Abstract
Using the way of weight functions and the idea of introducing parameters, and by means of Hermite-Hadamard’s inequality, a more accurate reverse half-discrete Hilbert-type inequality with the non-homogeneous kernel and a best constant factor is established. In addition, its best extension with parameters, the equivalent forms, as well as some particular cases are given.
Similar content being viewed by others
1 Introduction
If \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{n},b_{n}\geq0\), \(a=\{a_{n}\}_{n=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\| a\|_{p}=\{\sum_{n=1}^{\infty}a_{n}^{p}\}^{1/p}>0\), and \(\| b\|_{q}=\{\sum_{n=1}^{\infty}b_{n}^{q}\}^{1/q}>0\), then we have the following famous discrete Hilbert-type inequality (cf. [1]):
where the constant factor \([\pi/\sin(\pi/p)]^{2}\) is the best possible. Moreover the integral analogue of inequality (1) is given as follows (cf. [1]): If \(p>1\), \(\frac{1}{p}+\frac {1}{q}=1\), \(f(x),g(x)\geq0\), \(f\in L^{p}(0,\infty)\), \(g\in L^{q}(0,\infty)\), \(\| f\| _{p}=\{\int_{0}^{\infty}f^{p}(x)\,dx\}^{\frac{1}{p}}>0\), \(\| g\| _{q}=\{\int_{0}^{\infty}g^{q}(x)\,dx\}^{\frac{1}{q}}>0\), then
with the same best constant factor \([\pi/\sin(\pi/p)]^{2}\).
In 2006, Yang proved the following more accurate inequality of (1) (cf. [2]): If \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\frac {1}{2}\leq\alpha \leq1\), \(a_{n},b_{n}\geq0\), such that \(0<\| a\|_{p}<\infty\) and \(0<\| b\|_{q}<\infty\), then
where the constant factor \([\pi/\sin(\pi/p)]^{2}\) is still the best possible. Inequalities (1)-(3) are important in mathematical analysis and its applications [3]. There are lots of improvements, generalizations, and applications of inequalities (1)-(3); for more details, refer to [4–12].
At present, the research on the half-discrete Hilbert-type inequalities has gradually become in focus. We find a few results on the half-discrete Hilbert-type inequalities with the non-homogeneous kernel, which were published early (cf. [1], Theorem 351 and [13]). Recently, Yang gave some half-discrete Hilbert-type inequalities (cf. [14–17]). In 2011, Zhong proved a half-discrete Hilbert-type inequality with the non-homogeneous kernel as follows (cf. [18]): If \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0<\lambda\leq2\), \(a_{n},f(x)\geq0\), \(f(x)\) is a measurable function in \((0,\infty)\), such that \(0<\sum_{n=1}^{\infty}n^{p(1-\frac {\lambda}{2})-1}a_{n}^{p}<\infty\) and \(0<\int_{0}^{\infty}x^{q(1-\frac {\lambda}{2})-1}f^{q}(x)\,dx<\infty\), then
where the constant factor \((\frac{\pi}{\lambda})^{2}\) is the best possible.
In this article, using the way of weight functions and the idea of introducing parameters, and by means of Hadamard’s inequality, we give a more accurate reverse inequality of (4) with a best constant factor as follows: For \(p<0\), \(\frac{1}{p}+\frac{1}{q}=1\), we have
The main objective of this paper is to consider its best extension with parameters, the equivalent forms, as well as some particular cases.
