Let us suppose that f ( s ) = ∑ 1 ∞ a n e - λ n s , (1.1) with 0<λ1<λ2<… and λn→∞, (1.2) converges for s>0. If the series converges to sum A for s = 0, then f(s)→A as s→0+, but the converse will not hold without an additional, Tauberian, hypothesis. The usual form of such a hypothesis is a condition on {an}, but in the ‘high-indices’ theorem of Hardy and Littlewood [4], the hypothesis is a condition on {λn}. They showed that if λn+1/λn⩾q>1, for n⩾1, (1.3) then the existence and finiteness of lims→0+f(s) imply the convergence of ∑an. The condition (1.3) is usually referred to as the ‘Hadamard gap condition’ and a series (1.1) such that {λn} satisfies (1.3) is called a lacunary series, although this terminology is occasionally used for series satisfying other gap conditions. Zygmund has shown [15] that if the function f(s), defined as in (1.1) by a lacunary series, is of bounded variation in (0, ∞), then ∑an converges absolutely. Clearly, the converse implication holds. The result of Zygmund has the form ∑ 1 ∞ | a n | ⩽ A q ∫ 0 ∞ | f ′ ( s ) | d s (1.4) an uppercase letter with subscripts here denoting, as it will throughout this paper, a constant determined by the indicated subscripts. The result of Zygmund and that of Hardy and Littlewood are the extreme cases, corresponding to m = 1 and m = ∞ of the inequality ∑ 1 ∞ | a n | m λ n m - 1 ⩽ A q , m ∫ 0 ∞ | f ′ ( s ) m | d s (1.5) which holds for m⩾1 and which, with more general results, is to be found in the first part of this paper [8]. In all of these results, the interval of integration can be reduced to a finite interval (0, B q , λ 1 ).