Abstract
By applying the method of weight functions and the technique of real analysis, a multidimensional Hilbert-type integral inequality with multi-parameters and the best possible constant factor related to the gamma function is given. The equivalent forms and the reverses are obtained. We also consider the operator expressions and a few particular results related to the kernels of non-homogeneous and homogeneous.
Similar content being viewed by others
1 Introduction
Suppose that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f(x),g(y)\geq0\), \(f\in L^{p}(\mathbf{R}_{+})\), \(g\in L^{q}(\mathbf{R}_{+})\), \(\|f\|_{p}=(\int_{0}^{\infty }f^{p}(x)\,dx)^{\frac{1}{p}}>0\), \(\|g\|_{q}>0\). We have the following well-known Hardy-Hilbert integral inequality (cf. [1]):
where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is best possible. If \(a_{m},b_{n}\geq0\), \(a=\{a_{m}\}_{m=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\|a\|_{p}=(\sum_{m=1}^{\infty }a_{m}^{p})^{\frac{1}{p}}>0\), \(\|b\|_{q}>0\), then we still have the discrete variant of the above inequality with the same best constant \(\frac{\pi }{\sin(\pi/p)}\) as follows:
Inequalities (1) and (2) are important in the analysis and its applications (cf. [1–6]).
In 1998, by introducing an independent parameter \(\lambda\in(0,1]\), Yang [7] gave an extension of (1) at \(p=q=2\) with the kernel \(\frac{1}{(x+y)^{\lambda}}\). In 2009 and 2011, Yang [3, 4] gave some best extensions of (1) and (2) as follows.
If \(\lambda_{1},\lambda_{2},\lambda\in\mathbf{R}\), \(\lambda _{1}+\lambda _{2}=\lambda\), \(k_{\lambda}(x,y)\) is a non-negative homogeneous function of degree −λ, with
\(\phi(x)=x^{p(1-\lambda_{1})-1}\), \(\psi(y)=y^{q(1-\lambda _{2})-1}\), \(f(x),g(y)\geq0\),
\(g\in L_{q,\psi}(\mathbf{R}_{+})\), \(\|f\|_{p,\phi},\|g\|_{q,\psi}>0\), then we have the following inequality:
where the constant factor \(k(\lambda_{1})\) is best possible. Moreover, if \(k_{\lambda}(x,y)\) stays finite and \(k_{\lambda}(x,y)x^{\lambda _{1}-1}(k_{\lambda}(x,y)y^{\lambda_{2}-1})\) is decreasing with respect to \(x>0\) (\(y>0\)), then for \(a_{m},b_{n}\geq0\),
\(b=\{b_{n}\}_{n=1}^{\infty}\in l_{q,\psi}\), \(\|a\|_{p,\phi },\|b\|_{q,\psi }>0\), we have
where the constant factor \(k(\lambda_{1})\) is still best possible.
Clearly, for \(\lambda=1\), \(k_{1}(x,y)=\frac{1}{x+y}\), \(\lambda_{1}=\frac {1}{q}\), \(\lambda_{2}=\frac{1}{p}\), (3) reduces to (1), while (4) reduces to (2).
In 2006, Hong [8] first published a multidimensional Hilbert integral inequality by using the transfer formula, which is an extension of (3). Some other related results are given by [9–22], which provided some new methods to study these kinds of inequalities.
In this paper, by using the transfer formula and applying the method of weight functions and the technique of real analysis, we give a multidimensional Hilbert-type integral inequality with multi-parameters and the best possible constant factor related to the gamma function. The equivalent forms and the reverses are obtained. Furthermore, we also consider the operator expressions and a few particular results related to the kernels of non-homogeneous and homogeneous.
