Abstract
By means of the weight coefficients, using the idea of introduced parameters and the techniques of real analysis, a Mulholland-type inequality with a homogeneous kernel and an equivalent form are provided. Some equivalent statements of the best possible constant factor related to a few parameters are considered. Some particular inequalities and the operator expressions are obtained.
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1 Introduction
If \(p > 1,\frac{1}{p} + \frac{1}{q} = 1,a{}_{m},b_{n} \ge 0\), are such that \(0 < \sum_{m = 1}^{\infty} a_{m}^{p} < \infty,0 < \sum_{n = 1}^{\infty} b_{n}^{q} < \infty\), then we have the following Hardy–Hilbert-type inequality with the best possible constant factor pq (cf. [1], Theorem 341):
Also we have the following Mulholland inequality with the best possible constant \(\frac{\pi}{\sin (\pi /p)}\) (cf. [1], Theorem 343):
Assuming that \(f(x),g(y) \ge 0,0 < \int_{0}^{\infty} f^{p}(x)\,dx < \infty\), and \(0 < \int_{0}^{\infty} g^{q}(y)\,dy < \infty\), we have the following well known Hardy–Hilbert integral inequality (cf. [1], Theorem 316):
where the constant factor \(\frac{\pi}{\sin (\pi /p)}\) is the best possible.
Inequalities (1), (2), and (3) are important in analysis and its applications (cf. [2, 3]). Recently, some new extensions with parameters and applications were given in [4,5,6,7,8,9,10,11,12,13].
A half-discrete Hilbert-type inequality was provided in 1934 as follows (cf. [1], Theorem 351): If \(K(x)\) is decreasing on \((0,\infty ),0 < \varphi (s) < \int_{0}^{\infty} K(x)x^{s - 1} \,dx < \infty\), then we have the following inequality with the best possible constant factor \(\varphi ^{p}(\frac{1}{q})\):
Some extensions of (4) were provided in [14,15,16,17,18,19].
In 2016, by using the techniques of real analysis, Hong [20] considered some equivalent statements of the general form of (1) with the homogeneous kernel related to a few parameters and a best possible constant factor. Other similar results about the extended integral inequalities (3) were obtained in [21,22,23,24].
In this paper, following the approach of [20], by means of the weight coefficients, the idea of introduced parameters and using techniques of real analysis, a Mulholland-type inequality with the homogeneous kernel and its equivalent form are obtained in Lemma 2 and Theorem 1. The equivalent statements of the best possible constant factor related to a few parameters are considered in Theorem 2. Some particular cases and the operator expressions are provided by Remark 3 and Theorem 3.
2 An example and some lemmas
In what follows, we assume that \(p > 1,\frac{1}{p} + \frac{1}{q} = 1,s \in \mathrm{N} = \{ 1,2, \ldots \},0 < {c}_{1} \le c_{2} \le \cdots \le c_{s} < \infty, \lambda_{i} + \alpha, \lambda - \lambda_{i} + \alpha \in (0,1]\) (\(i = 1,2\)), \(a_{m},b_{n} \ge 0\), are such that
Example 1
For \(\mathrm{R}_{ +} = (0,\infty )\), we set
In view of Example 1 in [25], it follows that for \(\gamma = \lambda_{1},\lambda - \lambda_{2}\),
