Recognizing an Inner Function by Its Distribution of Values

The Nevanlinna counting function is used to give several different characterizations of inner functions. The results all assert that a holomorphic selfmap of the unit disc is inner if and only if its Nevanlinna counting function is, in various pointwise, asymptotic, and operator-theoretic senses, as large as it can be. Although many of the results in this note are known, the asymptotic and operator-theoretic ones (Theorems 4.1 and 5.4) appear to be new. The exposition is heavily expository, featuring a detailed review of those basic facts about the counting function required for the proofs. 1. The Nevanlinna Counting Function We work throughout with holomorphic functions φ defined on the open unit disc U with φ(U) ⊂ U. For such a “holomorphic selfmap of U,” if w ∈ φ(U) we let φ−1(w) denote the “multiplicity preimage” of w under φ, i.e. φ−1(w) = (z1(w), z2(w) . . . ), a finite or infinite list whose entries comprise the usual set-theoretic preimage, but ordered according to increasing moduli, with each entry repeated as many times as its multiplicity. Let nφ(w) be the length of the list φ −1(w) (possibly = ∞), and set