Some refinements and generalizations of Carleman's inequality

The constant is the best possible. There is a vast literature which deals with alternative proofs, various generalizations and extensions, and numerous variants and applications in analysis of inequality (1.1); see [1, 2, 3, 5, 7, 9, 8, 10, 13, 14, 15, 16, 17, 18, 19] and the references given therein. According to Hardy (see [6, Theorem 349]), Carleman’s inequality was generalized as follows. If an ≥ 0, λn ≥ 0, Λn = ∑n m=1λm (n∈N), and 0< ∑∞ n=1λnan <∞, then