Zalcman's lemma and normality concerning shared values of holomorphic functions and their total derivatives in several complex variables

Abstract

The paper is a continuation of a recent paper of ours, which considers the normality for a family of holomorphic functions concerning the total derivative in several complex variables. Here we extend the famous Zalcman's lemma concerning the total derivative from one complex variable to several complex variables, and obtain some normality criteria where complex values are shared by every function from the family and its $k$-th total derivative which may be seen as some generalizations of normality for holomorphic functions sharing values in one complex variable. In addition, the case of sharing analytic function by every function from the family and its $k$-th total derivative is also verified.