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.
