Abstract
By the use of the weight coefficients, the idea of introducing parameters and Hermite–Hadamard’s inequality, a more accurate reverse Mulholland-type inequality with parameters and the equivalent forms are given. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are also considered.
Similar content being viewed by others
1 Introduction
If \(p > 1,\frac{1}{p} + \frac{1}{q} = 1,a_{m},b_{n} \ge 0,0 < \sum_{m = 1}^{\infty } a_{m}^{p} < \infty\), and \(0 < \sum_{n = 1}^{\infty } b_{n} ^{q} < \infty \), then we have the following Hardy–Hilbert’s inequality with the best possible constant factor \(\pi /\sin (\frac{\pi }{p})\) (cf. [1], Theorem 315):
Mulholland’s inequality with the same best possible constant factor was provided as follows (cf. [1], Theorem 343, replacing \(\frac{a_{m}}{m},\frac{b _{n}}{n}\) by \(a_{m},b_{n}\)):
If \(f(x),g(y) \ge 0,0 < \int _{0}^{\infty } f^{p}(x)\,dx < \infty \), and \(0 < \int _{0}^{\infty } g^{q}(y)\,dy < \infty \), then we still have the following Hardy–Hilbert’s integral inequality (cf. [1], Theorem 316):
where the constant factor \(\pi /\sin (\frac{\pi }{p})\) is the best possible. Inequalities (1), (2), and (3) with their extensions are important in analysis and its applications (cf. [2,3,4,5,6,7,8,9,10,11,12]).
In 1934, a half-discrete Hilbert-type inequality was given as follows (cf. [1], Theorem 351): If \(K(t)\ (t > 0)\) is decreasing, \(p > 1, \frac{1}{p} + \frac{1}{q} = 1,0 < \phi (s) = \int _{0}^{\infty } K(t)t ^{s - 1} \,dt < \infty \), then we have
In the last ten years, some new extensions of (4) with their applications and the reverses were provided by [13,14,15,16,17].
In 2016, by the use of the technique of real analysis, Hong [18] considered some equivalent statements of the extensions of (1) with the best possible constant factor related to a few parameters. The other similar works about Hilbert-type integral inequalities were given by [19,20,21,22].
In this paper, following the way of [18], by the use of the weight coefficients, the idea of introducing parameters and Hermite–Hadamard’s inequality, a more accurate reverse Mulholland-type inequality with parameters and the equivalent forms are given in Theorem 1. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are considered in Theorem 2 and Remarks 1–2.
2 Some lemmas
In what follows, we assume that \(p < 0\ (0 < q < 1),\frac{1}{p} + \frac{1}{q} = 1,\xi,\eta \in [0,\frac{1}{2}], \mathrm{s} \in \mathrm{N} = \{ 1,2, \ldots \}, 0 < c_{1} \le \cdots \le c_{s}, 0 < \lambda _{i} < \lambda \le s,\lambda _{i} \le 1\ (i = 1,2)\), \(a_{m},b_{n} \ge 0\), such that
For \(\gamma = \lambda _{1},\lambda - \lambda _{2}\), we set
By Example 1 of [23], it follows that
In particular, for \(s = 1\), we have
for \(s = 2\), we have
Lemma 1
Define the following weight coefficients:
For \(\lambda _{2} \le 1\), we have
for \(\lambda _{1} \le 1\), we have
where \(\theta _{s}(\lambda _{1})\) is indicated by
Proof
Since for \(0 < \lambda _{2} \le 1,0 < \lambda \le s,y > \frac{3}{2}\), we find that
It follows that
By Hermite–Hadamard’s inequality (cf. [24]), we find
Setting \(u = \frac{\ln (m - \xi )}{\ln (y - \eta )}\), it follows that \(du = \frac{ - \ln (m - \xi )}{\ln ^{2}(y - \eta )}\frac{1}{y - \eta } \,dy\) and
namely (8) follows.
