Power Series of Several Variables with the Condition of the Logarithmical Convexity of the Coefficients

Abstract

In the paper, the generalization of the Hardy theorem for power series of several variables inverse to the power series with positive coefficients is considered. It is proved that if the sequence of coefficients {as} = $${{a}_{{{{s}_{1}},{{s}_{2}}, \ldots ,{{s}_{n}}}}}$$ of a power series satisfies the condition similar to the condition of the logarithmical convexity of the coefficients and the first coefficient a0 is sufficiently large starting from a certain place ||s|| ≥ K, then all the coefficients of the inverse power series are negative, except the first coefficient. The classical Hardy theorem corresponds to the case K = 0 and n = 1. These results are applicable in the Nevanlinna–Pick theory.