On the Approximation of Positive Functions by Power Original Research Article

The problem to be studied goes back to a question of Erdös and Kövari, concerning functions M(x), x ∈ R+0, which are positive and logarithmically convex in ln x. The question to find necessary and sufficient conditions for the existence of a power series N(x) = Σ cnxn, cn ≥ 0 with d1 ≤ M(x)/N(x) ≤ d2, x ≥ 0, where d1, d2 ∈ R+, has been treated by several authors. The present paper concerns a generalization of this problem regarding positive functions h(x), x ∈ R+0.