In the same way, for \(\lambda _{1} \le 1\), by Hermite–Hadamard’s inequality, we find
Setting \(u = \frac{\ln (x - \xi )}{\ln (n - \eta )}\), it follows that
By the decreasing property, we also find
Lemma 2
We have the following inequality:
Proof
By reverse Hölder’s inequality (cf. [24]), we obtain
Then, by (8) and (9), we have (11). □
Remark 1
By (11), for \(\lambda _{1} + \lambda _{2} = \lambda \), we find
and the following inequality:
In particular, for \(\xi = \eta = 0\), we have \(\tilde{\theta }_{s}( \lambda _{1},n) = O(\frac{1}{\ln ^{\lambda _{1}}n}) \in (0,1)\), and
Hence, (12) is a more accurate extension of (13).
Lemma 3
For \(0 < \varepsilon < q\lambda _{2}\), we have
Proof
There exist constants \(m,M > 0\) such that
By Hermite–Hadamard’s inequality, it follows that
Hence, (14) follows. □
Lemma 4
The constant factor \(k_{s}(\lambda _{1})\) in (12) is the best possible.
Proof
For \(0 < \varepsilon < q\lambda {}_{2}\), we set
If there exists a constant \(M \ge k_{s}(\lambda _{1})\) such that (12) is valid when replacing \(k_{s}(\lambda _{1})\) by M, then, in particular, we have
In view of (10) and (14), we obtain
By (8), setting \(\hat{\lambda }_{2} = \lambda _{2} - \frac{\varepsilon }{q} \in (0,\lambda )(\hat{\lambda }_{2} \le 1,\hat{\lambda }_{1} = \lambda _{1} + \frac{\varepsilon }{q})\), we find
Then we have
For \(\varepsilon \to 0^{ +} \), we find \(k_{s}(\lambda _{1}) \ge M\). Hence, \(M = k_{s}(\lambda _{1})\) is the best possible constant factor of (12).
Setting \(\tilde{\lambda }_{1}: = \frac{\lambda - \lambda _{2}}{p} + \frac{ \lambda _{1}}{q},\tilde{\lambda }_{2}: = \frac{\lambda - \lambda _{1}}{q} + \frac{\lambda _{2}}{p}\), we find
and we can rewrite (11) as follows:
□
Lemma 5
If \(\lambda \in (\lambda _{1} + (1 - q)\lambda _{2},(1 - p)\lambda _{1} + \lambda _{2})\), the constant factor \(k_{s}^{ \frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{\frac{1}{q}}(\lambda _{1})\) in (15) is the best possible, then we have \(\lambda = \lambda _{1} + \lambda _{2}\).
Proof
For \(\lambda _{1} + (1 - q)\lambda _{2} < \lambda \le \lambda _{1} + \lambda _{2}\), we obtain
for \(\lambda _{1} + \lambda _{2} < \lambda < (1 - p)\lambda _{1} + \lambda _{2}\), we still obtain
Hence, we have \(\tilde{\lambda }_{i} \in (0,\lambda )\ (i = 1,2)\), and then \(k _{s}(\tilde{\lambda }_{1}) \in \mathrm{R}_{ +} \).
If the constant factor \(k_{s}^{\frac{1}{p}}(\lambda - \lambda _{2})k _{s}^{\frac{1}{q}}(\lambda _{1})\) in (15) is the best possible, then, in view of (12), the unique best possible constant factor must be the form of \(k_{s}(\tilde{\lambda }_{1})\), namely
By reverse Hölder’s inequality, we find
We conclude that (16) keeps the form of equality if and only if there exist constants A and B such that they are not all zero and (cf. [24])
Assuming that \(A \ne 0\) (otherwise, \(B = A = 0\)), it follows that \(u^{\lambda - \lambda _{2} - \lambda _{1}} = \frac{A}{B}\) a.e. in \(\mathrm{R}_{ +} \), and then \(\lambda - \lambda _{2} - \lambda _{1} = 0\), namely \(\lambda = \lambda _{1} + \lambda _{2}\). □
3 Main results and particular cases
Theorem 1
Inequality (11) is equivalent to the following inequalities:
If the constant factor in (11) is the best possible, then so is the constant factor in (17) and (18).
