Multiplicative property of series used in the Nevanlinna-Pick problem

In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna — Pick theory. For example, if function $k(x, y)$ can be represented as power series $\sum_{n \geqslant 0} a_{n}(x\bar{y})^n$, $a_n > 0$, and reciprocal function $1/k(x, y)$ can be represented as power series $\sum_{n \geqslant 0}b_n(x\bar{y})^n$ such that $b_n < 0$, $n > 0$, then $k(x, y)$ is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc $\mathrm D$ with Nevanlinna — Pick property. The reproducing kernel $1/(1- x\bar{y})$ of the classical Hardy space $H^{2} (\mathrm D)$ is a prime example for our theorems.