A refinement to Jack’s lemma for holomorphic functions of several complex variables and their applications

The paper is devoted to two new versions of Jack’s lemma for holomorphic functions of several complex variables. The assumption in the first case are formulated in the terms of the real part of certain complex quantities related to Jack’s lemma, and in the second one in the terms of the argument of adequate quantities. All versions are presented common with their applications to the study of Bavrin’s families of s.c.v. holomorphic functions. By Bavrin’s families we mean some collections of holomorphic functions in the open unit polydisc satisfying certain geometric conditions for these functions and their Fréchet differentials of various order.