We proved criteria of boundedness of L-index in joint variables and established a connection between the classes of entire functions of bounded l j -index in each direction e j and functions of bounded L-index in joint variables. We deduce new sufficient conditions of boundedness of L-index in joint variables. The obtained restrictions describe the behaviour of logarithmic derivative in each variable and the distribution of zeros.

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We have generalized some criteria of boundedness of L-index in joint variables for analytic functions in the unit ball, where L : B n → R n + is a continuous vector-function, B n is the unit ball in C n. One of propositions gives an estimate of the coefficients of power series expansions by a dominating homogeneous polynomial for analytic functions in the unit ball. Also we provide growth estimates of these functions. They describe the behavior of maximum modulus of analytic function on a skeleton in a polydisc by behavior of the function L. Most of our results are based on polydisc exhaustion of the unit ball. Nevertheless, we have generalized criteria of boundedness of L-index in joint variables which describe local behavior of partial derivatives on sphere in C n. The proposition uses a ball exhaustion. An analog of Hayman's theorem is applied to investigation of boundedness of L-index in joint variables for analytic solutions in the unit ball of some linear higher-order systems of PDE's. There were found sufficient conditions providing the boundedness. Growth estimates of analytic solutions in the unit ball are also obtained.

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