Smirnov functions are analytic functions on the open unit disk \(\mathbb{D}\subset \mathbb{C}\) which can be written as the quotient of two bounded analytic functions, where the denominator is an outer function. The class of real Smirnov functions \(N^{+}_{\mathbb{R}}\) is composed by those Smirnov functions which have real boundary values almost everywhere.
In this paper the authors characterize all possible ranges \(\varphi(\mathbb{D})\) and all possible valences \(v_\varphi(\lambda)=\text{card} \{z\in\mathbb{D}: \varphi(z)=\lambda\}\) of functions \(\varphi\in N^{+}_{\mathbb{R}}\) when the valence is finite.
The main result on the range of non-constant functions \(\varphi\in N^{+}_{\mathbb{R}}\) states that \(\varphi(\mathbb{D})\) is either \[ \mathbb{C}_{+}\setminus F \quad\text{or}\quad \mathbb{C}_{-}\setminus G \quad\text{or}\quad \mathbb{C}\setminus (F\cup G \cup E), \] where \(\mathbb{C}_{+}\) and \(\mathbb{C}_{-}\) denote the upper and the lower open half planes, respectively, \( E\subsetneq \mathbb{R}\) is a closed set and \(F \subseteq \mathbb{C}_{+}\), \(G \subseteq \mathbb{C}_{-}\) are relatively closed sets of logarithmic capacity zero. Conversely, given any sets \(E, F\) and \(G\) satisfying the above conditions, there exist functions in \(N^{+}_{\mathbb{R}}\) with ranges \(\mathbb{C}_{+}\setminus F\), \(\mathbb{C}_{-}\setminus G\) and \(\mathbb{C}\setminus (F\cup G \cup E)\).
The main result on valences states that the valence of every \(\varphi\in N^{+}_{\mathbb{R}}\) with finite valence is given by the valence of a plane valence tree. Moreover, any valence arising from a plane valence tree is the valence of a real Smirnov function.