Sufficient conditions of boundedness of L-index and analog of Hayman's Theorem for analytic functions in a ball

Andriy Bandura, Oleh Skaskiv

Abstract


We generalize some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic in an unit ball functions.
Our propositions give an estimate maximum modulus of the analytic function on a skeleton in polydisc with the larger radii by
maximum modulus on a skeleton in the polydisc with the lesser radii.
An analog of Hayman's Theorem for the functions is obtained.
Also we established a connection between class of analytic in ball functions of bounded $l_j$-index in every direction $\mathbf{1}_j,$ $j\in\{1,\ldots,n\}$ and
class of analytic in ball of functions of bounded $\mathbf{L}$-index in joint variables, where
$\mathbf{L}(z)=(l_1(z),\ldots,l_n(z)),$
$l_j: \mathbb{B}^n\to \mathbb{R}_+$ is continuous function,
$\mathbf{1}_j=(0,\ldots,0, \underbrace{1}_{j-\mbox{th place}}, 0,\ldots,0)\in\mathbb{R}^n_{+},$ $z\in\mathbb{C}^n.$


Keywords


analytic function; unit ball; bounded L-index in joint variables; maximum modulus; partial derivative; bounded L-index in direction

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References


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DOI: http://dx.doi.org/10.24193/subbmath.2018.4.06

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