- Open Access
On an inequality suggested by Littlewood
© Gao; licensee Springer. 2011
- Received: 19 January 2011
- Accepted: 17 June 2011
- Published: 17 June 2011
We study an inequality suggested by Littlewood, our result refines a result of Bennett. 2000 Mathematics Subject Classification. Primary 26D15.
- infinite series
- non-negative sequences
The above problem was solved by Bennett , who proved the following more general result:
The special case p = 1, q = r = 2 in (1.2) leads to inequality (1.1) with K = 4 and Theorem 1.1 implies that K(p, q, r) is finite for any p ≥ 1, q > 0, r > 0 satisfying (p(q + r) - q)/p ≥ 1, a fact we shall use implicitly throughout this article. We note that Bennett only proved Theorem 1.1 for p, q, r ≥ 1 but as was pointed out in , Bennett's proof actually works for the p, q, r's satisfying the condition in Theorem 1.1. Another proof of inequality (1.2) for the special case r = q was provided by Bennett  and a close look at the proof there shows that it in fact can be used to establish Theorem 1.1.
Note that inequality (1.3) with constant 21/3 corresponds to the case p = 3, q = 2, r = 1 in (1.4). In , an even better constant was obtained but the proof there is incorrect. In [3, 6, 7], results were also obtained concerning inequality (1.2) under the extra assumption that the sequence (a n ) is non-decreasing.
The exact value of K(p, q, r) is not known in general. But note that K(1, q, 1) = 1 as it follows immediately from Theorem 1.1 that K(1, q, 1) ≤ 1 while on the other hand on setting a1 = 1, a n = 0, n ≥ 2 in (1.2) that K(1, q, 1) ≥ 1. Therefore, we may restrict our attention on (1.2) for p, r's not both being 1. In this article, it is our goal to improve the result in Theorem 1.1 in the following
On considering the values of C(p, q, δ) for δ = 1 and δ = q(p - 1)/(p(q + 1) - q), we readily deduce from Theorem 1.3 the following
We note that Theorem 1.3 together with Lemma 2.4 below shows that a bound for K(p, q, r) with p ≥ 1, q > 0, r > 0 satisfying (p(q + r) q)/p ≥ 1 can be obtained by a bound of K(p(1 + (r - 1)/q), q + r - 1, 1) and as (1.8) implies that K(p(1 + (r - 1)/q), q + r - 1, 1) ≤ (p(q + r) - q)/p, it is easy to see that the assertion of Theorem 1.1 follows from the assertions of Theorem 1.3 and Lemma 2.4.
We point out here that among the three expressions on the right-hand side of (1.8), each one is likely to be the minimum. For example, the middle one becomes the minimum when p = 2, q = 1 while it is easy to see that the last one becomes the minimum for p = q large enough and the first one becomes the minimum when q is being fixed and p → ∞. Moreover, it can happen that the minimum value in (1.5) occurs at a δ other than q(p - 1)/(p(q + 1) - q), 1. For example, when p = q = 6, the bound (1.8) gives K(6, 6, 1) ≤ 21/5 while one checks easily that C(6, 6, 1.15/1.2) < 21/5. We shall not worry about determining the precise minimum of (1.5) in this article.
We note that the special case p = 1, q = r = 2 of Theorem 1.3 leads to the following improvement on Bennet's result on the constant K of inequality (1.1):
Corollary 1.2. Inequality (1.1) holds with .
The constant is best possible.
The above lemma is the well-known Copson's inequality [8, Theorem 1.1], see also Corollary 3 to Theorem 2 of .
Inequality (2.1) follows from above upon noting that . ☐
The assertion of the lemma now follows on applying inequality (1.2) to both factors of the last expression above.☐
The assertion of the lemma now follows on applying inequality (1.2) to the second factor of the last expression above.☐
We obtain the proof of Theorem 1.3 via the following two lemmas:
Lemma 3.1. Let p ≥ 1, q > 0 be fixed. Under the same notions of Theorem 1.1, inequality (1.2) holds when r = 1 with K(p, q, 1) bounded by the right-hand side expression of (1.5).
Proof. We may assume that only finitely many a n 's are positive, say a n = 0 whenever n > N. We may also assume a1> 0. As the case p = 1 of the lemma is already contained in Theorem 1.1, we may further assume p > 1 throughout the proof. Moreover, even though the assertion that K(p, q, 1) ≤ (p(q + 1) - q)/p is already given in Theorem 1.1, we include a new proof here.
so that 0 < θ < 1 and the inequality in (3.1) follows from an application of Hölder's inequality.
where C(p, q, δ) is defined as in (1.6) and the minimum is taken over the δ 's satisfying (1.7).
by Lemma 2.2.
where C(p, q, δ) is defined as in (1.6) and the minimum is taken over the δ 's satisfying (1.7) and this completes the proof of Lemma 3.1. ☐
The assertion of the lemma for r ≥ 2 now follows on applying inequality (1.2) to the right-hand side expression above.
The assertion of the lemma for 1 ≤ r ≤ 2 now follows and this completes the proof. ☐
The bound for K(p, q, 1) follows from Lemma 3.1 and this completes the proof of Theorem 1.3.
Here K(p, q, r) is also the best possible constant such that inequality (4.1) holds for any non-negative sequence (a n ).
We point out that one can give another proof of Theorem 1.3 by studying (4.1) directly. As the general case r ≥ 1 can be reduced to the case r = 1 in a similar way as was done in the proof of Theorem 1.3 in Sect. 3, one only needs to establish the upper bound for K(p, q, 1) given in (1.5). For this, one can use an approach similar to that taken in Sect. 3, in replacing Lemmas 2.1 and 2.2 by the following lemmas. Due to the similarity, we shall leave the details to the reader.
The constant is best possible.
The above lemma is Corollary 6 to Theorem 2 of  and only the special case d = c is needed for the proof of Theorem 1.3.
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