Power series of one variable with condition of logarithmical convexity

Abstract

We obtain a new version of the Hardy theorem about power series reciprocal to the power series with positive coefficients in this work. We prove that, if the sequence {an}, n ≥ K is logarithmically convex, then the 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. These results are useful in the Nevanlinna–Pick theory. For instance, if the function k(x, y) can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{a}_{n}}{{{(x\bar {y})}}^{n}}} $$, an > 0, and the reciprocal function $$\frac{1}{{k(x,y)}}$$ can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{b}_{n}}{{{(x\bar {y})}}^{n}}} $$ such that bn 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc $$\mathbb{D}$$ with the Nevanlinna–Pick property. The reproducing kernel $$\frac{1}{{1 - x\bar {y}}}$$ of the classical Hardy space H2($$\mathbb{D}$$) is a prime example for our theorems. We also obtained new estimates on the growth of analytic functions reciprocal to analytic functions with positive Taylor coefficients.