Power series of one variable with condition of logarithmical convexity

We obtain a new version of Hardy theorem about power series reciprocal to the power series with positive coefficients. We prove that if the sequence ${a_n}$, $n \geqslant K$ is logarithmically convex, then reciprocal power series has only negative coefficients bn, $n > 0$ for any $K$ if the first coefficient $a_0$ is sufficiently large. The classical Hardy theorem corresponds to the case $K = 0$. 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. In addition, we get new estimates on growth of analytic functions reciprocal to analytic functions with positive Taylor coefficients.