Composition Operators on a Class of Analytic Function Spaces Related to Brennan’s Conjecture

Abstract

Brennan’s conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk $${\mathbb{D}}$$ onto a simply connected domain G and −1/3 < p < 1, then 1/(τ′) p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces $${L^2_a(\mu _p)}$$ on G and prove that Brennan’s conjecture is equivalent to the existence of compact composition operators on these spaces for every simply connected domain G and all $${p\in(-1/3,1)}$$ . Motivated by this result, we study the boundedness and compactness of composition operators in this setting.