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

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, $\mathbf{1}_j=(0,\ldots,0, \underbrace{1}_{j-\mbox{th place}}, 0,\ldots,0)\in\mathbb{R}^n_{+},$ $z\in\mathbb{C}^n.$