Abstract
By the use of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel is given, which is an extension of the more accurate Hardy–Hilbert’s inequality. An equivalent form is obtained. The equivalent statements of the best possible constant factor related to several parameters, the operator expressions and a few particular cases are considered.
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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 more accurate Hardy–Hilbert’s inequality with the best possible constant \(\frac{\pi }{\sin (\pi /p)}\) (cf. [1], Theorem 323):
For \(p = q = 2\), inequality (1) reduces to the more accurate Hilbert’s inequality. Since \(\frac{1}{m + n} < \frac{1}{m + n - 1}\), we still have the following Hardy–Hilbert’s inequality (cf. [1], Theorem 315):
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 integral analogue of (2), namely Hardy–Hilbert’s integral inequality:
with the best possible constant factor \(\frac{\pi }{\sin (\pi /p)}\) (cf. [1], Theorem 316).
By introducing an independent parameter \(\lambda > 0\), Yang [2, 3] gave an extension of (2) (for \(p = q = 2\)) with the kernel \(\frac{1}{(x + y)^{\lambda }}\) and the best possible constant factor \(B(\frac{\lambda }{2},\frac{\lambda }{2})\) (\(B(u,v): = \int _{0}^{\infty } \frac{t^{u - 1}}{(1 + t)^{u + v}} \,dt\ (u,v > 0)\) is the beta function) in 1998. Inequalities (1), (2) and (3) play an important role in analysis and its applications (cf. [4–15]).
The following half-discrete Hilbert-type inequality was provided in 1934 (cf. [1], Theorem 351): If \(K(x)\) (\(x > 0\)) is decreasing, \(p > 1,\frac{1}{p} + \frac{1}{q} = 1,0 < \phi (s) = \int _{0}^{\infty } K(x)x^{s - 1} \,dx < \infty \), \(a_{n} \ge 0, 0 < \sum_{n = 1}^{\infty } a_{n}^{p} < \infty \), then
Some new extensions and applications of (4) were obtained in recent years [16–21]. In 2016, by the use of the technique of real analysis, Hong et al. [22] provided some equivalent statements of the extensions of (1) with the best possible constant factor related to several parameters. Other results about the extensions of (1)–(4) were given by [23–37].
In this paper, following the approach of [22], by means of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane is given as follows: for \(r > 1,\frac{1}{r} + \frac{1}{s} = 1\),
which is an extension of (1). The general form of (4), as well as an equivalent form, is obtained. The equivalent statements of the best possible constant factor related to several parameters, the operator expressions and a few particular cases are considered.
2 Some lemmas
In what follows, we suppose that \(p > 1,\frac{1}{p} + \frac{1}{q} = 1, - \frac{1}{2} \le \xi,\eta \le \frac{1}{2}, - 1 < \alpha,\beta < 1,\lambda,\lambda _{1},\lambda _{2} \in \mathrm{R} = ( - \infty,\infty ), d: = \lambda - \lambda _{1} - \lambda _{2}\), \(k_{\lambda } (x,y)\ ( \ge 0)\) is a homogeneous function of degree −λ, satisfying
\(k_{\lambda } (x,y)x^{\lambda _{1} - 1}\) (resp. \(k_{\lambda } (x,y)y^{\lambda _{2} - 1}\)) is strictly decreasing and strictly convex with respect to \(x > 0\) (resp. \(y > 0\)), such that \(( - 1)^{i}\frac{\partial ^{i}}{\partial x^{i}}(k_{\lambda } (x,y)x^{\lambda _{1} - 1}) > 0,( - 1)^{i}\frac{\partial ^{i}}{\partial y^{i}}(k_{\lambda } (x,y)y^{\lambda _{2} - 1}) > 0\ (x,y > 0;i = 1,2)\), and
We still assume that \(a_{m},b_{n} \ge 0\ (|m|,|n| \in \mathrm{N} = \{ 1,2, \ldots \} )\), satisfy
where, \(\sum_{|j| = 1}^{\infty } =\cdots = \sum_{j = - 1}^{ - \infty }+ \cdots + \sum_{j = 1}^{\infty } \cdots (j = m,n)\).
