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Maximum modulus in a bidisc of analytic functions of bounded $$\mathbf{L}$$-index and an analogue of Hayman’s theorem. (English) Zbl 06997370
The authors’ abstract: “We generalize some criteria of boundedness of {L}-index in joint variables for in a bidisc analytic functions. Our propositions give an estimate the maximum modulus on a skeleton in a bidisc and an estimate of $$(p+1)$$th partial derivative by lower order partial derivatives (analogue of Hayman’s theorem).”
As the authors point out, their paper continues their previous investigations from [A. I. Bandura et al., Mat. Stud. 46, No. 1, 72–80 (2016; Zbl 1373.30043)], where they introduced a definition in a polydisc defined analytic function to be of bounded {L}-index in joint variables. As a matter of fact, the authors give now new analogues of criteria of boundedness of {L}-index in joint variables.
Especially, in Theorem 4.1 the authors give a necessary maximum modulus property for in the bidisc $${\mathbb{D}}^2$$ defined analytic functions $$F$$ of bounded {L}-index in joint variables. In addition, in Theorem 4.2 they show that a related property is sufficient for an analytic function $$F$$ in $${\mathbb{D}}^2$$ to have bounded {L}-index in joint variables. Moreover, and applying these results, the authors give in Theorem 5.1 an analogue of Hayman’s theorem, by giving a necessary and sufficient condition for in $${\mathbb{D}}^2$$ analytic function to have bounded {L}-index in joint variables. This condition gives an estimate of $$(p+1)$$th partial derivative by lower order partial derivatives.

MSC:
 32A10 Holomorphic functions of several complex variables 32A17 Special families of functions of several complex variables 32A30 Other generalizations of function theory of one complex variable 30D60 Quasi-analytic and other classes of functions of one complex variable
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References:
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