Abstract
In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of some published results. Moreover, the equivalent forms, the operator expressions and a few particular inequalities are considered.
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1 Introduction
If \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(a_{m},b_{n} \ge0\), \(a = \{ a_{m}\}_{m = 1}^{\infty} \in l^{p}\), \(b = \{ b_{n}\}_{n = 1}^{\infty} \in l^{q}\), \(\Vert a \Vert _{p} = (\sum_{m = 1}^{\infty} a_{m}^{p} )^{\frac{1}{p}} > 0\), \(\Vert b \Vert _{q} > 0\), then we have the following Hardy-Hilbert inequality with the best possible constant \(\frac{\pi}{\sin(\pi/p)}\):
and the following Hilbert-type inequality:
with the best possible constant factor pq (cf. [1], Theorem 315, Theorem 341). Inequalities (1) and (2) are important in the analysis and its applications (cf. [1–3]).
Assuming that \(\{ \mu_{m}\}_{m = 1}^{\infty}\), \(\{ \nu_{n}\}_{n = 1}^{\infty}\) are positive sequences,
we have the following Hardy-Hilbert-type inequality (cf. [1], Theorem 321):
For \(\mu_{i} = \nu_{j} = 1\) (\(i,j \in\mathbb{N}\)), inequality (3) reduces to (1).
In 2014, Yang and Chen [4] gave the following multidimensional Hilbert-type inequality: For \(i_{0},j_{0} \in\mathbb{N}\), \(\alpha,\beta> 0\),
\(0 < \lambda_{1} + \eta\le i_{0}\), \(0 < \lambda_{2} + \eta\le j_{0}\), \(\lambda_{1} + \lambda_{2} = \lambda\), \(a_{m},b_{n} \ge0\), we have
where\(\sum_{m} = \sum_{m_{i_{0}} = 1}^{\infty}\cdots\sum_{m_{1} = 1}^{\infty}\), \(\sum_{n} = \sum_{n_{j_{0}} = 1}^{\infty}\cdots\sum_{n_{1} = 1}^{\infty}\), the series on the right-hand side are positive, and the best possible constant factor \(K_{1}^{\frac{1}{p}}K_{2}^{\frac{1}{q}}\) is indicated by
For \(i_{0} = j_{0} = \lambda= 1\), \(\eta= 0\), \(\lambda_{1} = \frac{1}{q}\), \(\lambda_{2} = \frac{1}{p}\), inequality (4) reduces to (2). The other results on this type of inequalities were provided by [5–17].
In 2015, Shi and Yang [18] gave another extension of (2) as follows:
Some other results on Hardy-Hilbert-type inequalities were given by [19–25].
In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of (4) and (5). Moreover, the equivalent forms, the operator expressions and a few particular inequalities are considered.
2 Some lemmas
If \(\mu_{i}^{(k)} > 0\) (\(k = 1,\ldots,i_{0}\); \(i = 1,\ldots,m\)), \(\nu _{j}^{(l)} > 0\) (\(l = 1,\ldots,j_{0}\); \(j = 1,\ldots,n\)), then we set
We also set functions \(\mu_{k}(t): = \mu_{m}^{(k)}\), \(t \in(m - 1,m]\) (\(m \in \mathbb{N}\)); \(\nu_{l}(t): = \nu_{n}^{(l)}\), \(t \in(n - 1,n]\) (\(n \in \mathbb{N}\)), and
It follows that \(U_{k}(m) = U_{m}^{(k)}\) (\(k = 1,\ldots,i_{0}\); \(m \in \mathbb{N}\)), \(V_{l}(n) = V_{n}^{(l)}\) (\(l = 1,\ldots,j_{0}\); \(n \in \mathbb{N}\)), and for \(x \in(m - 1,m)\), \(U_{k}'(x) = \mu_{k}(x) = \mu_{m}^{(k)}\) (\(k = 1,\ldots,i_{0}\); \(m \in\mathbb{N}\)); for \(y \in(n - 1,n)\), \(V_{l}'(y) = \nu_{l}(y) = \nu_{n}^{(l)}\) (\(l = 1, \ldots,j_{0}\); \(n \in\mathbb{N}\)).
