−𝑟𝑟𝑑𝑟<∞. (2.20)

This result is more general than Corollary 2.3 because the crucial condition (2.15) is not needed. However, 𝐾 is concave whenever (4) is satisfied, and in that case, 𝐾′(𝑡)≤𝐾(𝑡)/𝑡, and then (2.20) implies (2.19).

We next analyze the necessity of condition (2.15). This is needed in two consecutive steps in the proof of Theorem 2.2. These steps together establish the asymptotic inequality¹₀𝑀²_∞1𝑟,𝑓𝑟1𝐾′log𝑟𝑑𝑟≲¹₀𝑀²_∞𝑟,𝑓′¹_𝑟𝐾1log𝑟𝑑𝑠𝑑𝑟 (2.21) for all 𝑓∈ℋ(𝔻). As usual, we write 𝐴≲𝐵, if there exists 𝐶>0 such that 𝐴≤𝐶𝐵, and the notation 𝐴≳𝐵 is understood in an analogous manner. In particular, if 𝐴≲𝐵 and 𝐴≳𝐵, then we write 𝐴≍𝐵. Set now 𝑣(𝑟)=∫¹_𝑟𝐾(log(1/𝑟)), so that (2.21) becomes¹₀𝑀²_∞𝑟,𝑓𝑣^′′(𝑟)𝑑𝑟≲¹₀𝑀²_∞𝑟,𝑓′𝑣(𝑟)𝑑𝑟. (2.22) We will show next that (2.22) does not necessarily remain true even for univalent functions 𝑓, if 𝐾, that induces 𝑣, does not satisfy (2.15). To see this, take 𝑣(𝑟)=1−𝑟(((𝑟)1log1/1−𝑟)),𝑣′≍−log(1/(1−𝑟)),𝑣^′′(𝑟)≍1(1−𝑟)log(1/(1−𝑟))²,𝑟⟶1⁻, (2.23) and 𝑓∈𝒰 with 𝑓²(𝑧)=1−log(1−𝑧), so that 𝑀²_∞1𝑟,𝑓≍log1−𝑟,𝑀²_∞≍1𝑟,𝑓′(1−𝑟)²log(1/1−𝑟),𝑟⟶1⁻. (2.24) For these choices, the right-hand side of (2.22) is finite, whereas the left-hand side is not. If 𝐾(𝑡)=−1/log𝑡, then 𝑣(𝑟)=∫¹_𝑟𝐾(log(1/𝑟))≍(1−𝑟)(log(1/(1−𝑟)))⁻¹, but (2.15) for this 𝐾 is equivalent to −2−log𝑡+𝑡log𝑡≤𝐶, which clearly fails as 𝑡→0⁺. Moreover, 𝑓∈𝒰∩ℬ, but (2.19) fails for 𝐾(𝑡)=−1/log𝑡. This justifies the assumption (2.15) in Theorem 2.2 and Corollary 2.3.

3. Proof of Theorem 2.2

We will prove the first assertion, the second one, then immediately follows by the proof.