2 Some lemmas
Lemma 1
If \(0<\lambda_{1}<1\), \(\lambda_{1}+\lambda_{2}=1\), then we have the following expression for the Beta function (cf. [1]):
Lemma 2
Suppose that \(\lambda>0\), \(0<\sigma<\lambda\leq1\), \(\beta _{1}\in(-\infty,\infty)\), \(0\leq\beta_{2}\leq\frac{1}{2}\), \(\delta\in \{-1,1\}\). We define the weight functions \(\omega_{\sigma}(n)\) and \(\widetilde{\omega}_{\sigma}(x)\) as follows:
Setting \(k_{\lambda}(\sigma):=\frac{1}{\lambda^{2}}[B(\frac{\sigma}{ \lambda},1-\frac{\sigma}{\lambda})]^{2}=[\frac{\pi}{\lambda\sin (\frac{\pi\sigma}{\lambda})}]^{2}\), we have the following inequalities:
Proof
Putting \(u=[(x-\beta_{1})^{\delta}(n-\beta_{2})]^{\lambda}\) in (7), by Lemma 1, we find
For fixed \(x\in(\beta_{1},\infty)\), setting
in view of \(0<\sigma<\lambda\leq1\), we find \([\frac{\ln u}{u^{\lambda}-1}]^{\prime}<0\), \([\frac{\ln u}{u^{\lambda}-1}]^{\prime\prime}>0\) (\(u>0\)) (cf. [19], Example 2.2.1) and then \(f^{\prime}(t)<0\), and \(f^{\prime \prime}(t)>0\). By the following Hermite-Hadamard inequality (cf. [20]):
and putting \(v=[(x-\beta_{1})^{\delta}(t-\beta_{2})]^{\lambda}\), it follows that
where
Since \(\lim_{v\rightarrow0^{+}}\frac{\ln v}{v-1}v^{\frac{\sigma}{2\lambda}}=\lim_{v\rightarrow\infty}\frac{\ln v}{v-1}v^{\frac {\sigma}{2\lambda}}=0\) and \(\frac{\ln v}{v-1}v^{\frac{\sigma}{2\lambda}}| _{v=1}=1\), in view of the bounded properties of continuous function, there exists a constant \(M>0\), such that \(0<\frac{\ln v}{v-1}v^{\frac{\sigma }{2\lambda}}\leq M\) (\(v\in(0,\infty)\)). For \(x\in(\beta_{1},\infty)\), we have
Hence we proved that (9) and (10) are valid. □
Lemma 3
Suppose that \(\frac{1}{p}+\frac{1}{q}=1\) (\(p\neq 0,1\)), \(0<\sigma<\lambda\leq1\), \(\beta_{1}\in(-\infty,\infty)\), \(0\leq \beta_{2}\leq\frac{1}{2}\), \(\delta\in\{-1,1\}\), \(a_{n}\geq0\), \(f(x)\) is a non-negative real measurable function in \((\beta_{1},\infty)\). Then
(i) for \(p>1\), we have the following inequalities:
where \(\omega_{\sigma}(n)\) and \(\widetilde{\omega}_{\sigma}(x)\) are defined by (7) and (8);
(ii) for \(0< p<1\) or \(p<0\), we have the reverses of (15) and (16).
Proof
(i) By (7)-(10) and the Hölder inequality [20], we have
By the Lebesgue term by term integration theorem [21], (17), and (9), we obtain
Hence (15) is valid. Using the Hölder inequality, in view of the Lebesgue term by term integration theorem and (9) again, we have
and therefore
Hence (16) is valid.
(ii) For \(0< p<1\) (\(q<0\)) or \(p<0\) (\(0< q<1\)), using the reverse Hölder inequality (cf. [20]) and in the same way, we obtain the reverses of (15) and (16). □
Definition 1
As the assumptions of Lemma 2 and Lemma 3, we define \(\phi(x):=(x-\beta_{1})^{p(1-\delta\sigma)-1}\), \(\widetilde{\phi}(x):=(1-\theta_{\lambda}(x))\phi(x)\), \(\psi(n):=(n-\beta _{2})^{q(1-\sigma)-1}\), and the following sets:
Note
If \(p>1\), then \(L_{p,\phi}(\beta_{1},\infty)\) and \(l_{q,\psi}\) are normed spaces; if \(0< p<1\) or \(p<0\), then both \(L_{p,\phi }(\beta_{1},\infty)\) and \(l_{q,\psi}\) are not normed spaces, but we still use the formal symbols in the following.
