Abstract
By applying the weight functions, the idea of introducing parameters, and Euler–Maclaurin summation formula, a new extended half-discrete Hilbert’s inequality with the homogeneous kernel and the beta, gamma function is given. The equivalent statements of the best possible constant factor related to a few parameters are considered. As applications, a corollary about the case of the non-homogeneous kernel and some particular cases are obtained.
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1 Introduction
If \(0 < \sum_{m = 1}^{\infty } a_{m}^{2} < \infty \) and \(0 < \sum_{n = 1} ^{\infty } b_{n}^{2} < \infty \), then we have the following discrete Hilbert’s inequality with the best possible constant factor π (cf. [1], Theorem 315):
Assuming that \(0 < \int _{0}^{\infty } f^{2}(x)\,dx < \infty \) and \(0 < \int _{0}^{\infty } g^{2}(y) \,dy < \infty \), we still have the following 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 the analysis and its applications (cf. [2–13]).
We still have the following half-discrete Hilbert-type inequality (cf. [1], Theorem 351): 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
In recent years, some new extensions of (3) have been provided by [14–19].
In 2006, by using the Euler–Maclaurin summation formula, Krnic et al. [20] gave an extension of (1) with the kernel \(\frac{1}{(m + n)^{ \lambda }}\) (\(0 < \lambda \le 4\)); and in 2019, according to the results 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] considered some equivalent statements of the extensions of (1) and (2) with a few parameters. Some similar interested works were provided by [24–26].
In this paper, according to the way of [21, 22], by the use of the weight functions, the idea of introducing parameters and the Euler–Maclaurin summation formula, a new extended half-discrete Hilbert’s inequality with the homogeneous kernel \(\frac{1}{(x + n)^{\lambda }}\) (\(0 < \lambda \le 26\)) and the beta, gamma function is given. The equivalent statements of the best possible constant factor related to a few parameters are considered. As applications, a corollary about the case of non-homogeneous kernel and some particular cases are also obtained.
2 Some lemmas
In what follows, we assume that \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(\lambda \in ( - 2,26]\), \(\lambda _{2} \in ( - 1,1]\), \(\lambda _{1},\lambda _{2} \in ( - 1,\lambda + 1)\), \(f(x) \ge 0\), \(f \in L^{1}(R_{ +} )\) (\(R _{ +} = (0,\infty )\)), \(a_{n} \ge 0\) (\(n \in \mathbb{N} = \{ 1,2, \ldots \} \)), \(\{ a_{n}\}_{n = 1}^{\infty } \in l^{1}\),
such that
By the definition of the gamma function, for \(\lambda ,x > 0\), \(n \in \mathbf{N}\), the following equality holds:
Lemma 1
For \(t > 0\), we have
Proof
Since \(\{ a_{n}\}_{n = 1}^{\infty } \in l^{1}\), we find \(\lim_{n \to \infty } A_{n} = \sum_{i = 1}^{\infty } a_{i} \in [0, \infty )\). Using Abel’s summation by parts formula and the inequality \(1 - e^{ - t} \le t\), we have (cf. [21])
namely, inequality (5) follows. For \(f \in L^{1}(R_{ +} )\), \(F(0) = 0\), \(F( \infty ) \in [0,\infty )\), we find
and then expression (6) follows. □
Lemma 2
For \(1 < s \le 28\), \(\sigma \in (0,2] \cap (0,s)\), define the following weight function:
We have the following inequality:
Proof
We set function \(g(t): = \frac{t^{\sigma - 1}}{(x + t)^{s}}\) (\(t > 0\)). Using the Euler–Maclaurin summation formula (cf. [20]), for \(\rho (t): = t - [t] - \frac{1}{2}\), we have
We obtain \(- \frac{1}{2}g(1) = \frac{ - 1}{2(x + 1)^{s}}\). Integrating by parts, it follows that
Since we find
and for \(0 < \sigma \le 2\), \(1 < s \le 28\),
still by the Euler–Maclaurin summation formula (cf. [20]), for \(s + 1 - \sigma > 0\), we have
Hence, we have \(h(\sigma ,s) > \frac{h_{1}(\sigma ,s)}{(x + 1)^{s}} + \frac{sh_{2}(\sigma ,s)}{(x + 1)^{s + 1}} + \frac{s(s + 1)h_{3}( \sigma ,s)}{(x + 1)^{s + 2}}\), where
and \(h_{3}(\sigma ,s): = \frac{1}{\sigma (\sigma + 1)(\sigma + 2)} - \frac{s + 2}{720}\).
