The Clunie–Hayman theorem in ℂ m and normality criteria concerning partial derivative

The main aim of this paper is to show that the Clunie and Hayman result [The spherical derivative of integral and meromorphic functions. Comment Math Helv. 1966;40:117–148], which is the relation between the maximum modulus and the spherical metric works in several complex variables, and using this result, we prove that a holomorphic function in Cm is of order almost one if it has a bounded spherical metric in Cm, which is an extension of the result of Chang and Zalcman [Meromorphic functions that share a set with their derivatives. J Math Anal Appl. 2008;338:1020–1028]. Furthermore, we obtain a normality criterion for a family of holomorphic functions in Cm concerning partial derivative, which extends the result of Fang and Xu [Normal families of holomorphic functions and shared values. Isr J Math. 2002;129:125–141] into higher dimensions.