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Transactions of the American Mathematical Society

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



Weak subordination and stable classes of meromorphic functions

Author: Kenneth Stephenson
Journal: Trans. Amer. Math. Soc. 262 (1980), 565-577
MSC: Primary 30D55; Secondary 30C80
MathSciNet review: 586736
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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\,\mathop w\limits_ < \,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

$\displaystyle F\, < \,G \qquad{\text{and}}\quad G\, \in \,\mathcal{X}\, \Rightarrow \,F\, \in \,\mathcal{X}.$ ($ (\ast)$)

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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Keywords: Inner function, composition, subordination, Hardy spaces, BMOA
Article copyright: © Copyright 1980 American Mathematical Society