2 Some lemmas
If \(m,n\in\mathbf{N}\) (N is the set of positive integers), \(\alpha ,\beta >0\), we set
Lemma 1
If \(s\in\mathbf{N}\), \(\gamma,M>0\), \(\Psi(u)\) is a non-negative measurable function in \((0,1]\), and
then we have the following transfer formula (cf. [6]):
where \(\Gamma(\cdot)\) is the gamma function defined by
In view of (5), since \(\mathbf{R}_{+}^{s}=\lim_{M\rightarrow \infty }D_{M}\), we have
By (6), (i) for
setting \(\Psi(u)=0\) (\(u\in(0,\frac{1}{M^{\gamma}})\)), it follows that
(ii) for
setting \(\Psi(u)=0\) (\(u\in(\frac{1}{M^{\gamma}},\infty)\)), we have
Remark 1
For \(\delta\in\{-1,1\}\), \(s\in\mathbf{N}\), \(\gamma,M>0\), setting \(E_{\delta}:=\{u>0;u^{\delta}\geq\frac{1}{M^{\delta\gamma }}\}\), in view of (7) and (8), it follows that
Lemma 2
For \(\delta\in\{-1,1\}\), \(s\in\mathbf{N}\), \(\gamma, \varepsilon>0\), we have
Proof
By (9), for \(\delta\in\{-1,1\}\), it follows that
Hence, we have (10). □
Definition 1
For \(m,n\in\mathbf{N}\), \(\alpha,\beta,\lambda _{1},\lambda_{2}>0\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\eta>-1\), \(\delta \in\{-1,1\}\), \(x=(x_{1},\ldots,x_{m})\in\mathbf{R}_{+}^{m}\), \(y=(y_{1},\ldots,y_{n})\in\mathbf{R}_{+}^{n}\), we define two weight functions \(\omega(\lambda_{1},y)\) and \(\varpi(\lambda_{2},x)\) as follows:
By (6), we find
Setting \(t=v^{\delta_{1}}\) in (13), for \(\delta=\pm1\), by simplification, it follows that
Lemma 3
For \(m,n\in\mathbf{N}\), \(\alpha,\beta,\lambda _{1},\lambda _{2},\widetilde{\lambda}_{1},\widetilde{\lambda}_{2}>0\), \(\lambda _{1}+\lambda_{2}=\widetilde{\lambda}_{1}+\widetilde{\lambda }_{2}=\lambda\), \(\eta>-1\), \(\delta\in\{-1,1\}\), we have
Proof
By (14), we have (15). By the same way, we can obtain (16).
In view of (9) and (13), we find
Setting \(F(u):=\int_{0}^{u}\frac{|\ln t|^{\eta}t^{\widetilde{\lambda}_{1}-1}}{(\max\{t,1\})^{\lambda}}\,dt\) (\(u\in(0,\infty)\)), it follows that \(F(u)\) is continuous in \((0,\infty)\). Since
there exists a constant \(L>0\) such that
Then we have
namely, \(\theta_{\widetilde{\lambda}_{1}}(y)=O(\|y\|_{\beta}^{-\frac{ \widetilde{\lambda}_{1}}{2}})\) (\(y\in\mathbf{R}_{+}^{n}\)). Hence, we have (17) and (18). □
Lemma 4
As the assumptions of Definition 1, if \(p\in\mathbf {R}\backslash\{0,1\}\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f(x)=f(x_{1},\ldots ,x_{m})\geq0\), \(g(y)=g(y_{1},\ldots,y_{n})\geq0\), then (i) for \(p>1\), we have the following inequality:
(ii) for \(0< p<1\), or \(p<0\), we have the reverse of (19).
Proof
(i) For \(p>1\), by Hölder’s inequality with weight (cf. [23]), it follows that
Then by Fubini’s theorem (cf. [24]), we have
Hence, (19) follows.
(ii) For \(0< p<1\), or \(p<0\), by the reverse Hölder inequality with weight (cf. [23]), we obtain the reverse of (20). Then by Fubini’s theorem, we still can obtain the reverse of (19). □
Lemma 5
As the assumptions of Lemma 4, then (i) for \(p>1\), we have the following inequality equivalent to (19):
(ii) for \(0< p<1\), or \(p<0\), we have the reverse of (22) equivalent to the reverse of (19).
Proof
(i) For \(p>1\), by Hölder’s inequality (cf. [23]), it follows that
On the other hand, assuming that (22) is valid, we set
Then it follows that
If \(J_{1}=0\), then (19) is trivially valid; if \(J_{1}=\infty\), then by (21), (19) keeps the form of equality (=∞). Suppose that \(0< J_{1}<\infty\). By (22), we have
Dividing out \(J_{1}^{p-1}\) in the above inequality, it follows that
and then (19) follows. Hence, (19) and (22) are equivalent.
(ii) For \(0< p<1\), or \(p<0\), by the same way, we have the reverse of (22) equivalent to the reverse of (19). □
3 Main results and operator expressions
Setting functions
we have the following.
