Power series of one variable with condition of logarithmical convexity

Abstract

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 {an}, n K is logarithmically convex, then reciprocal power series has only negative coefficients bn, n > 0 for any K if the first coefficient a0 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 Pn 0 an(x¯y)n, an > 0, and reciprocal function 1 k(x,y) can be represented as power series Pn 0 bn(x¯y)n such that bn < 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¯y of the classical Hardy space H2(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.