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