3 Main results and applications
Theorem 1
Suppose that \(p<0\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0<\sigma <\lambda\leq1\), \(\beta_{1}\in(-\infty,+\infty)\), \(0\leq\beta _{2}\leq \frac{1}{2}\), \(\delta\in\{-1,1\}\), \(f(x),a_{n}\geq0\), satisfying \(f\in L_{p,\phi}(\beta_{1},\infty)\), \(a=\{a_{n}\}_{n=1}^{\infty}\in l_{q,\psi }\), \(\| f\|_{p,\phi}>0\), \(\| a\|_{q,\psi}>0\). Then we have the following equivalent inequalities:
where the constant factor \(k_{\lambda}(\sigma)=[\frac{\pi}{\lambda \sin(\frac{\pi\sigma}{\lambda})}]^{2}\) is the best possible.
Proof
By the Lebesgue term by term integration theorem [21], we find that there are two expressions of I in (21). By (9), the reverse of (15) and \(0<\| f\|_{p,\phi}<\infty\), we have (22). By the reverse Hölder inequality, we find
Then by (22), (21) is valid. On the other hand, assuming that (21) is valid, setting
then by (21), we have
By (9), the reverse of (15), and \(0<\| f\|_{p,\phi }<\infty\), it follows that \(J>0\). If \(J=\infty\), then (22) is trivially valid; if \(J<\infty\), then \(0<\| a\|_{q,\psi }=J^{p-1}<\infty\). Thus the conditions of applying (21) are fulfilled and by (21), (26) takes a strict sign inequality. Thus we find
Hence, (22) is valid, which is equivalent to (21).
By (9), the reverse of (16) and \(0<\| a\|_{q,\psi }<\infty\), we obtain (23). By the reverse Hölder inequality again, we have
Hence (21) is valid by using (23). On the other hand, assuming that (21) is valid, setting
then by (21), we find
By (9), the reverse of (16) and \(0<\| a\|_{q,\psi }<\infty\), it follows that \(L>0\). If \(L=\infty\), then (23) is trivially valid; if \(L<\infty\), then \(0<\| f\|_{p,\phi }=L^{q-1}<\infty\), i.e. the conditions of applying (21) are fulfilled and by (30), we still have
Hence (23) is valid, which is equivalent to (21).
It follows that (21), (22), and (23) are equivalent.
We prove that the constant factor in (21) is the best possible. For \(0<\varepsilon<q\sigma\), we set \(E_{\delta}:=\{x|0<(x-\beta _{1})^{\delta }<1\}\), \(\widetilde{a}=\{\widetilde{a}_{n}\}_{n=1}^{\infty}\), and \(\widetilde{f}(x)\) as follows:
If there exists a positive number \(k\geq k_{\lambda}(\sigma)\), such that (21) is still valid as we replace \(k_{\lambda}(\sigma)\) by k, then in particular, on substitution of \(\widetilde{a}\) and \(\widetilde {f}(x)\), we have
For \(p<0\), \(0< q<1\), setting \(\widetilde{\sigma}=\sigma-\frac{\varepsilon }{q}\), we find by Lemma 2 that
Setting \(u=(x-\beta_{1})^{\delta}\), by calculation we obtain
In view of (32), (33) and \(0< q<1\), it follows that
For \(\varepsilon\rightarrow0^{+}\) in (36), we have \(k_{\lambda }(\sigma)\geq k\). Hence \(k=k_{\lambda}(\sigma)\) is the best value of (21). We confirm that the constant factor \(k_{\lambda}(\sigma)\) in (22) ((23)) is the best possible. Otherwise we can get the contradiction by (24) ((28)) that the constant factor in (21) is not the best possible. □
Theorem 2
Suppose that \(0< p<1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0<\sigma <\lambda\leq1\), \(\beta_{1}\in(-\infty,+\infty)\), \(0\leq\beta _{2}\leq \frac{1}{2}\), \(\delta\in\{-1,1\}\), \(f(x),a_{n}\geq0\), satisfying \(f\in L_{p,\widetilde{\phi}}(\beta_{1},\infty)\), \(a=\{a_{n}\} _{n=1}^{\infty }\in l_{q,\psi}\), \(\| f\|_{p,\widetilde{\phi}}>0\), \(\| a\| _{q,\psi}>0\). Then we have the following equivalent inequalities:
where the constant factor \(k_{\lambda}(\sigma)=[\frac{\pi}{\lambda \sin(\frac{\pi\sigma}{\lambda})}]^{2}\) is the best possible.