In particular, for \(s = 1\) (or \(c_{s} = \cdots c_{1}\)), we have \(k_{\lambda} (x,y) = \frac{(\min \{ x,c_{1}y\} )^{\alpha}}{(\max \{ x,c_{1}y\} )^{\alpha + \lambda}}\ ((x,y) \in \mathrm{R}_{ +}^{2})\), and
-
(a)
For \(\lambda_{2} + \alpha \le 1,\lambda - \lambda_{2} + \alpha > 0\), we find that \(1 - \lambda_{2} - \alpha \ge 0,1 + \lambda - \lambda_{2} + \alpha - \frac{i}{s}(\lambda + 2\alpha ) > 0\) \((i = 1, \ldots,s - 1), 1 + \lambda - \lambda_{2} + \alpha > 1 > 0\), and then, for fixed \(x > 0\),
$$\begin{aligned} &k_{\lambda} (x,y)\frac{1}{y^{1 - \lambda_{2}}}\\ &\quad = \frac{1}{y^{1 - \lambda_{2}}}\prod _{k = 1}^{s} \frac{(\min \{ c_{k}^{ - 1}x,y\} )^{\frac{\alpha}{s}}}{c_{k}^{\frac{\lambda}{s}}(\max \{ c_{k}^{ - 1}x,y\} )^{\frac{\lambda + \alpha}{s}}} \\ &\quad= \textstyle\begin{cases} \frac{1}{y^{1 - \lambda_{2} - \alpha}} \prod_{k = 1}^{s} \frac{(\min \{ c_{k}^{ - 1}x,y\} )^{\frac{\alpha}{s}}}{c_{k}^{\frac{\lambda}{s}}(c_{k}^{ - 1}x)^{\frac{\lambda + \alpha}{s}}},&0 < y \le c_{s}^{ - 1}x, \\ \frac{1}{y^{1 + \lambda - \lambda_{2} + \alpha - \frac{i}{s}(\lambda + 2\alpha )}}\frac{\prod_{k = i + 1}^{s} ( c_{k}^{ - 1}x)^{\frac{\alpha}{s}}}{\prod_{k = 1}^{s} c_{k}^{\frac{\lambda}{s}}\prod_{k = 1}^{i} ( c_{k}^{ - 1}x)^{\frac{\lambda + \alpha}{s}}},&c_{i + 1}^{ - 1}x < y \le c_{i}^{ - 1}x\ (i = 1, \ldots,s - 1), \\ \frac{1}{y^{1 + \lambda - \lambda_{2} + \alpha}} \prod_{k = 1}^{s} \frac{(c_{k}^{ - 1}x)^{\frac{\alpha}{s}}}{c_{k}^{\frac{\lambda}{s}}},&c_{1}^{ - 1}x < y < \infty \end{cases}\displaystyle \end{aligned}$$is decreasing for \(y > 0\) and strictly decreasing for \(y > c_{1}^{ - 1}x\).
-
(b)
In the same way, for \(\lambda_{1} + \alpha \le 1,\lambda - \lambda_{1} + \alpha > 0\), we find that for fixed \(y > 0\), \(k_{\lambda} (x,y)\frac{1}{x^{1 - \lambda_{1}}}\) is decreasing for \(x > 0\) and strictly decreasing for \(x > c_{s}y\).
Definition 1
Define the following weight coefficients:
Lemma 1
We have the following inequalities:
Proof
For fixed \(m \ge 2, \lambda_{2} + \alpha \le 1,\lambda - \lambda_{2} + \alpha > 0\), in view of Example 1(a), we find that the positive function
is strictly decreasing for \(t > 1\). By the decreasingness property, setting \(u = \frac{\ln m}{\ln t}\), we find that
Hence, we have (11).
In the same way, for fixed \(n \ge 2, \lambda_{1} + \alpha \le 1,\lambda - \lambda_{1} + \alpha > 0\), in view of Example 1(b), by the decreasingness property, setting \(u = \frac{\ln t}{\ln n}\), we find that
and then (12) follows. □
Lemma 2
We have the following inequality:
Proof
By Hölder’s inequality (cf. [26]), we obtain
Then by (11) and (12), in view of (5), we have (13). □
Remark 1
By (13), for \(\lambda_{1} + \lambda_{2} = \lambda\), we use the assumptions that \(\lambda_{i} + \alpha \in (0,1] \cap (0,\lambda + 2\alpha )\ (i = 1,2)\), which give
and find the following inequality:
Lemma 3
The constant factor \(k_{s}(\lambda_{1})\) in (15) is the best possible.
