Normal Families, Orders Of Zeros, And Omitted Values

Abstract

. For k a positive integer,\Omega a region in the complex plane, and ff a complex number, let M k(\Omega ; ff) denote the collection of all functions f meromorphic in\Omega such that each zero of f \Gamma ff has multiplicity at least k . Let D denote the unit disk in the complex plane. We give three conditions, each of which is sufficient for a subset F of M k (D; ff) to be a normal family. These conditions are: (1) for each compact subset K of D and for some fi ? 0 there exists a constant MK (fi) (depending on both K and fi ) such that, for each f 2 F , fz 2 K : jf(z)j ! fig ae fz 2 K : jf (k) (z)j MK (fi)g; (2) for ? 2=k and for each compact subset K of D there exists a constant C K; (depending on both K and ) such that ZZ fz2K:jf(z)j!1g jf (k) (z)j dx dy ! C K; for each f 2 F ; (3) for each compact subset K of D there is a constant M k (K) such that the product of the spherical derivatives of two or three consecutive derivatives of f , up to the derivative of order k \..