Inner functions as improving multipliers

If X ⊂ Y are two classes of analytic functions in the unit disk D and θ is an inner function, θ is said to be (X,Y )-improving, if every function f ∈ X satisfying f θ ∈ Y must actually satisfy f θ ∈ X. This notion has been recently introduced by K.M. Dyakonov. In this paper we study the (X,Y )-improving inner functions for several pairs of spaces (X,Y ). In particular, we prove that for any p ∈ (0, 1) the (Qp,BMOA)-improving inner functions and the (Qp,B)-improving inner functions are precisely the inner functions which belong to the space Qp. Here, B is the Bloch space. We also improve some results of Dyakonov on the subject regarding Lipschitz spaces and Besov spaces. © 2008 Elsevier Inc. All rights reserved