Proof
For \(0 < \varepsilon < \min \{ p(\lambda {}_{1} + \alpha ),q(\lambda_{2} + \alpha )\}\), we set
If there exists a positive constant \(M \le k_{s}(\lambda_{1})\), such that (15) is valid when replacing \(k_{s}(\lambda_{1})\) by M, then in particular, we have
We obtain
Since \(0 < (\lambda_{1} - \frac{\varepsilon}{p}) + \alpha \le 1,0 < (\lambda_{2} - \frac{\varepsilon}{q}) + \alpha \le 1\), by the decreasingness property, we find
For fixed \(x \ge 2\), setting \(u = \frac{\ln x}{\ln y}\), by Fubini theorem (cf. [27]), we find
Then we have
For \(\varepsilon \to 0^{ +} \), by Fatou lemma (cf. [27]), we find
Hence, \(M = k_{s}(\lambda_{1})\) is the best possible constant factor of (15). □
Remark 2
Setting \(\hat{\lambda}_{1}: = \frac{\lambda - \lambda_{2}}{p} + \frac{\lambda_{1}}{q},\hat{\lambda}_{2}: = \frac{\lambda - \lambda_{1}}{q} + \frac{\lambda_{2}}{p}\), we find \(\hat{\lambda}_{1} + \hat{\lambda}_{2} = \lambda\),
and then we rewrite (13) as follows:
Lemma 4
If the constant factor \(k_{s}^{\frac{1}{p}}(\lambda - \lambda_{2})k_{s}^{\frac{1}{q}}(\lambda_{1})\) in (13) is the best possible, then we have \(\lambda_{1} + \lambda_{2} = \lambda\).
Proof
If the constant factor \(k_{s}^{\frac{1}{p}}(\lambda - \lambda_{2})k_{s}^{\frac{1}{q}}(\lambda_{1})\) in (13) is the best possible, then so is the constant factor in (16). By (15), the unique best possible constant factor must be \(k_{s}(\hat{\lambda}_{1})\), namely,
By Hölder’s inequality (cf. [26]), we obtain
Observe that (17) keeps the form of equality if and only if there exist constants A and B such that they are not all zero and (cf. [26])
Assuming that \(A \ne 0\) (otherwise, \(B = A = 0\)), it follows that \(u^{\lambda - \lambda_{2} - \lambda_{1}} = \frac{B}{A}\) a.e. in \(\mathrm{R}_{ +} \), and then \(\lambda - \lambda_{2} - \lambda_{1} = 0\), namely, \(\lambda_{1} + \lambda_{2} = \lambda\). □
3 Main results
Theorem 1
Inequality (13) is equivalent to the following:
If the constant factor in (13) is the best possible, then so is the constant factor in (18).
Proof
Suppose that (18) is valid. By Hölder’s inequality, we have
On the other hand, assuming that (13) is valid, we set
If \(J = 0\), then (18) is naturally valid; if \(J = \infty \), then it is impossible to make (18) valid, hence \(J < \infty\). Suppose that \(0 < J < \infty\). By (13), we find
namely, (18) follows. Hence, inequality (13) is equivalent to (18).
If the constant factor in (13) is the best possible, then so is constant factor in (18). Otherwise, by (19), we would reach a contradiction that the constant factor in (13) is not the best possible. □
Theorem 2
The following statements 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 expressible as a single integral;
-
(iii)
\(k_{s}^{\frac{1}{p}}(\lambda - \lambda_{2})k_{s}^{\frac{1}{q}}(\lambda_{1})\) in (13) is the best possible constant;
-
(iv)
\(\lambda_{1} + \lambda_{2} = \lambda\).
If statement (iv) is true, namely, \(\lambda_{1} + \lambda_{2} = \lambda\), then we have (15) and the following equivalent inequality with the best possible constant factor \(k_{s}(\lambda_{1})\):
Proof
(i)⟹(ii). If \(k_{s}^{\frac{1}{p}}(\lambda - \lambda_{2})k_{s}^{\frac{1}{q}}(\lambda_{1})\) is independent of \(p,q\), then we find
namely, \(k_{s}^{\frac{1}{p}}(\lambda - \lambda_{2})k_{s}^{\frac{1}{q}}(\lambda_{1})\) is expressible by a single integral
(ii)⟹(iv). In (13), if \(k_{s}^{\frac{1}{p}}(\lambda - \lambda_{2})k_{s}^{\frac{1}{q}}(\lambda_{1})\) is expressible as a single integral \(k_{s}(\frac{\lambda - \lambda_{2}}{p} + \frac{\lambda_{1}}{q})\), then (17) keeps the form of equality. In view of the proof of Lemma 4, it follows that \(\lambda_{1} + \lambda_{2} = \lambda\).
