Univalent Functions in the Möbius Invariant QK Space

and Applied Analysis 3 where dΩ w stands for the Euclidean distance fromw to the boundary ofΩ, 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 supw∈ΩdΩ 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 Qp ∩U for any p > 0. Recall that Qp is a Möbius invariant subspace of B and consists of those f ∈ H D for which ∥ ∥f ∥ ∥2 Qp : sup a∈D ∫ D ∣ ∣f ′ z ∣ ∣gp z, a dA z < ∞, 2.8 where dA z is the Euclidean area element on D. In particular, Q1 BMOA and Qp 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 Qp spaces, see 20, 21 . For K : 0,∞ → 0,∞ , the Möbius invariant space QK consists of those f ∈ H D for which ∥f ∥2 QK : sup a∈D ∫ D ∣f ′ z ∣2K ( g z, a ) dA z < ∞. 2.9 Moreover, if the integral above tends to zero as a approaches to the boundary of D, then f ∈ QK,0. If K t t, then QK Qp, and therefore QK can be viewed as a generalized Qp space. For results on QK, 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 ∫1/e 0 K log 1/r r dr < ∞; 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 QK as well as the inclusions QK ⊂ B and QK,0 ⊂ B0. Conditions 3 and 4 are, of course, restrictions, yet, for example, K t t satisfies both of them for all 0 < p ≤ 1. Before proceeding further, we give an example related to QK spaces in order to illustrate the variety of spacesinduced by different choices of K. 4 Abstract and Applied Analysis Example 2.1. For 0 < p < ∞ and q ∈ R, consider the weight K t K ( p, q ) t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎩ : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ t ( log 1 t )q , 0 < t < e−q/p, e−q ( q p )q , e−q/p ≤ t < 1, 0 ≤ q < ∞, : ⎧ ⎪ ⎨ ⎪ ⎩ t ( log 1 t )q , 0 < t < 1 2 , 2−p ( log 2 )q , 1 2 ≤ t < 1, −∞ < q < 0. 2.10 Obviously, QK p,0 Qp for all 0 < p < ∞. Since K′ t tp−1 ( log 1 t )q−1( p log 1 t − 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 ∫1 0 K ( log 1 r )( 1 − r2 )−2 r dr 2.12 diverges for all q ∈ R, and therefore QK B by 22, Theorem 2.3 . Moreover, 22, Theorem 2.6 shows that Qp1 QK p1,−q1 QK p1,−q2 QK p2,q2 QK p2,q1 Qp2 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 Qp if and only if sup a∈D ∫1 0 M2 ∞ ( r, f ◦ φa − f a ) ( log 1 r )p−1 dr < ∞, 0 < p ≤ 1. 2.14 Of course, this is not a natural way to state the result because U ∩Qp U ∩ B for all p, but it appears to be useful for our purposes. The case p 1 of 2.14 , which corresponds to BMOA, reduces to 2.5 with p 2. An appropriate interpretation of 2.14 allows us to conclude what happenswithQK. Namely, since p log 1/r p−1 K′ log 1/r forK t t, condition 2.14 , with log 1/r p−1 being replaced byK′ log 1/r , gives a candidate for a characterization of Abstract and Applied Analysis 5 univalent functions inQK. 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 < ∞, 2.15and Applied Analysis 5 univalent functions inQK. 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 ∈ QK if and only if


Introduction
The aim of this paper is to characterize univalent functions f in the M öbius 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.

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 . 2.4 For more information on univalent functions in Hardy spaces, see 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 if 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 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 Moreover, if the integral above tends to zero as a approaches to the boundary of D, then 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: 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 ⊂ B 0 .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 spacesinduced by different choices of K.
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 if sup 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 to BMOA, 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

2.16
Moreover, f ∈ Q K,0 if and only if 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 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 and M 2 ∞ r, f • ϕ a − f a are of the same growth uniformly in a when measured in terms of L 1 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 .
Wulan 26 showed that every areally mean q-valent Bloch function belongs to 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 inequality

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 For these choices, the right-hand side of 2.22 is finite, whereas the left-hand side is not.

Proof of Theorem 2.2
We will prove the first assertion, the second one, then immediately follows by the proof.

Sufficiency of 2.16
Let f ∈ U. Recall first that M 2 r, h is nondecreasing for any h ∈ H D .Therefore we may use conditions 1 and 3 together with Fubini's theorem to obtain

3.1
If h ∈ U, then and an application of this inequality to h f • ϕ a − f a , together, with 3.1 , gives 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.

Necessity of 2.16
To prove this, we will need a special case of a result due to Pavlović and Peláez 25 , which states that for some constant C > 0. Assumption 2.15 is equivalent to 3.5 for ω r : 1/rK log 1/r .Therefore, 3.4 yields

3.6
We next show that where C is the constant in 2.15 .To prove 3.7 , consider the function By the assumptions 1 , 3 , and 4 on K, g r → 0 as r → 1 − .Moreover, 2.15 yields and thus g is decreasing, and therefore 3.7 holds.This, together with Fubini's theorem, shows that

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

4.4
Since f ∈ B 0 ⊂ B, the assumption 2.19 implies that I 2 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 2πrM 1 r, h , and hence M ∞ r, h ≤ πrM 1 r, h |f 0 |.This estimate, applied to h f • ϕ a − f a , together with the assumption f ∈ B 0 , yields

2
Abstract and Applied AnalysisFor 0 < p ≤ ∞, the Hardy space H p consists of those f ∈ H D for which -means of the restriction of f to the circle of radius r centered at the origin, and M ∞ r, f : max 0≤θ≤2π f re 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 if 1 0 19where 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∈Ω d Ω w < ∞, that is, if the image of D under f does not contain arbitrarily large discs.is the Euclidean area element on D. In particular, Q 1 BMOA and Q p B for all p > 1.Moreover, Q 0 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 byWalsh 18, see also the related results by Donaire et al.19.For the theory of Q p spaces, see20, 21 .For Let f ∈ U and assume that K satisfies condition 2.15 .Then, f ∈ Q K if and only if but 2.15 for this K is equivalent to −2 − log t t log t ≤ C, which clearly fails as t → 0 .Moreover, f ∈ U ∩ B, but 2.19 fails for K t −1/ log t.This justifies the assumption 2.15 in Theorem 2.2 and Corollary 2.3.
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.Assume that K satisfies 2.19 and let ε > 0. Since Q K,0 ⊂ B 0 , it suffices to show that U ∩ B 0 ⊂ Q K,0 .To see this, let f ∈ B 0 .By applying 4.2 we obtain .3 and thus f ∈ Q K by Theorem 2.2.4.2.Necessity of 2.19If