For \(s \in (1,28]\), \(\frac{s}{720} < \frac{1}{24}\), \(\sigma \in (0,2]\), it follows that
In fact, setting \(g(\sigma ): = 24 - 20\sigma + 7\sigma ^{2} - \sigma ^{3}(\sigma \in (0,2])\), we obtain
and then \(g(\sigma ) \ge g(2) = 4 > 0\) (\(\sigma \in (0,2]\)).
We still find that \(h_{2}(\sigma ,s) > \frac{1}{6} - \frac{1}{12} - \frac{30}{360} = 0\) and \(h_{3}(\sigma ,s) \ge \frac{1}{24} - \frac{30}{720} = 0\). Hence, we have \(h(\sigma ,s) > 0\), and then
namely, (8) follows. □
Lemma 3
Suppose that \(s \in (1,28]\), \(\mu ,\sigma \in (1,s)\), \(\sigma \in (0,2]\),
We have the following inequality:
Proof
For \(n \in \mathbf{N}\), setting \(x = nu\), we obtain the following weight function:
By Hölder’s inequality (cf. [27]), we obtain
Then, by (8) and (10), we have (9). □
Remark 1
For \(s = \lambda + 2\), \(\lambda \in ( - 1,26]\), \(\lambda _{1} = \mu - 1 \in (0,\lambda + 1)\), \(\lambda _{2} = \sigma - 1 \in (0,1] \cap (0,\lambda + 1)\), we can reduce (9) as follows:
3 Main results
Theorem 1
If \(\lambda \in (0,26]\), \(\lambda _{1},\lambda _{2} \in (0,\lambda + 1)\), \(\lambda _{2} \in (0,1]\), then we have the following inequality:
In particular, for \(\lambda _{1} + \lambda _{2} = \lambda \), we also have
where the constant factor \(\lambda _{1}\lambda _{2}B(\lambda _{1},\lambda _{2})\)is the best possible.
Proof
Using (4), (5), and (6), we find
In view of (11), we have (12).
In the case of \(\lambda _{1} + \lambda _{2} = \lambda \), we find
and then (13) follows.
For any \(0 < \varepsilon < \min \{ p\lambda _{1},q\lambda _{2}\}\), we set
We obtain from \(\lambda _{1},\lambda _{2} \in (0,\lambda + 1)\), \(\lambda _{2} \in (0,1]\), and \(0 < \varepsilon < \min \{ p\lambda _{1},q\lambda _{2}\}\) that \(\tilde{F}(x) = 0\) (\(0 < x < 1\)),
If there exists a positive constant M (\(M \le \lambda _{1}\lambda _{2}B( \lambda _{1},\lambda _{2})\)) such that (13) is valid when replacing \(\lambda _{1}\lambda _{2}B(\lambda _{1},\lambda _{2})\) by M, then in particular, by substitution of \(f(x) = \tilde{f}(x)\) and \(a_{n} = \tilde{a}_{n}\), we have
We find
In view of Fubini’s theorem (cf. [28]), it follows that
So we obtain
For \(\varepsilon \to 0^{ +} \) in the above inequality, in view of the continuity of the beta function, we find \(B(\lambda _{1},\lambda _{2}) \le \frac{M}{\lambda _{1}\lambda _{2}}\), namely \(\lambda _{1}\lambda _{2}B( \lambda _{1},\lambda _{2}) \le M\). Hence \(M = \lambda _{1}\lambda _{2}B( \lambda _{1},\lambda _{2})\) is the best possible constant factor of (13). □
Remark 2
We set \(\hat{\lambda }_{1}: = \frac{\lambda + 1 - \lambda _{2}}{p} + \frac{\lambda _{1} + 1}{q} - 1\), \(\hat{\lambda }_{2}: = \frac{\lambda _{2} + 1}{p} + \frac{\lambda + 1 - \lambda _{1}}{q} - 1\). It follows that
\(0 < \hat{\lambda }_{1}, \hat{\lambda }_{2} < \lambda + 1\), and then we reduce (12) as follows:
Theorem 2
Assuming that \(\lambda \in (0,26]\), \(\lambda _{1}, \lambda _{2} \in (0,\lambda + 1)\), \(\lambda _{2} \in (0,1]\), if the constant factor
in (15) is the best possible, then \(\lambda _{1} + \lambda _{2} = \lambda\).