Proof
Suppose that (17) is valid. By Hölder’s inequality, we have
Then, by (17), we obtain (11). On the other hand, assuming that (11) is valid, we set
If \(J = 0\), then (17) is naturally valid; if \(J = \infty \), then it is impossible that makes (17) valid, namely \(J < \infty \). Suppose that \(0 < J < \infty \). By (11), we have
namely (17) follows. Hence, inequality (11) is equivalent to (17).
Suppose that (18) is valid. By Hölder’s inequality, we have
Then, by (18), we obtain (11). On the other hand, assuming that (11) is valid, we set
If \(J_{1} = 0\), then (18) is naturally valid; if \(J_{1} = \infty \), then it is impossible that makes (18) valid, namely \(J_{1} < \infty \). Suppose that \(0 < J_{1} < \infty \). By (11), we have
namely (18) follows. Hence, inequality (11) is equivalent to (17) and (18).
If the constant factor in (11) is the best possible, then so is the constant factor in (17) and (18). Otherwise, by (19) (or (20)), we would reach a contradiction that the constant factor in (11) is not the best possible. □
Theorem 2
If \(\lambda \in (\lambda _{1} + (1 - q)\lambda _{2},(1 - p)\lambda _{1} + \lambda _{2})\), then the following statements (i), (ii), (iii), and (iv) are equivalent:
-
(i)
\(k_{s}^{\frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{\frac{1}{q}}( \lambda _{1})\) is independent of \(p,q\);
-
(ii)
\(k_{s}^{\frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{\frac{1}{q}}( \lambda _{1})\) is expressed by a single integral;
-
(iii)
\(k_{s}^{\frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{\frac{1}{q}}( \lambda _{1})\) in (10) is the best possible constant factor;
-
(iv)
\(\lambda = \lambda _{1} + \lambda _{2}\).
If statement (iv) follows, namely \(\lambda = \lambda _{1} + \lambda _{2}\), then we have (12) and the following equivalent inequalities with the best possible constant factor \(k_{s}(\lambda _{1})\):
Proof
(i)⇒(ii). By (i), we have
namely \(k_{s}^{\frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{\frac{1}{q}}( \lambda _{1})\) is expressed by a single integral
(ii)⇒(iv). If \(k_{s}^{\frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{ \frac{1}{q}}(\lambda _{1})\) is expressed by a convergent single integral \(k _{s}(\frac{\lambda - \lambda _{2}}{p} + \frac{\lambda _{1}}{q})\), then (16) keeps the form of equality. In view of the proof of Lemma 5, it follows that \(\lambda = \lambda _{1} + \lambda _{2}\).
(iv)⇒(i). If \(\lambda = \lambda _{1} + \lambda _{2}\), then \(k_{s}^{ \frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{\frac{1}{q}}(\lambda _{1}) = k_{s}(\lambda _{1})\), which is independent of \(p,q\). Hence, it follows that (i) ⇔ (ii) ⇔ (iv).
(iii)⇒(iv). By Lemma 5, we have \(\lambda = \lambda _{1} + \lambda _{2}\).
(iv)⇒(iii). By Lemma 4, for \(\lambda = \lambda _{1} + \lambda _{2}\), \(k_{s}^{\frac{1}{p}}(\lambda - \lambda _{2})k_{s}^{\frac{1}{q}}(\lambda _{1})( = k_{s}(\lambda _{1}))\) is the best possible constant factor of (11). Therefore, we have (iii) ⇔ (iv).
Hence, statements (i), (ii), (iii), and (iv) are equivalent. □
Remark 2
For \(\lambda = 1,\lambda _{1} = \lambda _{2} = \frac{1}{2}\),
in (12), (21), and (22), we have the following equivalent inequalities with the best possible constant factor \(\hat{k}_{s}(\frac{1}{2})\):
4 Conclusions
In this paper, by the use of the weight coefficients, the idea of introducing parameters and Hermite–Hadamard’s inequality, a more accurate reverse Mulholland-type inequality with parameters and the equivalent forms are given in Theorem 1. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are considered in Theorem 2 and Remarks 1–2. The lemmas and theorems provide an extensive account of this type of inequalities.