Lemma 1
For any \(\gamma > 0\), we have the following inequalities:
Proof
Since \(( - 1)^{i}\frac{d^{i}}{dt^{i}}\frac{1}{(t - |\xi |)^{\gamma + 1}} > 0\ (t > \frac{3}{2};i = 1,2)\), for \(\frac{3}{2} \ge 1 + |\xi |\), by Hermite–Hadamard’s inequality (cf. [38]) and using the decreasing property of series, we find
and then we have (6).
The lemma is proved. □
Definition 1
We set
and define the following weight coefficients:
Lemma 2
The following inequalities are valid:
Proof
For fixed \(|m| \in \mathrm{N}\), we set
where from for \(y > - \eta, k^{(1)}(m, - y) = k_{\lambda } (|m - \xi | + \alpha (m - \xi ),(1 - \beta )(y + \eta ))\). We find
In view of the assumptions, \(k^{(1)}(m, - y)(y + \eta )^{\lambda _{2} - 1}\) (resp. \(k^{(2)}(m,y)(y - \eta )^{\lambda _{2} - 1}\)) is strictly decreasing and strictly convex with respect to \(y \in ( - \eta,\infty )\) (resp. \(y \in (\eta,\infty )\)). By Hermite–Hadamard’s inequality and using the decreasing property of series, for \(\frac{1}{2} \ge \pm \eta \), we obtain
Setting \(u = \frac{(1 - \beta )(y + \eta )}{|m - \xi | + \alpha (m + \xi )}\) (resp. \(u = \frac{(1 + \beta )(y - \eta )}{|m - \xi | + \alpha (m + \xi )}\)) in the first (resp. second) integral of (11), we obtain
Hence, we have (9).
In the same way, setting \(v = \frac{1}{u}\), we obtain
and then (10) follows.
The lemma is proved. □
Lemma 3
If \(\lambda _{1} + \lambda _{2} = \lambda\) (or \(d = 0\)), then for any \(\varepsilon > 0\), we have
Proof
By (7) (for \(\lambda _{1} + \lambda _{2} = \lambda \)) and (12), replacing \(\lambda _{2}\) (resp. \(\lambda _{1}\)) by \(\lambda _{2} - \frac{\varepsilon }{q}\) (resp. \(\lambda _{1} + \frac{\varepsilon }{q}\)), we have
Then we find
where we denote
In the following, we estimate \(H_{1}\). Still using the decreasing property of series, for fixed \(x > - \xi, \frac{2}{1 - \eta } \ge 1{}(\eta = \alpha,\beta )\), setting \(u = \frac{(1 - \beta )(y + \eta )}{(1 - \alpha )(x + \xi )}\), we obtain
In the same way, we can find that
In view of the above results, we have
and then (13) follows.
The lemma is proved. □
Lemma 4
The following inequality holds:
Proof
By Hölder’s inequality with weight (cf. [38]), (7) and (8), we obtain
Then by (9) and (10), we have (14).
The lemma is proved. □
Remark 1
(i) By (14), for \(\lambda _{1} + \lambda _{2} = \lambda \) (or \(d = 0\)), we find
and the following more accurate Hilbert-type inequality in the whole plane:
In particular, for \(\alpha = \beta = \xi = \eta = 0,a_{ - m} = a_{m},b_{ - n} = b_{n}\ (m,n \in \mathrm{N})\) in (15), we have
(ii) For \(\lambda = 1,\lambda _{1} = \frac{1}{q},\lambda _{2} = \frac{1}{p}\) in (16), we have
for \(\lambda = 1,\lambda _{1} = \frac{1}{p},\lambda _{2} = \frac{1}{q}\) in (16), we have the dual form of (17) as follows:
for \(p = q = 2\), both (17) and (18) reduce to the following Hilbert-type inequality:
(iii) For \(\alpha = \beta = 0,\xi = \eta = \frac{1}{2},\lambda = 1,k_{1}(m,n) = \frac{1}{m + n},\lambda _{1} = \frac{1}{r},\lambda _{2} = \frac{1}{s}\ (r > 1,\frac{1}{r} + \frac{1}{s} = 1)\), (15) reduces to (5). Hence, (14) and (15) are general extensions of (5).
