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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.