Proof
As regards to the assumptions, we find \(0 < \hat{\lambda }_{1}\), \(\hat{\lambda }_{2} < \lambda + 1\). By (13), the unified best possible constant factor in (15) must be of the following form:
namely, it follows that
By Hölder’s inequality (cf. [27]), we obtain
We observe 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 \(Au^{ \lambda - \lambda _{2}} = Bu^{\lambda _{1}}\) a.e. in \(R_{ +} \). Assuming that \(A \ne 0\), it follows that \(u^{\lambda - \lambda _{2} - \lambda _{1}} = \frac{B}{A}\) a.e. in \(R_{ +} \), namely \(\lambda - \lambda _{2} - \lambda _{1} = 0\), and then \(\lambda _{1} + \lambda _{2} = \lambda \). □
Theorem 3
If \(\lambda \in (0,26]\), \(\lambda _{1},\lambda _{2} \in (0,\lambda + 1)\), \(\lambda _{2} \in (0,1]\), then the following statements are equivalent:
- (i)
\(B^{\frac{1}{p}}(\lambda _{2} + 1,\lambda + 1 - \lambda _{2})B ^{\frac{1}{q}}(\lambda _{1} + 1,\lambda + 1 - \lambda _{1})\)is independent ofp, q;
- (ii)
\(B^{\frac{1}{p}}(\lambda _{2} + 1,\lambda + 1 - \lambda _{2})B^{\frac{1}{q}}(\lambda _{1} + 1,\lambda + 1 - \lambda _{1})\)is expressible as a single integral;
- (iii)
\(\lambda _{1} + \lambda _{2} = \lambda \);
- (iv)
The constant factor
$$ \frac{\varGamma (\lambda + 2)}{\varGamma (\lambda )}B^{\frac{1}{p}}(\lambda _{2} + 1,\lambda + 1 - \lambda _{2})B^{\frac{1}{q}}(\lambda _{1} + 1, \lambda + 1 - \lambda _{1}) $$in (12) is the best possible.
Proof
(i) ⇒ (ii). We find
which is a single integral. (ii) ⇒ (iii). Suppose that \(B ^{\frac{1}{p}}(\lambda _{2} + 1,\lambda + 1 - \lambda _{2})B^{ \frac{1}{q}}(\lambda _{1} + 1,\lambda + 1 - \lambda _{1})\) is expressible as a single integral \(\int _{0}^{\infty } \frac{1}{(1 + u)^{\lambda + 2}}u^{\frac{\lambda + 1 - \lambda _{2}}{p} + \frac{\lambda _{1} + 1}{q} - 1}\,du\). Then (16) keeps the form of equality. By the proof of Theorem 2, we have \(\lambda _{1} + \lambda _{2} = \lambda \). (iii) ⇒ (i). If \(\lambda _{1} + \lambda _{2} = \lambda \), then
which is a single integral.
(iii) ⇒ (iv). By Theorem 1, for \(\lambda _{1} + \lambda _{2} = \lambda \), the constant factor
in (12) is the best possible. (iv) ⇒ (iii). By Theorem 2, we have \(\lambda _{1} + \lambda _{2} = \lambda \).
Hence, statements (i), (ii), (iii), and (iv) are equivalent. □
Remark 3
If \(\mu + \sigma = s\),then inequality (9) reduces to
We confirm that the constant factor \(B(\mu ,\sigma )\) in (17) is the best possible. Otherwise, we would reach a contradiction by (14) that the constant factor in (13) is not the best possible.