Lemma 5
The constant factor \(\frac{2k_{\lambda } (\lambda _{2})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\) in (15) is the best possible.
Proof
For any \(\varepsilon > 0\), we set
If there exists a constant \(M( \le \frac{2k_{\lambda } (\lambda _{2})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}})\), such that (15) is valid when replacing \(\frac{2k_{\lambda } (\lambda _{2})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\) by M, then in particular, in view of \(\lambda _{1} + \lambda _{2} = \lambda \), by Lemma 1, we have
In view of the above result and (13), we have
For \(\varepsilon \to 0^{ +} \), by Fatou lemma (cf. [39]), we find
namely, \(\frac{2k_{\lambda } (\lambda _{2})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}} \le M\), which means that \(M = \frac{2k_{\lambda } (\lambda _{2})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\) is the best possible constant factor of (15).
The lemma is proved. □
Remark 2
(i) In view of Lemma 5, the constant factors in (16)–(19) are also the best possible.
(ii) Setting \(\hat{\lambda }_{1}: = \frac{\lambda - \lambda _{2}}{p} + \frac{\lambda _{1}}{q} = \lambda _{1} + \frac{d}{p},\hat{\lambda }_{2}: = \frac{\lambda - \lambda _{1}}{q} + \frac{\lambda _{2}}{p} = \lambda _{2} + \frac{d}{q}\), we find
and then by Hölder’s inequality (cf. [38]), it follows that
We can rewrite (14) as follows:
Lemma 6
If the constant factor \(\frac{2k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\) in (14) (or (21)) is the best possible, then we have \(\lambda _{1} + \lambda _{2} = \lambda \).
Proof
If the constant factor \(\frac{2k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\) in (14) (or (21)) is the best possible, then by (21) and (15) (for \(\lambda _{i} = \hat{\lambda } {}_{i}\ (i = 1,2)\)), we have the following inequality:
namely, \(k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1}) \le k_{\lambda } (\hat{\lambda }_{2})\), which means that (20) is an equality.
We observe that (20) is an equality if and only if there exist constants A and B, such that they are not both zero and (cf. [38])
Assuming that \(A \ne 0\), it follows that \(u^{\lambda _{2} + \lambda _{1} - \lambda } = \frac{B}{A}\) a.e. in \(\mathrm{R}_{ +} \), and then \(\lambda _{2} + \lambda _{1} - \lambda = 0\), namely, \(\lambda _{1} + \lambda _{2} = \lambda \).
The lemma is proved. □
3 Main results
Theorem 1
Inequality (14) is equivalent to the following more accurate Hilbert-type inequality in the whole plane:
Proof
Suppose that (22) is valid. By Hölder’s inequality (cf. [38]), we find
Then by (22), we obtain (14). On the other hand, assuming that (14) is valid, we set
Then we have
If \(L = 0\), then (22) is naturally valid; if \(L = \infty \), then it is impossible that (22) is valid, namely, \(L < \infty \). Suppose that \(0 < L < \infty \). By (14), it follows that
namely, (22) follows, which is equivalent to (14).
The theorem is proved. □
Theorem 2
The following statements are equivalent:
-
(i)
Both \(k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})\) and \(k_{\lambda } (\frac{\lambda _{2}}{p} + \frac{\lambda - \lambda _{1}}{q})\) are independent of \(p,q\);
-
(ii)
\(k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})\le k_{\lambda } (\frac{\lambda _{2}}{p} + \frac{\lambda - \lambda _{1}}{q})\);
-
(iii)
\(\lambda _{1} + \lambda _{2} = \lambda\);
-
(iv)
\(\frac{2k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\) is the best possible constant factor of (14);
-
(v)
\(\frac{2k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\) is the best possible constant factor of (22).
