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

Brennan's conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk 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(μ _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 ε (-1/3,1). Motivated by this result, we study the boundedness and compactness of composition operators in this setting.