Application of Hayman's theorem to directional differential equations with analytic solutions in the unit ball

In this paper, we investigate analytic solutions of higher order linear non -homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded L -index in direction, where L : B-n -> R+ is a continuous function such that L(z) > beta|b| /1- |z| for all z is an element of B-n , b is an element of C-n \ {0} be a fxed direction, beta > 1 is some constant. Our proofs are based on application of inequalities from analog of Hayman's theorem for analytic functions in the unit ball. There are presented growth estimates of their solutions which contains parameters depending on the coefficients of the equations. Also we obtained sufficient conditions that every analytic solution of the equation has bounded L-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc.