Analytic Functions in the Unit Ball

A concept of boundedness of the L-index in joint variables is considered for function which are analytic in a ball. There are proved criteria of boundedness of the L-index in joint variables which describe local behavior of partial derivatives and maximum modulus on a skeleton of a polydisc properties of power series expansion. Also we obtained analog of Hayman's Theorem. As a result they are applied to study linear higher-order systems of partial differential equations and to deduce sufficient conditions of boundedness of the L-index in joint variables for their analytic solutions and to estimate its growth. Mainly we used an exhaustion of the unit ball by polydiscs and by balls of lesser radii. Also growth estimates of analytic functions of bounded L-index in joint variables are obtained. Note that the concept of bounded L-index in joint variables have few advantages in the comparison with traditional approaches to study properties of analytic solutions of differential equations. In particular if an analytic solution has bounded index then it immediately yields its growth estimates an uniform distribution of its zeros a certain regular behavior of the solution etc.