MULTIVALENT FUNCTIONS AND QK SPACES

We give a criterion for q-valent analytic functions in the unit disk to belong to QK, a Mobius-invariant space of functions analytic in the unit disk in the plane for a nondecreasing function K:[0,∞)→[0,∞), and we show by an example that our condition is sharp. As corollaries, classical results on univalent functions, the Bloch space, BMOA, and Qp spaces are obtained.


Introduction.
For analytic univalent function f in the unit disk ∆, Pommerenke [8] proved that f ∈ Ꮾ if and only if f ∈ BMOA, which easily implies a result of Baernstein II [4] about univalent Bloch functions: if g(z) ≠ 0 is an analytic univalent function in ∆, then log g ∈ BMOA. We know that Pommerenke's result mentioned above was generalized to Q p spaces for all p, 0 < p < ∞, by Aulaskari  Here, Q p and its subspace Q p,0 , 0 < p < ∞, denote the spaces of analytic functions f in ∆ defined, respectively, as follows (cf. [1,3]): where g(z, a) = log 1/|ϕ a (z)| is a Green's function in ∆ with pole at a ∈ ∆, and ϕ a (z) = (a − z)/(1 −āz) is a Möbius transformation of ∆. We know that Q 1 = BMOA, the space of all analytic functions of bounded mean oscillation (cf. [5]), and for each p ∈ (1, ∞), the space Q p is the Bloch space Ꮾ (cf. [1]), which is defined as follows: Similar to the above we have Q 1,0 = VMOA, the space of all analytic functions of vanishing mean oscillation (cf. [5]), and Q p,0 = Ꮾ 0 for all p ∈ (1, ∞), where Ꮾ 0 denotes the little Bloch space defined by In the present paper, we consider a more general space Q K (see below) and show that all the above-mentioned results are true for space Q K . Our contribution gives an extended version of Pommerenke's theorem, which is also a slight improvement of all the above results, and the proof presented here is independently developed.
Let K : [0, ∞) → [0, ∞) be a right-continuous and nondecreasing function. Recall that the space Q K consists of analytic functions f in ∆ for which Modulo constants, Q K is a Banach space under the norm defined in (1.5). It is clear that Q K is Möbius-invariant and a subspace of the Bloch space Ꮾ (cf. [6]). For 0 < p < ∞, K(t) = t p gives the space Q p . Choosing K(t) = 1, we get the Dirichlet space Ᏸ. By [6, Proposition 2.1] we know that if the integral is divergent, then the space Q K is trivial; that is, the space Q K contains only constant functions. From now on, we assume that the function K : is rightcontinuous and nondecreasing and that the integral (1.7) is convergent. Without loss of generality, we can assume that K(1) > 0. For a general theory for Q K spaces, see [6,11].

Main results.
A function f analytic in the unit disk is said to be q-valent if the equation f (z) = w has never more than q solutions. Let where q is a positive number, we say that f is areally mean q-valent or circumferentially mean q-valent, respectively (cf. [7, pages 38 and 144]). It is clear that if f is circumferentially mean q-valent, then f is areally mean q-valent. Note that if (1.1) holds, f will be areally mean q-valent in ∆ for some q > 0. We know that if f is univalent, then f must be areally and circumferentially mean 1-valent. Thus, it is natural to conjecture that Pommerenke's result and Theorem 1.1 are also true for the areally and circumferentially mean q-valent functions.
We know that the space Q K can be nontrivial if K is not too big at infinity (see condition (1.7)). For such functions K, the properties of Q K depend essentially on the behavior of K near the origin. From [6, Theorems 2.3 and 2.5], we know that A natural idea is to look for an integral condition which is weaker than that given by (2.4) for some special f . For the areally mean q-valent case, we present the main result in this paper as follows.
for 0 < r < 1, but the converse is not true. For example, K(t) = t gives that (2.5) holds but (2.4) fails. By [6, Theorems 2.3 and 2.5], (2.5) is also necessary for Theorem 2.1(i) and (ii) in case f is an areally mean q-valent function in ∆.
In the light of the following example it is impossible to drop the assumption of areally mean q-valence of the functions f in Theorem 2.1. Indeed, choose K 1 (t) = t 2α−1 and It is easy to see that f 1 ∈ Ꮾ and (2.5) holds for K 1 . Since f 1 has a gap series representation, f 1 is not an areally mean q-valent in ∆. The following argument shows that f ∉ Q K 1 .

Theorem 2.2. Let f be a circumferentially mean q-valent and nonvanishing function in ∆. If (2.5) holds, then log f ∈ Q K .
It is clear that the integral in (2.5) is convergent for K(t) = t p , p > 0. Thus, we have the following result which extends Theorem 1.1.

Proofs.
In the proofs of Theorems 2.1 and 2.2, we need two lemmas, the first one can be considered as a generalization of a result of Pommerenke (cf. [9, page 174]). where M(r , f ) = sup |z|=r |f (z)|, 0 < r < 1.
Proof. If 1/2 < r < 1, we obtain Since f is areally mean q-valent, we deduce that Now we turn to give the proofs of our main theorems.
Proof of Theorem 2.1. We first prove (i). Since Q K ⊂ Ꮾ, it suffices to prove that if a Bloch function f is areally mean q-valent in ∆, then f ∈ Q K . We use the change of variable w = ϕ a (z) to deduce that It is known that if g ∈ Ꮾ, then Choosing g = f • ϕ a − f (a) and observing that g Ꮾ = f Ꮾ , we obtain It follows from (3.5) and Lemma 3.1 that On the other hand, we have (3.9) Combining the upper bounds given by (3.8), (3.9), and (2.5), we see that f ∈ Q K , which proves part (i) of Theorem 2.1.
To prove (ii), we assume that f is an areally mean q-valent function in ∆ which is also in Ꮾ 0 . By Lemma 3.2(i), it suffices to prove that f ∈ Q K,0 . By Lemma 3.2(ii), there exists an r 0 , 1/2 < r 0 < 1, such that By the proof of part (i) and assumption (2.5), we see that which means that log f is areally mean q 1 -valued in ∆ for some q 1 > 0. It follows from Theorem 2.1 that log f ∈ Q K .

Further discussion.
In [10] we studied the conditions for analytic univalent Bloch function f to belong to Q K spaces. The log-order of the function K(r ) is defined as We note that Theorem 4.1 can be viewed as a consequence of Theorem 2.1. In fact, conditions (i) and (ii) of Theorem 4.1 show that the space Q K is not trivial. That is, the integral (1.7) is convergent in this case. Suppose that K(t) = O((t log 1/t) p ), t → 0. There exist an r 0 ∈ (1/2, 1) and a constant C > 0 such that both log 1/r ≤ 2(1 − r ) and