Proof
By (9), the reverse of (15) and \(0<\| f\| _{p,\widetilde{\phi}}<\infty\), we have (38). Using the reverse Hölder inequality, we obtain the reverse form of (24) as follows:
On the other hand, if (37) is valid, setting \(a_{n}\) as (25), then (26) still holds with \(0< p<1\). By (37), we have
Then by (9), the reverse of (18) and \(0<\| f\|_{p,\widetilde{\phi}}<\infty\), it follows that
If \(J=\infty\), then (38) is trivially valid; if \(J<\infty\), then \(0<\| a\|_{q,\psi}=J^{p-1}<\infty\), i.e. the conditions of applying (37) are fulfilled and by (41), we still have
Hence (38) is valid, which is equivalent to (37).
By the reverse of (16), in view of \(\widetilde{\omega}_{\sigma }(x)>k_{\lambda}(\sigma)(1-\theta_{\lambda}(x))\) and \(q<0\), we have
then (39) is valid. By the reverse Hölder inequality again, we have
Hence (37) is valid by (39). On the other hand, if (37) is valid, setting
then by the reverse of (16) and \(0<\| a\|_{q,\psi}<\infty\), it follows that
If \(\widetilde{L}=\infty\), then (39) is trivially valid; if \(\widetilde{L}<\infty\), then \(0<\| f\|_{p,\widetilde{\phi}}=\widetilde{L}^{q-1}<\infty\), i.e. the conditions of applying (37) are fulfilled and we have
Hence (39) is valid, which is equivalent to (37). It follows that (37), (38), and (39) are equivalent.
If there exists a positive number \(K\geq k_{\lambda}(\sigma)\), such that (37) is still valid as we replace \(k_{\lambda}(\sigma)\) by K, then in particular, we have
where \(\widetilde{a}=\{\widetilde{a}_{n}\}_{n=1}^{\infty}\) and \(\widetilde{f}\) are taken as (31) (\(0<\varepsilon<p(\lambda-\sigma)\)). We find
Since by (35) and setting \(u=[(x-\beta_{1})^{\delta}(n-\beta _{2})]^{\lambda}\), it follows that
by (35), (43), and (44), we have (notice that \(q<0\))
For \(\varepsilon\rightarrow0^{+}\) in (45), we obtain \(k_{\lambda }(\sigma)=[\frac{1}{\lambda}\sin( \frac{\pi\sigma}{\lambda })]^{2}\geq K\). Hence \(k_{\lambda}(\sigma)=K\) is the best value of (37). We confirm that the constant factor \(k_{\lambda}(\sigma)\) in (38) ((39)) is the best possible. Otherwise we can get the contradiction by (40) ((42)) that the constant factor in (37) is not the best possible. □
Remark 1
(i) For \(\beta_{1}=\beta_{2}=0\), \(\sigma=\frac{\lambda }{2}\), \(\delta=1\) in (21), we have the reverse of (4). In particular, for \(\lambda=1\), \(p=q=2 \) in the reverse of (4), we have
(ii) For \(\beta_{1}=0\), \(\beta_{2}=\frac{1}{2}\), \(\sigma=\frac{\lambda }{2}\), \(\delta=1\) in (21), and (22), it follows from (5) that
In particular, for \(\lambda=1\), \(p=q=2\) in (5), we obtain
which is a more accurate inequality than (46).
