Multivalent functions and QK spaces

We give a criterion for q-valent analytic functions in the unit disk to belong to Q K , a Möbius-invariant space of functions analytic in the unit disk in the plane for a nonde-creasing 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 Q p spaces are obtained. 1. 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 et al. (cf. [2, Theorem 6.1]). Their result can be stated as follows.