3.1. Sufficiency of (2.16)

Let 𝑓∈𝒰. Recall first that 𝑀₂(𝑟,ℎ) is nondecreasing for any ℎ∈ℋ(𝔻). Therefore we may use conditions (1) and (3) together with Fubini’s theorem to obtain𝐼𝑓,𝐾,𝑎∶=_𝔻𝑓∘𝜑_𝑎^′(𝑧)²𝐾1log|𝑧|(𝑧)≍1𝑑𝐴2𝜋_𝔻𝑓∘𝜑_𝑎^′(𝑧)²|𝑧|𝐾1log|𝑧|(𝑧)=𝑑𝐴₁₀𝑀²₂𝑟,𝑓∘𝜑_𝑎−𝑓(𝑎)^′1_𝑟1𝑠1𝐾′log𝑠𝑑𝑠𝑟²≤𝑑𝑟₁₀𝑀²₂𝑟,𝑓∘𝜑_𝑎−𝑓(𝑎)^′1_𝑟1𝐾′log𝑠𝑑𝑠𝑟𝑑𝑟=¹_0Δ(0,𝑠)𝑓∘𝜑_𝑎−𝑓(𝑎)^′(𝑧)²(𝑧)1𝑑𝐴𝐾′log𝑠𝑑𝑠. (3.1) If ℎ∈𝒰, then1𝜋_Δ(0,𝑠)ℎ^′(𝑧)²(𝑧)≤1𝑑𝐴𝜋_{|𝑤|≤𝑀∞(𝑠,ℎ)}𝑑𝐴(𝑤)=2^𝑀∞₀^(𝑠,ℎ)𝑡𝑑𝑡=𝑀²_∞(𝑠,ℎ), (3.2) and an application of this inequality to ℎ=𝑓∘𝜑_𝑎−𝑓(𝑎), together, with (3.1), gives 𝐼𝑓,𝐾,𝑎≲¹₀𝑀²_∞𝑠,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝐾′log𝑠𝑑𝑠. (3.3) Therefore a univalent function 𝑓 belongs to 𝑄_𝐾 if (2.16) is satisfied. Note that this part of the proof uses the univalence of 𝑓, but does not rely on (2.15). Moreover, if 𝑓 is areally mean 𝑞-valent, then the reasoning in (3.2) remains valid as soon as the right-hand side of the inequality is multiplied by 𝑞.

3.2. Necessity of (2.16)

To prove this, we will need a special case of a result due to Pavlović and Peláez [25], which states that¹₀𝑀²_∞(𝑟,ℎ)𝜔(𝑟)𝑑𝑟≍|ℎ(0)|²+¹₀𝑀²_∞∫𝑟,ℎ′^1𝑟𝜔(𝑠)𝑑𝑠²𝜔(𝑟)𝑑𝑟,ℎ∈ℋ(𝔻), (3.4) provided 𝜔∶[0,1)→(0,∞) is differentiable and satisfies𝜔′(𝑟)𝜔(𝑟)^21𝑟𝜔(𝑥)𝑑𝑥≤𝐶,0<𝑟<1, (3.5) for some constant 𝐶>0. Assumption (2.15) is equivalent to (3.5) for 𝜔(𝑟)∶=1/𝑟𝐾′(log1/𝑟). Therefore, (3.4) yields ¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝑟1𝐾′log𝑟𝑑𝑟≲¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎′𝑟𝐾log(1/𝑟)²𝐾′log(1/𝑟)𝑑𝑟. (3.6) We next show that𝑟𝐾log(1/𝑟)²()≤()𝐾′log1/𝑟𝐶+2¹_𝑟𝐾1log𝑠𝑑𝑠, (3.7) where 𝐶 is the constant in (2.15). To prove (3.7), consider the function 𝑔(𝑟)∶=(𝐶+2)¹_𝑟𝐾1log𝑠𝐾()𝑑𝑠−𝑟log1/𝑟²().𝐾′log1/𝑟 (3.8) By the assumptions (1), (3), and (4) on 𝐾, 𝑔(𝑟)→0 as 𝑟→1⁻. Moreover, (2.15) yields (𝑟)1𝑔′=−𝐶𝐾log𝑟−𝐾′log(1/𝑟)+𝐾^′′log(1/𝑟)𝐾log(1/𝑟)²𝐾′log(1/𝑟)²≤0, (3.9) and thus 𝑔 is decreasing, and therefore (3.7) holds. This, together with Fubini's theorem, shows that ¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝑟1𝐾′log𝑟≲𝑑𝑟₁₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎′¹_𝑟𝐾1log𝑠=𝑑𝑠𝑑𝑟₁₀𝐾1log𝑠^𝑠₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎^′𝑑𝑟𝑑𝑠, (3.10) from which the inequality ∫^𝑠₀𝑀²_∞(𝑟,ℎ)𝑑𝑟≤𝜋𝑀²₂(𝑠,ℎ), valid for all ℎ∈ℋ(𝔻) (see [9, page 841] or [7, Hilfssatz 1]), yields ¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝐾′log𝑟𝑑𝑟≲¹₀𝑀²₂𝑠,𝑓∘𝜑_𝑎′𝐾1log𝑠≤𝑓𝑑𝑠≍𝐼𝑓,𝐾,𝑎²_𝑄𝐾. (3.11) Thus, (2.16) is satisfied if 𝑓∈𝑄_𝐾.

