Abstract
By applying the weight functions, the idea of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality involving the upper limit functions is given. The constant factor related to the gamma function is proved to be the best possible in a condition. A corollary about the case of the nonhomogeneous kernel and some particular inequalities are obtained.
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1 Introduction
Assuming that \(0 < \sum_{m = 1}^{\infty } a_{m}^{2} < \infty \) and \(0 < \sum_{n = 1}^{\infty } b_{n}^{2} < \infty \), we have the following Hilbert’s inequality with the best possible constant factor π (cf. [1], Theorem 315):
If \(0 < \int _{0}^{\infty } f^{2}(x)\,dx < \infty \) and \(0 < \int _{0}^{\infty } g^{2}(y) \,dy < \infty \), then we still have the following integral analogue of (1), named Hilbert’s integral inequality (cf. [1], Theorem 316):
where the constant factor π is the best possible. Inequalities (1) and (2) play an important role in analysis and its applications (cf. [2–13]).
The following half-discrete Hilbert-type inequality was provided: If \(K(x)\ (x > 0)\) is a decreasing function, \(p > 1,\frac{1}{p} + \frac{1}{q} = 1,0 < \phi (s) = \int _{0}^{\infty } K(x)x^{s - 1} \,dx < \infty \), \(f(x) \ge 0, 0 < \int _{0}^{\infty } f^{p} (x)\,dx < \infty \), then (cf. [1], Theorem 351)
In recent years, some new extensions of (3) were provided by [14–19].
In 2006, by using Euler–Maclaurin summation formula, Krnic et al. [20] gave an extension of (1) with the kernel \(\frac{1}{(m + n)^{\lambda }}\ (0 < \lambda \le 4)\). In 2019, following the result of [20], Adiyasuren et al. [21] considered an extension of (1) involving the partial sums. In 2016–2017, by applying the weight functions, Hong [22, 23] obtained some equivalent statements of the extensions of (1) and (2) with a few parameters. A few similar works were provided by [24–38].
In this paper, following the idea of [21], by using the weight functions, the way of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality with the kernel \(\frac{1}{(x{}_{1} + \cdots + x_{n})^{\lambda }}\ (\lambda > 0)\) involving the upper limit functions is given. The constant factor related to the gamma function is proved to be the best possible in a condition. A corollary about the case of the nonhomogeneous kernel and some particular inequalities are obtained.
2 Some lemmas
In what follows, we assume that \(n \in \mathrm{N}\backslash \{ 1\}: = \{ 2,3, \ldots \},p_{i},r_{i} > 1\ (i = 1, \ldots,n),\sum_{i = 1}^{n} \frac{1}{p_{i}} = 1,\lambda > 0\), \(c_{\lambda }: = (1 - \sum_{j = 1}^{n} \frac{1}{r_{j}} )\lambda, f_{i}(x)\) (\(i = 1, \ldots,n\)) are nonnegative measurable functions in \(R_{ +} = (0,\infty )\) such that \(f_{i}(x) = o(e^{tx})\ (t > 0;x \to \infty )\), and for any \(A = (0,a)\ (a > 0), f_{i} \in L^{1}(A)\), the upper limit functions are defined by \(F_{i}(x): = \int _{0}^{x} f_{i}(t)\,dt\ (x \ge 0)\), satisfying
By the definition of the gamma function, for \(x_{i} > 0\ (i = 1, \ldots,n)\), the following expression holds:
Lemma 1
For \(t > 0\), we have the following expressions:
Proof
In view of \(F_{i}(0) = 0\),we find
If \(F_{i}(\infty ) = \mathrm{constant}\), then \(\lim_{x \to \infty } \frac{F_{i}(x)}{e^{tx}} = 0\) and (5) follows; if \(F_{i}(\infty ) = \infty \), since \(f_{i}(x) = o(e^{tx})\ (x \to \infty )\), we find
and then (5) follows, too.
The lemma is proved. □
Lemma 2
For \(x_{i} > 0\ (i = 1, \ldots,n)\), the following expression holds:
Proof
We have
and then (6) follows.
The lemma is proved. □
Lemma 3
For \(n \in \mathrm{N}\backslash \{ 1\} \), defining the following weight functions:
we have
In particular, for \(\sum_{i = 1}^{n} \frac{1}{r_{i}} = 1\), we have
Proof
For \(j \ne i\), setting \(u_{j} = \frac{x_{j}}{x_{i}}\) in (7), we have
Then by Lemma 9.15 and (9.1.19) (cf. [2], p. 341–342), we obtain (8).
