Weak Subordination and Stable Classes of Meromorphic Functions

Abstract

This paper introduces the notion of weak subordination: If $F$ and $G$ are meromorphic in the unit disc $\mathcal {U}$, then $F$ is weakly subordinate to $G$, written $F < G$, provided there exist analytic functions $\phi$ and $\omega :\mathcal {U} \to \mathcal {U}$, with $\phi$ an inner function, so that $F \circ \phi = G \circ \omega$. A class $\mathcal {X}$ of meromorphic functions is termed stable if $F \stackrel {w}{\prec } G$ and $G \in \mathcal {X} \Rightarrow F \in \mathcal {X}$. The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume $F$ and $G$ are meromorphic and $G$ has nontangential limits, a.e. Assume further that $F(\mathcal {U}) \cap G(\mathcal {U}) \ne \emptyset$ and $G({e^{i\theta }}) \notin F(\mathcal {U})$, a.e. This is denoted by $F < G$. Burkholder proved for several classes $\mathcal {X}$ that \begin{equation}\tag {$(\ast )$}F < G \qquad {\text {and}}\quad G \in \mathcal {X} \Rightarrow F \in \mathcal {X}.\end{equation} The main result of this paper is the Theorem: $F < G \Rightarrow F{ \prec ^w}G$. In particular, implication (*) holds for all stable classes $\mathcal {X}$. The paper goes on to study various stable classes, which include BMOA, ${H^p}$, $0 < p \leqslant \infty$, ${N_{\ast }}$, the space of functions of bounded characteristic, and the ${M^\Phi }$ spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.