Abstract
Let f be a meromorphic function. In this chapter we describe the work of Joseph Miles, which completes the work in the last chapter concerning representations of f as the quotient of entire functions with small Nevan-linna characteristic. Miles showed that every set Z of finite λ-density is λ-balanceable. As a consequence of this and the work of Rubel and Taylor in the last chapter, there exist absolute constants A and B such that if f is any meromorphic function in the plane, then f can be expressed as f 1/f 2 where f 1 and f 2are entire functions such that T(r, f i ) ≤ AT(Br, f) for i = 1, 2 and r > 0. It is implicit in the method of proof that for any B > 1 there is a corresponding A for which the desired representation holds for all f. Miles’ proof is ingenious, intricate, and deep. Miles also showed that, in general, B cannot be chosen to be 1 by giving an example of a meromorphic f such that if f = f 1/f 2, where f 1and f 2 are entire, then T(r, f 2) ≠ O(T(r, f)). We do not give this example here.
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© 1996 Springer Science+Business Media New York
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Rubel, L.A., Colliander, J.E. (1996). The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions. In: Entire and Meromorphic Functions. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0735-1_14
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DOI: https://doi.org/10.1007/978-1-4612-0735-1_14
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94510-1
Online ISBN: 978-1-4612-0735-1
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