RUDIN ORTHOGONALITY PROBLEM AND THE NEVANLINNA COUNTING FUNCTION

Let phi be a holomorphic function taking the open unit disk U into its elf. We show that the set of nonnegative powers of phi is orthogonal i n L-2(partial derivative U) if and only if the Nevanlinna counting fun ction of phi, N-phi, is essentially radial. As a corollary, we obtain that the orthogonality of {phi(n) : n = 0, 1, 2,...} for a univalent p hi implies phi(z) = cut for some constant alpha. We also show that if (phi(n) : n = 0, 1, 2,...) is orthogonal, then the closure of phi(U) m ust be a disk.