The product and existence theorems for analytic functions in a polydisc of bounded $L$-index in direction

For functions analytic in the unit polydisc with bounded $L$-index in a direction there are presented three various results.The product theorem specifies that the product of analytic functions of bounded $L$-index in direction belongs to the same class. Here $L$ is some positive continuous function which is defined in the unit polydics and its value at any point from the polydisc is greater than reciprocal of distance from the point to skeleton of the polydisc.The existence theorem demonstrates the generality of the class: for every analytic function with bounded multiplicities of zeros at every slice in given direction from the unit polydisc there exists such a positive continuous function $L$ that the primary analytic function has bounded $L$-index in the same direction.And the last theorem claims that every analytic function in the unit polydisc has bounded $L$-index in any direction in any domain compactly embedded in the unit polydisc. All the results presented are generalizations to the polydisc case of known results for entire functions of several complex variables.