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On a more accurate multidimensional Mulholland-type inequality
Journal of Inequalities and Applications volume 2014, Article number: 322 (2014)
In this paper, by using the way of weight coefficients and technique of real analysis, a more accurate multidimensional discrete Mulholland-type inequality with the best possible constant factor is given, which is an extension of the Mulholland inequality. The equivalent form, the operator expression with the norm as well as a few particular cases are also considered.
Suppose that , , , , , , . We have the following Hardy-Hilbert integral inequality (cf. ):
where the constant factor is the best possible. Assuming that , , , , , we have the following Hardy-Hilbert inequality with the same best constant (cf. ):
In 1998, by introducing an independent parameter , Yang  gave an extension of (1) for . Yang  gave some extensions of (1) and (2) as follows: If , , is a non-negative homogeneous function of degree −λ, with
, , ,
, , then
where the constant factor is the best possible. Moreover, if is finite and is decreasing with respect to (), then for ,
, , it follows that
where the constant factor is still the best possible.
About half-discrete Hilbert-type inequalities with the non-homogeneous kernels, Hardy et al. provided a few results in Theorem 351 of . But they did not prove that the constant factors are the best possible. However, Yang  gave a result with the kernel () by introducing a variable and proved that the constant factor is the best possible. In 2011 Yang  gave a half-discrete Hardy-Hilbert inequality with the best possible constant factor. Zhong et al. [27–33] investigated several half-discrete Hilbert-type inequalities with particular kernels. Using the way of weight functions and the techniques of discrete and integral Hilbert-type inequalities with some additional conditions on the kernel, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree and a best constant factor is obtained as follows:
In this paper, by using the way of weight coefficients and technique of real analysis, a more accurate multidimensional discrete Mulholland-type inequality with the best possible constant factor is given, which is an extension of (3). The equivalent form, the operator expression with the norm as well as a few particular cases are also considered.
2 Some lemmas
Lemma 1 If (; ), then for , ,
Proof We find
Then we have (7). □
If (N is the set of positive integers), , we set
Lemma 2 If , , is a non-negative measurable function in , and
then we have (cf. )
Lemma 3 If , , , , then
Proof For , we set
Then by the decreasing property and (10), it follows that
In the following, by mathematical induction we prove that, for any ,
For , by the Hermite-Hadamard inequality (cf. ), it follows that
and then (12) is valid. Assuming that (12) is valid for , then for s, we set
There exist constants , such that
By the assumption of mathematical induction for , we find
By Lemma 1 and the Hermite-Hadamard inequality (cf. ), we obtain
Hence we prove that (12) is valid for . Therefore, we have (11). □
Lemma 4 If C is the set of complex numbers and , () are different points, the function is analytic in except for (), and is a zero point of whose order is not less than 1, then for , we have
where . In particular, if () are all poles of order 1, setting (), then
Proof By [, p.118], we have (13). We find
In particular, since , it is obvious that
Then by (13), we obtain (14). □
Example 1 For , we set
For , , , by (14), we find
In particular, for , we obtain
Definition 1 For , , , , , , , , , , , we define two weight coefficients and as follows:
where and .
Lemma 5 Let the assumptions as in Definition 1 be fulfilled. Then:
and is indicated by (15);
for , , setting , , we have(20)
Proof By Lemma 1, the Hermite-Hadamard inequality (cf. ), (10), and (15), it follows that
Hence, we have (17). In the same way, we have (18).
By the decreasing property and (10), similarly to the proof of (11), we find
Hence, we have (20) and (21). □
3 Main results and operator expressions
Setting () and (), wherefrom
we have the following.
Theorem 1 If , , , , , , , , , then for , , , , we have
where the constant factor
is the best possible ( is indicated by (15)).
Proof By the Hölder inequality (cf. ), we have
Then by (17) and (18), we have (23).
For , , , we set
Then by (11) and (20)-(22), we obtain
If there exists a constant , such that (23) is valid when replacing by K, then we have
For , we find
and then . Hence, is the best possible constant factor of (23). □
Theorem 2 With the assumptions of Theorem 1, for , we have the following inequality with the best constant factor :
which is equivalent to (23).
Proof We set as follows:
Then it follows that . If , then (27) is trivially valid, since ; if , then it is a contradiction since the right hand side of (27) is finite. Suppose that . Then by (23), we find
namely, , and then (27) follows.
On the other hand, assuming that (27) is valid, by the Hölder inequality, we have
Then by (27), we have (23). Hence (27) and (23) are equivalent.
By the equivalency, the constant factor in (27) is the best possible. Otherwise, we would reach a contradiction by (28) that the constant factor in (23) is not the best possible. □
For , we define two real weight normal discrete spaces and as follows:
With the assumptions of Theorem 2, in view of , we have the following definition.
Definition 2 Define a multidimensional Hilbert-type operator as follows: For , there exists an unique representation , satisfying for ,
For , we define the following formal inner product of Ta and b as follows:
Then by Theorem 1 and Theorem 2, for , we have the following equivalent inequalities:
It follows that T is bounded since
Since the constant factor in (32) is the best possible, we have:
Corollary 1 With the assumptions of Theorem 2, T is defined by Definition 2, it follows that
Remark 1 (i) Setting () and (), then putting in (23) and (27), we have the following equivalent inequalities with the best constant factor :
Hence, (23) and (27) are more accurate inequalities than (35) and (36).
Putting , , () and (), in (32), we have the following new inequality:(37)
In particular, for , , in (37), we can deduce (4). Hence, (23) is an extension of (4).
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This work is supported by the National Natural Science Foundation of China (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2013KJCX0140).
The authors declare that they have no competing interests.
QC participated in the design of the study and performed the numerical analysis. BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
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Chen, Q., Yang, B. On a more accurate multidimensional Mulholland-type inequality. J Inequal Appl 2014, 322 (2014). https://doi.org/10.1186/1029-242X-2014-322
- Mulholland-type inequality
- weight coefficient
- equivalent form