Lemma 1
cf. [21]
Suppose that \(g(t)\) (>0) is decreasing in \(\mathbb{R}_{ +}\) and strictly decreasing in \([n_{0},\infty)\) (\(n_{0} \in \mathbb{N}\)), satisfying \(\int_{0}^{\infty} g(t)\,dt \in\mathbb{R}_{ +}\). We have
Lemma 2
If \(i_{0} \in\mathbb{N}\), \(\alpha,M > 0\), \(\Psi(u)\) is a non-negative measurable function in \((0,1]\), and
then we have the following transfer formula (cf. [26]):
Lemma 3
For \(i_{0},j_{0} \in\mathbb{N}\), \(\mu_{m}^{(k)} \ge \mu_{m + 1}^{(k)}\) (\(m \in\mathbb{N}\), \(k = 1,\ldots,i_{0}\)), \(\nu_{n}^{(l)} \ge\nu_{n + 1}^{(l)}\) (\(n \in\mathbb{N}\); \(l = 1,\ldots,j_{0}\)), \(\alpha,\beta> 0\), \(\varepsilon > 0\), we have
Proof
For \(M > i_{0}^{1/\alpha}\), we set
By (12), it follows that
Then by (10) and the above result, we find
For \(i_{0} = 1\), \(0 < \sum_{\{ m \in\mathbb{N}^{i_{0}};m_{i} = 1\}} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} < \infty\); for \(i_{0} \ge2\), \(\mu^{(i)} = \max_{m}\mu_{m}^{(i)}\), \(b = \sum_{i = 1}^{i_{0}} \mu^{(i)}\), in the same way, we find
Then we have
Hence, we have (13). In the same way, we have (14). □
Definition 1
For \(\alpha,\beta> 0\), \(0 < \lambda_{1} + \eta \le i_{0}\), \(0 < \lambda_{2} + \eta\le j_{0}\), \(\lambda_{1} + \lambda_{2} = \lambda\), we define two weight coefficients \(w(\lambda_{1},n)\) and \(W(\lambda_{2},m)\) as follows:
Example 1
With regard to the assumptions of Definition 1, we set
Then, (i) for fixed \(y > 0\),
is decreasing in \(\mathbb{R}_{ +}\) and strictly decreasing in \(([y] + 1,\infty)\). In the same way, for fixed \(x > 0\), \(k_{\lambda} (x,y)\frac{1}{y^{j_{0} - \lambda_{2}}}\) is decreasing in \(\mathbb{R}_{ +}\) and strictly decreasing in \(([x] + 1,\infty)\). We still have
(ii) For \(b > 0\), we have
Hence, for \(m - 1 < x_{i} < m\) (\(i = 1,\ldots,i_{0}\); \(m \in\mathbb {N}\)), we have \(\Vert U(m) \Vert _{\alpha} > \Vert U(x) \Vert _{\alpha}\) and
for \(m < x_{i} < m + 1\) (\(i = 1,\ldots,i_{0}\); \(m \in\mathbb{N}\)), we have \(\Vert U(m) \Vert _{\alpha} < \Vert U(x) \Vert _{\alpha}\) and
Lemma 4
With regard to the assumptions of Definition 1, (i) we have
where
(ii) for \(\mu_{m}^{(k)} \ge \mu_{m + 1}^{(k)}\) (\(m \in\mathbb{N}\)), \(\nu_{n}^{(l)} \ge\nu_{n + 1}^{(l)}\) (\(n \in\mathbb{N}\)), \(U_{\infty}^{(k)} = V_{\infty}^{(l)}\) (\(k = 1,\ldots,i_{0}\), \(l = 1, \ldots,j_{0}\)), \(0 < \lambda_{1} + \eta\le i_{0}\), \(\lambda_{2} + \eta> 0\), \(0 < \varepsilon< p\lambda_{1}\) (\(p > 1\)), we have
where, for \(c: = \max_{1 \le k \le i_{0}}\{ \mu_{1}^{(k)}\}\) (>0),
Proof
(i) By (10), (12) and Example 1(ii), for \(0 < \lambda_{1} + \eta\le i_{0}\), \(\lambda> 0\), it follows that
Hence, we have (18). In the same way, we have (19).
(ii) By (10) and in the same way, for \(c = \max_{1 \le k \le i_{0}}\{ \mu_{1}^{(k)}\}\) (>0), we have
For \(M > ci_{0}^{1/\alpha}\), we set
By (12), it follows that
Hence, we have
For \(\Vert V_{n} \Vert _{\beta} \ge ci_{0}^{1/\alpha}\), we obtain
3 Main results
Setting functions
and the following normed spaces:
we have the following.
