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

We generalize some criteria of boundedness of 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 1(j), j is an element of {1, ... ,n} and class of analytic in ball of functions of bounded L-index in joint variables, where L(z) = (l(1)(z), ... ,l(n) (z)), l(j): B-n -> R(+ )is continuous function, i(j )=( )(0, ... ,0, [GRAPHICS] , 0, ... ,0) is an element of R+(n), z is an element of C-n.