A subcollection of algebras in a collection of Banach spaces

Let D(p, r) with 1 ^ p < oo and -oo < r < 4-oo denote the Banach space consisting of certain analytic functions f(z) defined in the unit disk. A function f(z) = ~=o a>nZ n is a member of D(p, r) if and only if (n + l) r I an \ p < . We define the norm of / in D(p, r) by By the product of two functions / and g in D(p, r) we shall mean their product as functions, i.e., [fg](z)=f(z)g(z). The purpose of this paper is to discover which of the spaces D(p, r) are algebras. THEOREM 1. // D(p, r) is an algebra, then there exists a real O with \\\ ^c||/||||flr|| for every f,geD(p,r).