Theorem 1
If \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(\alpha ,\beta> 0\), \(\lambda> 0\), \(0 < \lambda_{1} + \eta\le i_{0}\), \(0 < \lambda_{2} + \eta\le j_{0}\), \(\lambda_{1} + \lambda_{2} = \lambda\), then for \(a_{m},b_{n} \ge0\), \(a = \{ a_{m}\} \in l_{p, \Phi}\), \(b = \{ b_{n}\} \in l_{q, \Psi}\), \(\Vert a \Vert _{p, \Phi}, \Vert b \Vert _{q,\Psi} > 0\), we have the following equivalent inequalities:
where
Proof
By Hölder’s inequality with weight (cf. [27]), we have
Then by (18) and (19), we have (23). We set
Then we have \(J = \Vert b \Vert _{q,\Psi}^{q - 1}\). Since the right-hand side of (24) is finite, it follows \(J < \infty\). If \(J = 0\), then (24) is trivially valid; if \(J > 0\), then by (23), we have
namely (24) follows.
On the other hand, assuming that (24) is valid, by Hölder’s inequality (cf. [27]), we have
Then by (24) we have (23), which is equivalent to (24). □
Theorem 2
With regard to the assumptions of Theorem 1, if \(\mu_{m}^{(k)} \ge\mu_{m + 1}^{(k)}\) (\(m \in\mathbb{N}\)), \(\nu _{n}^{(l)} \ge \nu_{n + 1}^{(l)}\) (\(n \in\mathbb{N}\)), \(U_{\infty}^{(k)} = V_{\infty}^{(l)} = \infty\) (\(k = 1,\ldots,i_{0}\), \(l = 1,\ldots,j_{0}\)), then the constant factor \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1})\) in (23) and (24) is the best possible.
Proof
For \(0 < \varepsilon< p(\lambda_{1} + \eta )\), \(\tilde{\lambda}_{1} = \lambda_{1} - \frac{\varepsilon}{p}\) (\(\in ( - \eta, - \eta+ i_{0})\)), \(\tilde{\lambda}_{2} = \lambda_{2} + \frac {\varepsilon}{ p}\) (\(> - \eta\)), we set
Then by (13) and (14), we obtain
If there exists a constant \(K \le K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1})\) such that (23) is valid when replacing \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1})\) by K, then we have \(\varepsilon\tilde{I} < \varepsilon K \Vert \tilde{a} \Vert _{p,\Phi} \Vert \tilde{b} \Vert _{q,\Psi}\), namely
For \(\varepsilon\to0^{ +}\), we find
and then \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1}) \le K\). Hence, \(K = K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1})\) is the best possible constant factor of (23). The constant factor in (24) is still the best possible. Otherwise, we would reach a contradiction by (26) that the constant factor in (23) is not the best possible. □
4 Operator expressions
With regard to the assumptions of Theorem 2, in view of
we can set the following definition.
Definition 2
Define a multidimensional Hilbert’s operator \(T:l_{p,\Phi} \to l_{p,\Psi^{1 - p}}\) as follows: For any \(a \in l_{p,\Phi}\), there exists a unique representation \(Ta = c \in l_{p,\Psi^{1 - p}}\), satisfying
For \(b \in l_{q,\Psi}\), we define the following formal inner product of Ta and b as follows:
Then by Theorems 1 and 2, we have the following equivalent inequalities:
It follows that T is bounded with
Since the constant factor \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda _{1})\) in (30) is the best possible, we have
Remark 1
(i) For \(\mu_{i} = \nu_{j} = 1\) (\(i,j \in\mathbb {N}\)), (23) reduces to (4). Hence, (23) is an extension of (4).
(ii) For \(\eta= 0\), \(0 < \lambda_{1} \le i_{0}\), \(0 < \lambda_{2} \le j_{0}\), (23) reduces to the following inequality:
In particular, for \(i_{0} = j_{0} = \lambda= 1\), \(\lambda_{1} = \frac{1}{q}\), \(\lambda_{2} = \frac{1}{p}\), (33) reduces to (5). Hence, (33) is also an extension of (5); so is (23).
(iii) For \(\eta= - \lambda\), \(\lambda_{1},\lambda_{2} < 0\), (23) reduces to the following inequality:
(iv) For \(\lambda= 0\), \(\lambda_{2} = - \lambda_{1}\) (\(- \eta< \lambda_{1} < \eta\)), (23) reduces to the following inequality:
The above particular inequalities are also with the best possible constant factors.
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Acknowledgements
This work is supported by the National Natural Science Foundation (No. 61370186, No. 61640222), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). We are grateful for their help.
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BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. JZ participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Zhong, J., Yang, B. An extension of a multidimensional Hilbert-type inequality. J Inequal Appl 2017, 78 (2017). https://doi.org/10.1186/s13660-017-1355-6
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DOI: https://doi.org/10.1186/s13660-017-1355-6