The lemma is proved. □
Lemma 4
We have the following inequality:
Proof
By (6) and Hölder’s integral inequality (cf. [39]), we obtain
If (11) takes the form of an equality, then there exist constants \(C_{i},C_{k}\ (i \ne k)\) such that they are not all zero and
namely, \(C_{i}x_{i}^{p_{i}(1 - \frac{\lambda }{r_{i}})}F_{i}^{p_{i}}(x_{i}) = C_{k}x_{k}^{p_{k}(1 - \frac{\lambda }{r_{k}})}F_{k}^{p_{k}}(x_{k}) = C\text{ a.e. in }\mathrm{R}_{ +} \). Assuming that \(C_{i} \ne 0\), we have
which contradicts the fact that \(0 < \int _{0}^{\infty } x_{i}^{p_{i}(1 - \frac{\lambda }{r_{i}}) - c_{\lambda } - 1} F_{i}^{p_{i}}(x_{i})\,dx_{i} < \infty \), in view of \(\int _{0}^{\infty } x_{i}^{ - c_{\lambda } - 1} \,dx_{i} = \infty \). Then by (8) and (11), we have (10).
The lemma is proved. □
Remark 1
Replacing λ (resp. \(\frac{\lambda }{r_{i}}\)) by \(\lambda + n\) (resp. \(\frac{\lambda }{r_{i}} + 1\)) in (10), we have
where we denote
3 Main results and a corollary
Theorem 1
We have the following inequality:
In particular, for \(\sum_{i = 1}^{n} \frac{1}{r_{i}} = 1\), we have
and the following inequality:
Proof
The theorem is proved. □
Theorem 2
The constant factor \(\frac{1}{\Gamma (\lambda )}\prod_{i = 1}^{n} \frac{\lambda }{r_{i}}\Gamma (\frac{\lambda }{r_{i}})\) in (14) is the best possible.
Proof
For any \(0 < \varepsilon < \lambda \min_{1 \le i \le n}\{ \frac{p_{i}}{r_{i}}\}\), we set
We obtain that \(\tilde{f}_{i}(x_{i}) = o(e^{tx_{i}})\ (t > 0;x_{i} \to \infty )\), and \(\tilde{F}_{i}(x_{i}) \equiv 0\ (0 < x_{i} \le 1)\),
If there exists a positive constant \(M(M \le \frac{1}{\Gamma (\lambda )}\prod_{i = 1}^{n} \frac{\lambda }{r_{i}}\Gamma (\frac{\lambda }{r_{i}}))\) such that (14) is valid when replacing \(\frac{1}{\Gamma (\lambda )}\prod_{i = 1}^{n} \frac{\lambda }{r_{i}}\Gamma (\frac{\lambda }{r_{i}})\) by M, then in particular, by substitution of \(f_{i}(x_{i}) = \tilde{f}_{i}(x_{i})\text{ and }F_{i}(x_{i}) = \tilde{F}_{i}(x_{i})\), we have
In view of Lemma 9.1.4 (9.1.5) in [2], we find
Hence, we have
For \(\varepsilon \to 0^{ +} \), we find
which yields that the constant factor \(M = \frac{1}{\Gamma (\lambda )}\prod_{i = 1}^{n} \frac{\lambda }{r_{i}}\Gamma (\frac{\lambda }{r_{i}})\) in (14) is the best possible.
The theorem is proved. □
Setting \(x = \frac{1}{x_{1}},f(x) = x^{\lambda - 2}f_{1}(\frac{1}{x})\) in I of (14), we have
For \(f_{1}(t) = t^{\lambda - 2}f(\frac{1}{t})\), we find
Then, replacing back x (resp. \(f(x)\)) by \(x_{1}\) (resp. \(f_{1}(x_{1})\)), we have
Corollary 1
If \(\tilde{F}_{1}(x_{1}) = \int _{0}^{x_{1}} t^{\lambda - 2}f_{1} (\frac{1}{t})\,dt\),
then we have the following inequality with the nonhomogeneous kernel:
where the constant factor \(\frac{1}{\Gamma (\lambda )}\prod_{i = 1}^{n} \frac{\lambda }{r_{i}}\Gamma (\frac{\lambda }{r_{i}})\) in (15) is the best possible.
Remark 2
(i) For \(n = 2\), (14) reduces to (cf. [40])
and (15) reduces to the following new inequality:
(ii) For \(r_{i} = p_{i}\ (i = 1, \ldots,n)\), (14) reduces to
and (15) reduces to
The constant factors in the above inequalities are the best possible.
4 Conclusions
In this paper, following the idea of [21], by the use of the weight functions, the way of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality with the kernel \(\frac{1}{(x_{1} + \cdots + x_{n})^{\lambda }}\ (\lambda > 0)\) involving the upper limit functions is given in Theorem 1. In a condition, the best possible constant factor related to the gamma function and a few parameters is proved in Theorem 2. A corollary about the case of nonhomogeneous kernel and some particular inequalities are obtained in Corollary 1 and Remark 2. The lemmas and theorems provide an extensive account of this type of inequalities.
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Acknowledgements
The authors thank the referee for his useful propose to reform the paper.
Funding
This work is supported by the National Natural Science Foundation (No. 61772140), and Characteristic innovation project of Guangdong Provincial Colleges and universities in 2020 (No. 2020KTSCX088). 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. On a multiple Hilbert-type integral inequality involving the upper limit functions. J Inequal Appl 2021, 17 (2021). https://doi.org/10.1186/s13660-021-02551-9
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DOI: https://doi.org/10.1186/s13660-021-02551-9