Analytic in a unit polydisc functions of bounded $L$-index in direction

The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for class analytic functions the unit polydisc, where $L$ some continuous function such that every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For from this we obtain sufficient and necessary conditions providing boundedness direction. They describe local behavior maximum modulus derivatives $F$ on slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at center circle, $t\in\mathbb{C}.$ Other criterion describes similar minimum via these functions. We proved an analog logarithmic desribing estimate derivative outside exceptional set $L$. generated union all discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, $z^0$ zero point $F$. also indicates distribution uniform over discs. In one-dimensional case, assertion many applications to theory differential equations infinite products, Blaschke product, Naftalevich-Tsuji product. Analog Hayman's Theorem deduced polydisc. It definition it possible remove factorials denominators. This allows investigate properties solutions directional equations.