On normality and $\phi$-normality of holomorphic functions in several complex variables

In this paper, we investigate $\phi$-normal functions and normal families of holomorphic functions concerning total derivatives in ${\mathbb{C}}^n\mathrm{.}$ More precisely, we prove a sufficient condition for a holomorphic function defined on an open unit ball in ${\mathbb{C}}^n$ satisfying certain conditions involving higher order partial derivatives to be $\phi$-normal. Furthermore, by using differential inequalities involving total differential polynomials in ${\mathbb{C}}^n$, we establish some normality criteria for holomorphic functions in ${\mathbb{C}}^n$ which generalize some known results.