Remark 2
For \(\delta=-1\), \(\mu=\lambda-\sigma\) (>0) in Theorem 1, setting \(\varphi(x):=(x-\beta_{1})^{p(1-\mu)-1}\), and \(F(x):=(x-\beta_{1})^{\lambda}f(x)\), we have the following equivalent inequalities with the homogeneous kernel and the best possible constant factor \(k_{\lambda}(\sigma)\):
In the same way, for \(\delta=-1\), \(\mu=\lambda-\sigma\) (>0) in Theorem 2, setting \(\varphi(x)=(x-\beta_{1})^{p(1-\mu)-1}\), and \(F(x)=(x-\beta_{1})^{\lambda}f(x)\), we still can find some new equivalent reverse inequalities with the best possible constant factor.
References
Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1952)
Yang, B: On a more accurate Hardy-Hilbert’s type inequality and its applications. Acta Math. Sin. 49(2), 363-368 (2006)
Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
Pachpatte, BG: On some new inequalities similar to Hilbert’s inequality. J. Math. Anal. Appl. 226, 166-179 (1998)
Gao, M, Yang, B: On the extended Hilbert’s inequality. Proc. Am. Math. Soc. 126(3), 751-759 (1998)
Hu, K: On Hardy-Littlewood-Pólya inequality. Acta Math. Sci. 20, 684-687 (2000)
Lu, Z: Some new generalizations of Hilbert’s integral inequalities. Indian J. Pure Appl. Math. 33(5), 691-704 (2002)
Kuang, J, Debnath, L: On new generalizations of Hilbert’s inequality and their applications. J. Math. Anal. Appl. 245, 248-265 (2000)
Sulaiman, WT: On Hardy-Hilbert’s integral inequality. J. Inequal. Pure Appl. Math. 5(2), 25 (2004)
Zhong, W, Yang, B: A reverse Hilbert’s type integral inequality with some parameters and the equivalent forms. Pure Appl. Math. 24(2), 401-407 (2008)
Yang, B: On a Hilbert-type operator with a symmetric homogeneous kernel of -1-order and applications. J. Inequal. Appl. 2007, Article ID 47812 (2007)
Zhong, J, Yang, B: On a relation to four basic Hilbert-type integral inequalities. Appl. Math. Sci. 4(19), 923-930 (2010)
Yang, B: A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl. Math. 20(3), 319-328 (2005)
Yang, B: A half-discrete Hilbert’s inequality. J. Guangdong Univ. Educ. 31(3), 1-8 (2011)
Xie, Z, Zeng, Z: A new half-discrete Hilbert’s inequality with the homogeneous kernel of degree \(-4\mu\). J. Zhanjiang Norm. Coll. 32(6), 13-19 (2011)
Yang, B: A new half-discrete Mulholland-type inequality with parameters. Ann. Funct. Anal. 3(1), 142-150 (2012)
Chen, Q, Yang, B: On a more accurate half-discrete Mulholland’s inequality and an extension. J. Inequal. Appl. 2012, Article ID 70 (2012). doi:10.1186/1029-242X-2012-70
Zhong, W: A mixed Hilbert-type inequality and its equivalent forms. J. Guangdong Univ. Educ. 31(5), 18-22 (2011)
Yang, B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)
Kuang, J: Applied Inequalities. Shandong Science Technic Press, Jinan (2010)
Kuang, J: Real and Functional Analysis. Higher Education Press, Beijing (2002)
Acknowledgements
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).
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. AW 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 is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Wang, A., Yang, B. A more accurate reverse half-discrete Hilbert-type inequality. J Inequal Appl 2015, 85 (2015). https://doi.org/10.1186/s13660-015-0613-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0613-8