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Transactions of the American Mathematical Society

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Commuting analytic functions


Author: Carl C. Cowen
Journal: Trans. Amer. Math. Soc. 283 (1984), 685-695
MSC: Primary 30D05; Secondary 39B10
DOI: https://doi.org/10.1090/S0002-9947-1984-0737892-8
MathSciNet review: 737892
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Abstract: Let $ f$ and $ g$ (not conformal automorphisms of the unit disk) be analytic mappings of the unit disk into itself. We say $ f$ and $ g$ commute if $ f \circ g = g \circ f$. This paper characterizes those functions $ g$ that commute with a given function $ f$. Several corollaries of this characterization give qualitative information about $ g$ given similar information about $ f$, and examples are given in each case to show the limitations of the conclusions. Some of the qualitative properties considered are univalence, fixed point sets, and whether two such $ g$ must commute with each other.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0737892-8
Keywords: Functional equation, commuting functions
Article copyright: © Copyright 1984 American Mathematical Society

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