References
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Yang, B.C.: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)
Krnić, M., Pečarić, J.: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8(1), 29–51 (2005)
Perić, I., Vuković, P.: Multiple Hilbert’s type inequalities with a homogeneous kernel. Banach J. Math. Anal. 5(2), 33–43 (2011)
Huang, Q.L.: A new extension of Hardy–Hilbert-type inequality. J. Inequal. Appl. 2015, 397 (2015)
He, B.: A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor. J. Math. Anal. Appl. 431, 889–902 (2015)
Xu, J.S.: Hardy–Hilbert’s inequalities with two parameters. Adv. Math. 36(2), 63–76 (2007)
Xie, Z.T., Zeng, Z., Sun, Y.F.: A new Hilbert-type inequality with the homogeneous kernel of degree −2. Adv. Appl. Math. Sci. 12(7), 391–401 (2013)
Zhen, Z., Raja Rama Gandhi, K., Xie, Z.T.: A new Hilbert-type inequality with the homogeneous kernel of degree −2 and with the integral. Bull. Math. Sci. Appl. 3(1), 11–20 (2014)
Xin, D.M.: A Hilbert-type integral inequality with the homogeneous kernel of zero degree. Math. Theory Appl. 30(2), 70–74 (2010)
Azar, L.E.: The connection between Hilbert and Hardy inequalities. J. Inequal. Appl. 2013, 452 (2013)
Adiyasuren, V., Batbold, T., Krnic, M.: Hilbert-type inequalities involving differential operators, the best constants and applications. Math. Inequal. Appl. 2015, 18 (2015)
Rassias, M., Yang, B.C.: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75–93 (2013)
Yang, B.C., Krnic, M.: A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0. J. Math. Inequal. 6(3), 401–417 (2012)
Rassias, M., Yang, B.C.: A multidimensional half—discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263–277 (2013)
Rassias, M., Yang, B.C.: On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800–813 (2013)
Yang, B.C., Debnath, L.: Half-Discrete Hilbert-Type Inequalities. World Scientific, Singapore (2014)
Hong, Y., Wen, Y.: A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor. Ann. Math. 37A(3), 329–336 (2016)
Hong, Y.: On the structure character of Hilbert’s type integral inequality with homogeneous kernel and applications. J. Jilin Univ. Sci. Ed. 55(2), 189–194 (2017)
Hong, Y., Huang, Q.L., Yang, B.C., Liao, J.L.: The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications. J. Inequal. Appl. 2017, 316 (2017)
Xin, D.M., Yang, B.C., Wang, A.Z.: Equivalent property of a Hilbert-type integral inequality related to the beta function in the whole plane. J. Funct. Spaces 2018 Article ID 2691816 (2018)
Hong, Y., He, B., Yang, B.C.: Necessary and sufficient conditions for the validity of Hilbert type integral inequalities with a class of quasi-homogeneous kernels and its application in operator theory. J. Math. Inequal. 12(3), 777–788 (2018)
Yang, B.C.: On a more accurate multidimensional Hilbert-type inequality with parameters. Math. Inequal. Appl. 18(2), 429–441 (2015)
Kuang, J.C.: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004)
Funding
This work is supported by the National Natural Science Foundation (No. 61772140), Natural Science Foundation of Jishou University (No. Jd16012), and Science and Technology Planning Project Item of Guangzhou City (No. 201707010229). We are grateful for their help.
Author information
Authors and Affiliations
Contributions
BY carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. LH and HL participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
He, L., Liu, H. & Yang, B. On a more accurate reverse Mulholland-type inequality with parameters. J Inequal Appl 2019, 183 (2019). https://doi.org/10.1186/s13660-019-2139-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-019-2139-y