Generalization of a theorem of Clunie and Hayman

Abstract

Clunie and Hayman proved that if the spherical derivative ‖f ′‖ of an entire function satisfies ‖f ′‖(z) = O(|z|σ) then T (r, f) = O(rσ+1). We generalize this to holomorphic curves in projective space of dimension n omitting n hyperplanes in general position. MSC 32Q99, 30D15. Introduction We consider holomorphic curves f : C → P; for the general background on the subject we refer to [7]. The Fubini–Study derivative ‖f ‖ measures the length distortion from the Euclidean metric in C to the Fubini–Study metric in P. The explicit expression is ‖f ‖ = ‖f‖ ∑ i<j |f ′ ifj − fif ′ j| , where (f0, . . . , fn) is a homogeneous representation of f (that is the fj are entire functions which never simultaneously vanish), and ‖f‖ = n