Eventual positivity for analytic functions
Let \(P\) and \(f\) be real power series with radius of convergence at least one, the former having radius of convergence exactly one and nonnegative coefficients. Under some circumstances (occurring, for example, if the coefficients of \(P\) are bounded above and below by multiples of \(1/(1+ i)^2\) and the absolute values of the coefficients of \(f\) are bounded above by a multiple of their counterparts in \(P\)), we show that \(f\) being strictly positive on the unit interval is sufficient in order that there exists \(n\) such that \(P^n f\) has nonnegative coefficients in its Maclaurin expansion (necessity of the condition is trivial). For other choices of \(P\) the corresponding results may fail, and we investigate how much is still true.
scientific article; zbMATH DE number 861916