4. Proof of Corollary 2.3

4.1. Sufficiency of (2.19)

Since 𝑄_𝐾⊂ℬ, it suffices to show that 𝒰∩ℬ⊂𝑄_𝐾. To see this, let 𝑓∈ℬ. An application of the inequality 𝑀_∞𝑟,𝑔≤𝑔_ℬ12log1+𝑟+𝑔(01−𝑟),𝑔∈ℬ, (4.1) to the function 𝑔=𝑓∘𝜑_𝑎−𝑓(𝑎)∈ℬ gives𝑀_∞𝑟,𝑓∘𝜑_𝑎−𝑓(𝑎)≤𝑓∘𝜑_𝑎−𝑓(𝑎)_ℬ12log1+𝑟=𝑓1−𝑟^ℬ12log1+𝑟1−𝑟 (4.2) for all 𝑎∈𝔻. This yields sup_𝑎∈𝔻¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎−f(𝑎)1𝐾′log𝑟≤1𝑑𝑟4𝑓²_ℬ¹₀log1+𝑟1−𝑟²1𝐾′log𝑟𝑑𝑟<∞, (4.3) and thus 𝑓∈𝑄_𝐾 by Theorem 2.2.

4.2. Necessity of (2.19)

If 𝐾 satisfies condition (2.15) and 𝑄_𝐾∩𝒰=ℬ∩𝒰, then the univalent Bloch function log((1+𝑧)/(1−𝑧)), together with Theorem 2.2, shows that (2.19) holds.

4.3. The Case 𝑄_𝐾,0∩𝒰=ℬ₀∩𝒰

Assume that 𝐾 satisfies (2.19) and let 𝜀>0. Since 𝑄_𝐾,0⊂ℬ₀, it suffices to show that 𝒰∩ℬ₀⊂𝑄_𝐾,0. To see this, let 𝑓∈ℬ₀. By applying (4.2) we obtain ¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝐾′log𝑟≤𝑑𝑟_𝑅0𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝐾′log𝑟+𝑓𝑑𝑟_2ℬ14¹_𝑅log1+𝑟1−𝑟²1𝐾′log𝑟𝑑𝑟=𝐼₁(𝑅)+𝐼₂(𝑅). (4.4) Since 𝑓∈ℬ₀⊂ℬ, the assumption (2.19) implies that 𝐼₂(𝑅)<𝜀/2 for all 𝑅 sufficiently large. Fix such an 𝑅. The image of the circle of radius 𝑟 centered at the origin under a univalent function ℎ−ℎ(0) is a Jordan curve with zero in its inner domain. The length of this image is 2𝜋𝑟𝑀₁(𝑟,ℎ′), and hence 𝑀_∞(𝑟,ℎ)≤𝜋𝑟𝑀₁(𝑟,ℎ′)+|𝑓(0)|. This estimate, applied to ℎ=𝑓∘𝜑_𝑎−𝑓(𝑎), together with the assumption 𝑓∈ℬ₀, yields 𝐼₁(𝑅)≤𝜋^𝑅₀12𝜋₀^2𝜋𝑓^′𝜑_𝑎𝑟𝑒^𝑖𝜃𝜑^′_𝑎𝑟𝑒^𝑖𝜃𝑑𝜃²1𝐾′log𝑟=1𝑟𝑑𝑟4𝜋^𝑅₀₀^2𝜋𝑓^′𝜑_𝑎𝑟𝑒^𝑖𝜃1−𝜑_𝑎𝑟𝑒^𝑖𝜃2𝑑𝜃²𝐾′log(1/𝑟)1−𝑟²²𝜀𝑟𝑑𝑟<2 (4.5) for all sufficiently large |𝑎|. Therefore, ¹₀𝑀²_∞𝑟,𝑓∘𝜑_𝑎(𝑎)1−𝑓𝐾′log𝑟𝑑𝑟⟶0,|𝑎|⟶1⁻, (4.6) and hence 𝑓∈𝑄_𝐾,0 by Theorem 2.2.

Acknowledgments

This research was supported in part by the MEC-Spain MTM2008-05891, the Academy of Finland 121281, and the European Networking Programme HCAA of the European Science Foundation.