We investigate the characteristic properties of functions of bounded of $L$-index in joint variables which are analytic in $\mathbb{D}×\mathbb{C}$. The obtained propositions are analogs of known criterion for analytic functions in the unit ball, in the polydisc and for entire functions of several variables.They described estimates maximum modulus of the function of bounded of $L$-index in joint variables in a bidisc. Particularly, we obtained analog of Hayman’s Theorem for this functions class. The theorem has applications in analytic theory of differential equations to analytic solutions in the unit ball, in the polydisc and to entire solutions. Also we posed two unsolved problems about growth estimates for these functions and its applications to system of partial differential equations. 

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