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Results in Mathematics

, Volume 52, Issue 1–2, pp 55–61 | Cite as

An Addition to the Tumura–Clunie Theorem

  • Jürgen GrahlEmail author
Article
  • 70 Downloads

Abstract.

Exploiting Hua’s extension of the Tumura–Clunie theorem and some general results on differential polynomials, we show that if P is an arbitrary differential polynomial of degree at most n − 1 with constant coefficients, f is an entire function and ψ := f n + P[f] is nonvanishing in \({\mathbb{C}}\), then f itself has a Picard exceptional value and satisfies certain differential equations. Under additional assumptions, f has the form f(z) = e az+b . We give some counterexamples to show that these results are sharp in some sense.

Keywords.

Differential polynomial Tumura–Clunie theorem entire function Picard type theorem differential equation 

Mathematics Subject Classification (2000).

Primary 30D35 Secondary 30D20, 34M05 

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Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Institut für Mathematik Am HublandUniversität WürzburgWürzburgGermany

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