Power series of several variables with condition of logarithmical convexity

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence $\{a_s\} = a_{s_1,s_2,\ldots,s_n}$ , $||s|| \geqslant K$ satisfies condition of logarithmically convexity and the first coefficient $a_0$ is sufficiently large then reciprocal power series has only negative coefficients ${b_s} = b_{s_1},s_2,\ldots,s_n$ , except $b_{0,0,\ldots,0}$ for any $K$. The classical Hardy theorem corresponds to the case $K = 0, n = 1$. 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 $D$ with Nevanlinna—Pick property. The reproducing kernel $1/(1-x\bar{y})$ of the classical Hardy space $H^2 (D)$ is a prime example for our theorems.