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New criteria of boundedness of $$L$$-index in joint variables for entire functions. (Ukrainian. English summary) Zbl 1374.30095
Summary: In the paper we obtain analogues of some criteria of $$L$$-index boundedness in joint variables for an entire function $$F:\mathbb C^n\to\mathbb C$$ and a function $$L(z)=(l_1(z),\dots,l_n(z)), z\in\mathbb C^n$$. Formerly such theorems were proved by M. T. Bordulyak [Mat. Stud. 4, 53–58 (1995; Zbl 1023.32500)], M. T. Bordulyak and M. M. Sheremeta [Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 9, 10–13 (1993)] for the case $$L(z)=(l_1(| z_1|),\dots,l_n(| z_n|))$$. We also present new criteria of $$L$$-index boundedness in joint variables, which describe the local behaviour of partial derivatives of an entire function in a polydisc.

##### MSC:
 30D60 Quasi-analytic and other classes of functions of one complex variable 32A10 Holomorphic functions of several complex variables 32A40 Boundary behavior of holomorphic functions of several complex variables
##### Keywords:
entire function; $$L$$-index boundedness; polydisc
Zbl 1023.32500
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