(iv)⟹(i). If \(\lambda_{1} + \lambda_{2} = \lambda\), 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, we have (i)⟺(ii)⟺(iv).
(iii)⟹(iv). By Lemma 4, we have \(\lambda_{1} + \lambda_{2} = \lambda\).
(iv)⟹(iii). By Lemma 3, for \(\lambda_{1} + \lambda_{2} = \lambda\),
is the best possible constant factor of (13). Therefore, we have (iii) ⟺(iv).
Hence, statements (i), (ii), (iii) and (iv) are equivalent. □
Remark 3
(i) For \(\lambda = 1,\alpha = 0,\lambda_{1} = \frac{1}{q},\lambda_{2} = \frac{1}{p}\) in (15) and (20), we have the following equivalent inequalities with the best possible constant factor
(ii) For \(\lambda = 1,\alpha = 0,\lambda_{1} = \frac{1}{p},\lambda_{2} = \frac{1}{q}\) in (15) and (20), we have the following equivalent dual inequalities with the best possible constant factor
(iii) For \(p = q = 2\), both (21) and (23) reduce to
and both (22) and (24) reduce to the equivalent form of (25) as follows:
where \(\tilde{k}_{s}(\frac{1}{2})\) is the best possible constant factor given by
In particular, for \(s = 1\) in (25) and (26), we have the following equivalent inequalities with the best possible constant factor \(\frac{4}{c_{1}^{1/2}}\):
4 Operator expressions
We set the following functions:
where
Define the following real normed spaces:
Assuming that \(a = \{ a_{m}\}_{m = 2}^{\infty} \in l_{p,\varphi } \), and setting
we can rewrite (13) as
namely, \(c \in l_{p,\psi^{1 - p}}\).
Definition 2
Define a Mulholland-type operator \(T:l_{p,\varphi } \to l_{p,\psi^{1 - p}}\) as follows: For any \(a \in l_{p,\varphi }\), there exists a unique representation \(c \in l_{p,\psi^{1 - p}}\). Define the formal inner product of Ta and \(b \in l_{q,\psi} \), and the norm of T as follows:
Theorem 3
If \(a \in l_{p,\varphi },b \in l_{q,\psi}, \Vert a \Vert _{p,\varphi }, \Vert b \Vert _{q,\psi} > 0\), then we have the following equivalent inequalities:
Moreover, \(\lambda_{1} + \lambda_{2} = \lambda\), if and only if
5 Conclusions
In this paper, by means of the weight coefficients, using the idea of introduced parameters and techniques of real analysis, a Mulholland-type inequality with the homogeneous kernel and an equivalent form are given in Theorem 1. The equivalent statements of the best possible constant factor related to some parameters are considered in Theorem 2. Some particular cases and the operator expressions are obtained in Remark 3 and Theorem 3. The lemmas and theorems provide an extensive account of such 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)
Yang, B.C.: Hilbert-Type Integral Inequalities. Bentham Science Publishers Ltd., The United Arab Emirates (2009)
Yang, B.C.: On the norm of an integral operator and applications. J. Math. Anal. Appl. 321, 182–192 (2006)
Xu, J.S.: Hardy–Hilbert’s inequalities with two parameters. Adv. Math. 36(2), 63–76 (2007)
Yang, B.C.: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 325, 529–541 (2007)
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)
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. 18, 111–124 (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)
Huang, Z.X., Yang, B.C.: On a half-discrete Hilbert-type inequality similar to Mulholland’s inequality. J. Inequal. Appl. 2013, 290 (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., Chen, Q.: On a Hardy–Hilbert-type inequality with parameters. J. Inequal. Appl. 2015, 339 (2015)
Kuang, J.C.: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004)
Kuang, J.C.: Real and Functional Analysis (Continuation) (Second Volume). Higher Education Press, Beijing (2015)
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.
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BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. HM participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Mo, H., Yang, B. Equivalent properties of a Mulholland-type inequality with a best possible constant factor and parameters. J Inequal Appl 2019, 123 (2019). https://doi.org/10.1186/s13660-019-2083-x
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DOI: https://doi.org/10.1186/s13660-019-2083-x