The level sets of the moduli of functions of bounded characteristic

Abstract:

For $f$ a nonconstant meromorphic function on $\Delta = \{ |z| < 1\}$ and $r \in (\inf |f|,\sup |f|)$, let $\mathcal {L}(f,r) = \{ z \in \Delta :|f(z)| = r\}$. In this paper, we study the components of $\Delta \backslash \mathcal {L}(f,r)$ along with the level sets $\mathcal {L}(f,r)$. Our results include the following: If $f$ is an outer function and $\Omega$ a component of $\Delta \backslash \mathcal {L}(f,r)$, then $\Omega$ is a simply-connected Jordan region for which $({\text {fr}}\;\Omega ) \cap \{ |z| = 1\}$ has positive measure. If $f$ and $g$ are inner functions with $\mathcal {L} (f,r) = \mathcal {L} (g,s)$, then $g = \eta {f^\alpha }$, where $|\eta | = 1$ and $\alpha > 0$. When $g$ is an arbitrary meromorphic function, the equality of two pairs of level sets implies that $g = c{f^\alpha }$, where $c \ne 0$ and $\alpha \in ( - \infty ,\infty )$. In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.