Generalization of a theorem of Clunie and Hayman

Barrett, Matthew; Eremenko, Alexandre

doi:10.1090/S0002-9939-2011-11033-5

Generalization of a theorem of Clunie and Hayman
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by Matthew Barrett and Alexandre Eremenko PDF

Proc. Amer. Math. Soc. 140 (2012), 1397-1402 Request permission

Abstract:

Clunie and Hayman proved that if the spherical derivative $\| f’\|$ of an entire function satisfies $\| f’\|(z)=O(|z|^\sigma )$, then $T(r,f)=O(r^{\sigma +1}).$ We generalize this to holomorphic curves in projective space of dimension $n$ omitting $n$ hyperplanes in general position.

References

Riccardo Benedetti and Jean-Jacques Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. MR 1070358
François Berteloot and Julien Duval, Sur l’hyperbolicité de certains complémentaires, Enseign. Math. (2) 47 (2001), no. 3-4, 253–267 (French, with French summary). MR 1876928
W. Cherry and A. Eremenko, Landau’s theorem for holomorphic curves in projective space and the Kobayashi metric on hyperplane complement, Pure and Appl. Math. Quarterly 7 (2011), no. 1, 199–221.
J. Clunie and W. K. Hayman, The spherical derivative of integral and meromorphic functions, Comment. Math. Helv. 40 (1966), 117–148. MR 192055, DOI 10.1007/BF02564366
M. Coste, An introduction to semialgebraic geometry, Inst. editoriali e poligrafici internazionali, Pisa, 2000.
Alexandre Eremenko, Brody curves omitting hyperplanes, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 565–570. MR 2731707, DOI 10.5186/aasfm.2010.3534
Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677, DOI 10.1007/978-1-4757-1945-1
Ch. Pommerenke, Estimates for normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I No. 476 (1970), 10. MR 0285710
Masaki Tsukamoto, On holomorphic curves in algebraic torus, J. Math. Kyoto Univ. 47 (2007), no. 4, 881–892. MR 2413072, DOI 10.1215/kjm/1250692296