Theorem 1
Suppose that \(m,n\in\mathbf{N}\), \(\alpha,\beta ,\lambda _{1},\lambda_{2}>0\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\eta>-1\), \(\delta \in \{-1,1\}\), \(p\in\mathbf{R}\backslash\{0,1\}\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f(x)=f(x_{1},\ldots,x_{m})\geq0\), \(g(y)=g(y_{1},\ldots,y_{n})\geq 0\),
(i) For \(p>1\), we have the following equivalent inequalities with the best possible constant factor \(K(\lambda_{1})\):
where we define the constant factor as follows:
(ii) For \(0< p<1\), or \(p<0\), we still have the equivalent reverses of (24) and (25) with the same best constant factor \(K(\lambda_{1})\).
Proof
(i) For \(p>1\), by the conditions, we can prove that (20) takes the form of strict inequality. Otherwise, if (20) takes the form of equality for \(y\in\mathbf{R}_{+}^{n}\), then there exist constants A and B, which are not all zero, satisfying
If \(A=0\), then \(B=0\), which is impossible; if \(A\neq0\), then (26) reduces to
which contradicts the fact that \(0<\|f\|_{p,\Phi}<\infty\). In fact, by (9), it follows that \(\int_{\mathbf{R}_{+}^{m}}\|x\|_{\alpha }^{-m}\,dx=\infty \). Hence, (20) takes the form of strict inequality. So does (19). By (15) and (16), we have (25).
In view of (23) (putting \(\omega(\lambda_{1},y)=1\)), we still have
Then by (27) and (25), we have (24). It is evident that by Lemma 5 and the assumptions, (24) and (25) are also equivalent.
For \(0<\varepsilon<\frac{p\lambda_{1}}{2}\), we set \(\widetilde{f}(x)\), \(\widetilde{g}(y)\) as follows:
Then, for \(\widetilde{\lambda}_{1}=\lambda_{1}-\frac{\varepsilon }{p}\in(\frac{\lambda_{1}}{2},\lambda)\) (\(\subset(0,\lambda)\)), by (10) we find
and then by (17) and (18) it follows that
If there exists a constant \(K\leq K(\lambda_{1})\), such that (24) is valid when replacing \(K(\lambda_{1})\) by K, then in particular we have
and then \(K(\lambda_{1})\leq K(\varepsilon\rightarrow0^{+})\). Hence \(K=K(\lambda_{1})\) is the best possible constant factor of (24).
By the equivalency, we can prove that the constant factor \(K(\lambda_{1})\) in (25) is best possible. Otherwise, we would reach a contradiction by (27) that the constant factor \(K(\lambda_{1})\) in (24) is not best possible.
(ii) For \(0< p<1\), or \(p<0\), by the same way, we still can obtain the equivalent reverses of (24) and (25) with the same best constant factor. □
As the assumptions of Theorem 1, for \(p>1\), in view of \(J< K(\lambda _{1})\|f\|_{p,\Phi}\), we give the following definition.
Definition 2
We define a multidimensional Hilbert-type integral operator
as follows:
For \(f\in\mathbf{L}_{p,\Phi}(\mathbf{R}_{+}^{m})\), there exists a unique representation \(Tf\in\mathbf{L}_{p,\Psi^{1-p}}(\mathbf{R}_{+}^{n})\), satisfying
For \(g\in\mathbf{L}_{q,\Psi}(\mathbf{R}_{+}^{n})\), we define the following formal inner product of Tf and g as follows:
Then by Theorem 1, for \(p>1\), \(0<\|f\|_{p,\Phi},\|g\|_{q,\Psi}<\infty\), we have the following equivalent inequalities:
It follows that T is bounded with
Since the constant factor \(K(\lambda_{1})\) in (31) is best possible, we have
4 Some corollaries
We also set functions
For \(\delta=-1\) in Theorem 1, setting \(F(x)=\|x\|_{\alpha }^{\lambda }f(x)\), by simplification, we have the following.
