Abstract
By using the way of weight functions and Jensen-Hadamard's inequality, a more accurate half-discrete Mulholland's inequality with a best constant factor is given. The extension with multi-parameters, the equivalent forms as well as the operator expressions are considered.
Mathematics Subject Classication 2000: 26D15; 47A07.
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1 Introduction
Assuming that , we have the following Hilbert's integral inequality (cf. [1]):
where the constant factor π is the best possible. Moreover, for , we still have the following discrete Hilbert's inequality
with the same best constant factor π. Inequalities (1) and (2) are important in analysis and its applications (cf. [2–4]) and they still represent the field of interest to numerous mathematicians. Also we have the following Mulholland's inequality with the same best constant factor (cf. [1, 5]):
In 1998, by introducing an independent parameter λ ∈ (0, 1], Yang [6] gave an extension of (1). By generalizing the results from [6], Yang [7] gave some best extensions of (1) and (2) as follows: If is a non-negative homogeneous function of degree -λ satisfying , , , then
where the constant factor k(λ1) is the best possible. Moreover if k λ (x, y) is finite and is decreasing for x > 0(y > 0), then for we have
where, k(λ1) is still the best value. Clearly, for , inequality (4) reduces to (1), while (5) reduces to (2). Some other results about Hilbert-type inequalities are provided by [8–16].
On half-discrete Hilbert-type inequalities with the general non-homogeneous kernels, Hardy et al. provided a few results in Theorem 351 of [1]. But they did not prove that the the constant factors in the inequalities are the best possible. However Yang [17] gave a result with the kernel by introducing an interval variable and proved that the constant factor is the best possible. Recently, Yang [18] gave the following half-discrete Hilbert's inequality with the best constant factor B(λ1, λ2)(λ1 > 0, 0 < λ2 ≤ 1, λ1 + λ2 = λ):
In this article, by using the way of weight functions and Jensen-Hadamard's inequality, a more accurate half-discrete Mulholland's inequality with a best constant factor similar to (6) is given as follows:
Moreover, a best extension of (7) with multi-parameters, some equivalent forms as well as the operator expressions are also considered.
2 Some lemmas
Lemma 1 If , setting weight functions ω(n) and ϖ(x) as follows:
then we have
Proof. Applying the substitution to (8), we obtain
Since by the conditions and for fixed ,
is decreasing and strictly convex in , then by Jensen-Hadamard's inequality (cf. [1]), we find
namely, (10) follows. □ ▪
Lemma 2 Let the assumptions of Lemma 1 be fulfilled and additionally, is a non-negative measurable function in . Then we have the following inequalities:
Proof. By Hälder's inequality cf. [1] and (10), it follows
Then by Beppo Levi's theorem (cf. [19]), we have
that is, (11) follows. Still by Hölder's inequality, we have
Then by Beppo Levi's theorem, we have
and then in view of (10), inequality (12) follows. □ ▪
3 Main results
We introduce two functions
wherefrom, , and .
Theorem 3 If , then we have the following equivalent inequalities:
where the constant B(λ1, λ2) is the best possible in the above inequalities.
Proof. By Beppo Levi's theorem (cf. [19]), there are two expressions for I in (13). In view of (11), for ϖ(x) < B(λ1, λ2), we have (14). By Hälder's inequality, we have
Then by (14), we have (13). On the other-hand, assuming that (13) is valid, setting
then Jp-1 = ||a|| q , Ψ. By (11), we find J < ∝. If J = 0, then (14) is valid trivially; if J > 0, then by (13), we have
that is, (14) is equivalent to (13). By (12), since [ϖ(x)]1-q>[B(λ1, λ2)]1-q, we have (15). By Hälder's inequality, we find
Then by (15), we have (13). On the other-hand, assuming that (13) is valid, setting
then Lq-1 = ║f║ p , Φ.. By (12), we find L < ∝. If L = 0, then (15) is valid trivially; if L > 0, then by (13), we have
That is, (15) is equivalent to (13). Hence inequalities (13), (14) and (15) are equivalent.
For 0 < ε <pλ1, setting , and
if there exists a positive number k(≤ B(λ1, λ2)), such that (13) is valid as we replace B(λ1, λ2) with k, then in particular, it follows
W find
that is, A(ε) = O(1) (ε → 0+). Hence by (18) and (19), it follows
and B(λ1, λ2) ≤ k(ε → 0+). Hence, k = B(λ1, λ2) is the best value of (13).
Due to the equivalence, the constant factor B(λ1, λ2) in (14) and (15) is the best possible. Otherwise, we can imply a contradiction by (16) and (17) that the constant factor in (13) is not the best possible. □ ▪
Remark 1 (i) Define the first type half-discrete Mulholland's operator as follows: for , we define as
Then by (14), it follows and then T is a bounded operator with ║T║ ≤ B(λ1, λ2). Since by Theorem 1, the constant factor in (14) is the best possible, we have ║T║ = B(λ1, λ2).
(ii) Define the second type half-discrete Mulholland's operator as follows: For a ∈ l q, ψ, define as
Then by (15), it follows and then is a bounded operator with . Since by Theorem 1, the constant factor in (15) is the best possible, we have .
Remark 2 We set in (13), (14) and (15). (i) if , then we deduce (7) and the following equivalent inequalities:
(ii) if α = 1, then we have the following half-discrete Mulholland's inequality and its equivalent forms:
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Acknowledgements
This work is supported by the Guangdong Science and Technology Plan Item (No. 2010B010600018), and the Guangdong Modern Information Service industry Develop Particularly item 2011 (No. 13090).
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QC conceived of the study, and participated in its design and coordination. BY wrote and reformed the article. All authors read and approved the final manuscript.
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Chen, Q., Yang, B. On a more accurate half-discrete mulholland's inequality and an extension. J Inequal Appl 2012, 70 (2012). https://doi.org/10.1186/1029-242X-2012-70
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DOI: https://doi.org/10.1186/1029-242X-2012-70