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Buck, Matthew M. (2013)
Languages: English
Types: Unknown
Subjects:

Classified by OpenAIRE into

arxiv: Mathematics::Complex Variables
Nevanlinna Theory is a powerful quantitative tool used to study the growth and behaviour of meromorphic functions on the complex plane. It plays an important role in value distribution theory, including generalising Picard's theorem that an entire function which omits two finite values is constant. The Nevanlinna Characteristic T(r,f) is a measure of a function's growth, and its associated counting function estimates how often certain values are taken. Using these tools, as well as other forms of modern complex analysis, we investigate several problems relating to differential polynomials in meromorphic functions. We also present a result relating to integer-valued meromorphic functions.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 2 Pairs of non-homogeneous linear di erential polynomials 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Proofs of the theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.1 Initial steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.2 Proof of Theorem 2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.3 Proof of Theorem 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.4 Proof of Theorem 2.2.3 . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.5 Proof of Theorem 2.2.4 . . . . . . . . . . . . . . . . . . . . . . . . 42
    • 3 Non-linear homogeneous di erential polynomials in f and f(k) 44 3.1 Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
    • 5 A normal families result for homogeneous di erential polynomials 71 5.1 Introduction and result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Proof of Theorem 5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
    • 6 Integer points of meromorphic functions 83 6.1 Introduction and result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.2 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3 Proof of Theorem 6.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.4 Appendix - How small is (d)? . . . . . . . . . . . . . . . . . . . . . . . . 94 6.5 Appendix - A thought on further work . . . . . . . . . . . . . . . . . . . . 96 Lemma 6.2.3 Let m 0, s 1, 0 < " least m distinct zeros in jzj 1)N (r; f ):
    • [1] W. Bergweiler, On the zeros of certain homogeneous di erential polynomials, (Archiv der Mathematik, 64, no. 3, 1995, pp199-202).
    • [2] W. Bergweiler, Bloch's principle (Computational Methods and Function Theory, 6, no. 1, 2006, pp77-108).
    • [3] W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families (Journal d'Analyse Mathematique, 91, 2003, pp353-367).
    • [4] W. Bergweiler and J. K. Langley, Multiplicities in Hayman's Alternative (Journal of the Australian Mathematical Society, 78, 2005, pp37-57).
    • [10] J. Clunie, On integral and meromorphic functions, (Journal of the London Mathematical Society, 37, 1962, pp17-27).
    • [30] N. Steinmetz, On the zeros of a certain Wronskian, (Bulletin of the London Mathematical Society, 20, 1988, pp525-531).
    • [31] K. Tohge, On the zeros of a homogeneous di erential polynomial of a meromorphic function, (Kodai Mathematical Journal, 16, no. 3, 1993, pp398-415).
    • [32] M. Waldschmidt, Integer valued entire functions on Cartesian products, (Number Theory in Progress, 1, 1999, pp553-576).
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