Weights induced by homogeneous polynomials

Let B be the unit ball and S the unit sphere in Cⁿ (n ≥ 2). Let σ be the unique normalized rotation-invariant Borel measure on S and m the normalized area measure on C. We first prove that if Λ is a holomorphic homogeneous polynomial on Cⁿ normalized so that Λ maps B onto the unit disk U in C and if µ = σ[(Λ/s)^-1], then µ < m and the Radon-Nikodym derivative dµ/dm is radial and positive on U. Then we obtain the asymptotic behavior of dµ/dm for a certain, but not small, class of functions Λ. These results generalize two recent special cases of P. Ahern and P. Russo. As an immediate consequence we enlarge the class of functions for which Ahern-Rudin’s Paley-type gap theorems hold.