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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Weak subordination and stable classes of meromorphic functions
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by Kenneth Stephenson PDF
Trans. Amer. Math. Soc. 262 (1980), 565-577 Request permission


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.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 262 (1980), 565-577
  • MSC: Primary 30D55; Secondary 30C80
  • DOI:
  • MathSciNet review: 586736