On The Product of Two Power Series

Abstract

We consider the product of two power series with positive coefficients: .What conditions will ensure that the coefficients wn shall be either (i) monotonic, or (ii) logarithmically convex? By the latter, we mean that wn2 ≤ Wn-1Wn+1 for n = 1, 2, … In investigating this question, which was suggested by a special example, we have found it convenient to express the conditions in terms of the ratios of un, vn to certain binomial coefficients, rather than in terms of un, vn themselves.