4 A corollary and some particular cases
Replacing x by \(\frac{1}{x}\) in (12), setting \(g(x) = x^{\lambda - 2}f( \frac{1}{x})\), we define
Then we obtain the following inequality with the non-homogeneous kernel:
It is obvious that inequality (18) is equivalent to (12).
In view of Theorem 3, we have the following.
Corollary 1
Assuming that \(\lambda \in (0,26]\), \(\lambda _{1}, \lambda _{2} \in (0,\lambda + 1)\), \(\lambda _{2} \in (0,1]\), the constant factor
in (18) is the best possible if and only if \(\lambda _{1} + \lambda _{2} = \lambda \). In the case of \(\lambda _{1} + \lambda _{2} = \lambda \), (18) reduces to the following inequality with the best possible constant factor \(\lambda _{1}\lambda _{2}B(\lambda _{1},\lambda _{2})\):
which is equivalent to (13).
Remark 4
(i) In (13) and (19), for \(0 < \lambda \le \min \{ p,26 \}\), \(\lambda _{1} = \frac{\lambda }{q}\), \(\lambda _{2} = \frac{\lambda }{p}\) (≤1), we have the following equivalent inequalities:
if \(0 < \lambda \le \min \{ q,26\}\), \(\lambda _{1} = \frac{\lambda }{p}\), \(\lambda _{2} = \frac{\lambda }{q}\) (≤1), then we have the following equivalent inequalities:
In particular, for \(p = q = 2\), \(0 < \lambda \le 2\), both inequalities (20) and (22) reduce to
and both (21) and (23) reduce to the equivalent form of (24) as follows:
(ii) In (13) and (19), for \(\frac{1}{p} < \lambda \le 26\), \(\lambda _{1} = \lambda - \frac{1}{p}\), \(\lambda _{2} = \frac{1}{p}\) (<1), we have the following equivalent inequalities:
if \(\frac{1}{q} < \lambda \le 26\), \(\lambda _{1} = \lambda - \frac{1}{q}\), \(\lambda _{2} = \frac{1}{q}\) (<1), then we have the following equivalent inequalities:
In particular, for \(p = q = 2\), \(\frac{1}{2} < \lambda \le 26\), both inequalities (26) and (28) reduce to
and both (27) and (29) reduce to the equivalent form of (30) as follows:
(iii) In (13) and (19), for \(1 < \lambda \le 26\), \(\lambda _{1} = \lambda - 1\), \(\lambda _{2} = 1\), we have the following equivalent inequalities:
if \(1 < \lambda \le 2\), \(\lambda _{1} = 1\), \(\lambda _{2} = \lambda - 1\) (≤1), we have the following equivalent inequalities:
In particular, for \(\lambda = 2\), both (32) and (34) reduce to
both (33) and (35) reduce to the equivalent form of (36) as follows:
The constant factors in the above inequalities are the best possible.
5 Conclusions
In this paper, according to the way of [21, 22], by applying the weight functions, the idea of introduced parameters, and the Euler–Maclaurin summation formula, a new extended half-discrete Hilbert’s inequality with the homogeneous kernel and the beta, gamma function is given in Theorem 1. The preliminaries are obtained in Theorem 2. The equivalent statements of the best possible constant factor related to some parameters are proved in Theorem 3. As applications, a corollary about the case of the non-homogeneous kernel and some particular cases are considered in Corollary 1 and Remark 4. The lemmas and theorems provide an extensive account of this type of inequalities.
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The authors thank the referee for his useful suggestions to reform the paper.
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This work is supported by the National Natural Science Foundation (Nos. 11961021, 11561019) and Hechi University Research Fund for Advanced Talents (No. 2019GC005). We are grateful for this help.
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BY carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. XH and RL participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Huang, X.S., Luo, R. & Yang, B. On a new extended half-discrete Hilbert’s inequality involving partial sums. J Inequal Appl 2020, 16 (2020). https://doi.org/10.1186/s13660-020-2293-2
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DOI: https://doi.org/10.1186/s13660-020-2293-2