Corollary 1
Suppose that \(m,n\in\mathbf{N}\), \(\alpha,\beta ,\lambda _{1},\lambda_{2}>0\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\eta>-1\), \(p\in \mathbf{R}\backslash\{0,1\}\), \(\frac{1}{p}+\frac{1}{q}=1\), \(F(x)=F(x_{1},\ldots,x_{m})\geq0\), \(g(y)=g(y_{1},\ldots,y_{n})\geq 0\), \(0<\|F\|_{p,\widetilde{\Phi}},\|g\|_{q,\Psi}<\infty\). (i) For \(p>1\), we have the following equivalent inequalities with the non-homogeneous kernel and the best possible constant factor \(K(\lambda_{1})\):
(ii) for \(0< p<1\), or \(p<0\), we still have the equivalent reverses of (33) and (34) with the same best constant factor \(K(\lambda_{1})\).
For \(\delta=1\) in Theorem 1, we have the following.
Corollary 2
Suppose that \(m,n\in\mathbf{N}\), \(\alpha,\beta ,\lambda _{1},\lambda_{2}>0\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\eta>-1\), \(p\in \mathbf{R}\backslash\{0,1\}\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f(x)=f(x_{1},\ldots,x_{m})\geq0\), \(g(y)=g(y_{1},\ldots,y_{n})\geq 0\), \(0<\|f\|_{p,\widehat{\Phi}},\|g\|_{q,\Psi}<\infty\). (i) For \(p>1\), we have the following equivalent inequalities with the homogeneous kernel of degree −λ and the best possible constant factor \(K(\lambda _{1})\):
(ii) for \(0< p<1\), or \(p<0\), we still have the equivalent reverses of (35) and (36) with the same best constant factor \(K(\lambda_{1})\).
References
Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934)
Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
Yang, B: Hilbert-Type Integral Inequalities. Bentham Science Publishers, Sharjah (2009)
Yang, BC: Discrete Hilbert-Type Inequalities. Bentham Science Publishers, Sharjah (2011)
Yang, BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)
Yang, BC: Hilbert-type integral operators: norms and inequalities. In: Pardalos, PM, Georgiev, PG, Srivastava, HM (eds.) Nonlinear Analysis: Stability, Approximation, and Inequalities, pp. 771-859. Springer, New York (2012)
Yang, BC: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778-785 (1998)
Hong, Y: On multiple Hardy-Hilbert integral inequalities with some parameters. J. Inequal. Appl. 2006, Article ID 94960 (2006)
Yang, BC, Brnetić, I, Krnić, M, Pečarić, JE: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 8(2), 259-272 (2005)
Krnić, M, Pečarić, JE: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 67(3-4), 315-331 (2005)
Yang, BC, Rassias, TM: On the way of weight coefficient and research for Hilbert-type inequalities. Math. Inequal. Appl. 6(4), 625-658 (2003)
Yang, BC, Rassias, TM: On a Hilbert-type integral inequality in the subinterval and its operator expression. Banach J. Math. Anal. 4(2), 100-110 (2010)
Azar, L: On some extensions of Hardy-Hilbert’s inequality and applications. J. Inequal. Appl. 2009, 546829 (2009)
Arpad, B, Choonghong, O: Best constant for certain multilinear integral operator. J. Inequal. Appl. 2006, 28582 (2006)
Kuang, JC, Debnath, L: On Hilbert’s type inequalities on the weighted Orlicz spaces. Pac. J. Appl. Math. 1(1), 95-103 (2007)
Hong, Y: On Hardy-Hilbert integral inequalities with some parameters. J. Inequal. Pure Appl. Math. 6(4), 92 (2005)
Zhong, WY, Yang, BC: On multiple Hardy-Hilbert’s integral inequality with kernel. J. Inequal. Appl. 2007, Article ID 27962 (2007)
Yang, BC, Krnić, M: On the norm of a multi-dimensional Hilbert-type operator. Sarajevo J. Math. 7(20), 223-243 (2011)
Rassias, MT, Yang, BC: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263-277 (2013)
Rassias, MT, Yang, BC: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75-93 (2013)
Yang, BC, Chen, Q: A multidimensional discrete Hilbert-type inequality. J. Math. Inequal. 8(2), 267-277 (2014)
Rassias, MT, Yang, BC: On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800-813 (2014)
Kuang, JC: Applied Inequalities. Shangdong Science Technic Press, Jinan (2004)
Kuang, JC: Introduction to Real Analysis. Hunan Education Press, Changsha (1996)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 61370186), and 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. ZH participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Huang, Z., Yang, B. A multidimensional Hilbert-type integral inequality. J Inequal Appl 2015, 151 (2015). https://doi.org/10.1186/s13660-015-0673-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0673-9