Sufficient conditions of boundedness of L-index in joint variables

Abstract

A concept of boundedness of L-index in joint variables (see in Bordulyak M.T. The space of entire in C n functions of bounded L-index, Mat. Stud., 4 (1995), 53-58. (in Ukrainian)) is generalised for L(z) = (l 1 (z), . . . , l n (z)), z C n . 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.