If the statement (iii) follows, namely, \(\lambda _{1} + \lambda _{2} = \lambda \) (or \(d = 0\)), then we have the following inequality equivalent to (15) with the best possible constant factor \(\frac{2k_{\lambda } (\lambda _{2})}{(1 - \beta ^{2})^{1/p}(1 - \alpha ^{2})^{1/q}}\):
In particular, for \(\alpha = \beta = \xi = \eta = 0,a_{ - m} = a_{m},b_{ - n} = b_{n}\ (m,n \in \mathrm{N})\) in (25), we have the following inequality equivalent to (16) with the best possible constant factor \(k_{\lambda } (\lambda _{2})\):
Proof
(i) ⇒ (ii). Since \(k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})\) is independent of \(p,q\), we find
Then by Fatou lemma (cf. [39]), we have the following inequality:
(ii) ⇒ (iii). If \(k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})\le k_{\lambda } (\frac{\lambda _{2}}{p} + \frac{\lambda - \lambda _{1}}{q})\), then (20) is an equality. By the proof of Lemma 6, it follows that \(\lambda _{1} + \lambda _{2} = \lambda \).
(iii) ⇒ (i). If \(\lambda _{1} + \lambda _{2} = \lambda \), then we have
Both \(k_{\lambda }^{\frac{1}{p}}(\lambda _{2})k_{\lambda }^{\frac{1}{q}}(\lambda - \lambda _{1})\) and \(k_{\lambda } (\frac{\lambda _{2}}{p} + \frac{\lambda - \lambda _{1}}{q})\) are independent of \(p,q\).
Hence, we have (i) ⇔ (ii) ⇔ (iii).
(iii) ⇔ (iv). By Lemmas 5 and 6, we obtain the conclusions.
(iv) ⇔ (v). If the constant factor in (14) is the best possible, then so is the constant factor in (22). Otherwise, by (23), we would reach a contradiction that the constant factor in (14) is not the best possible. On the other hand, if the constant factor in (22) is the best possible, then so is the constant factor in (14). Otherwise, by (24), we would reach a contradiction that the constant factor in (22) is not the best possible.
Therefore, the statements (i)–(v) are equivalent.
The theorem is proved. □
4 Operator expressions
We define functions
where
Define the following real normed spaces:
Assuming that \(a \in l_{p,\phi } \) and setting
we can rewrite (22) as follows:
namely, \(c \in l_{p,\psi ^{1 - p}}\).
Definition 2
Define a Hilbert-type operator \(T:l_{p,\phi } \to l_{p,\psi ^{1 - p}}\) as follows: For any \(a \in l_{p,\phi }\), there exists a unique representation \(Ta = c \in l_{p,\psi ^{1 - p}}\), satisfying for any \(|n| \in \mathrm{N}\), \(Ta(n) = c_{n}\). Define the formal inner product of Ta and \(b \in l_{q,\psi } \), and the norm of T, as follows:
Theorem 3
If \(a \in l_{p,\phi },b \in l_{q,\psi }, \Vert a \Vert _{p,\phi }, \Vert b \Vert _{q,\psi } > 0\), then we have the following equivalent inequalities:
Moreover, \(\lambda _{1} + \lambda _{2} = \lambda \) if and only if the constant factor
in (27) and (28) is the best possible, namely,
Example 1
For \(\lambda > 0,0 < \sigma \le 1,\lambda _{i} \in (0,\lambda ) \cap (0,1]\ (i = 1,2)\), setting \(k_{\lambda } (x,y) = \frac{1}{(x^{\sigma } + y^{\sigma } )^{\lambda /\sigma }}\ (x,y > 0)\) yields that
\(k_{\lambda } (x,y)x^{\lambda _{1} - 1}\) (resp. \(k_{\lambda } (x,y)y^{\lambda _{2} - 1}\)) is strictly decreasing and strictly convex with respect to \(x > 0\) (resp. \(y > 0\)), such that
By Theorem 3, it follows that \(\lambda _{1} + \lambda _{2} = \lambda \) if and only if
Example 2
For \(0 < \lambda \le 1,\lambda _{i} \in (0,\lambda ) \cap (0,1]\ (i = 1,2)\), setting \(k_{\lambda } (x,y) = \frac{\ln (x/y)}{x^{\lambda } - y^{\lambda }}\ (x,y > 0)\) yields that
\(k_{\lambda } (x,y)x^{\lambda _{1} - 1}\) (resp. \(k_{\lambda } (x,y)y^{\lambda _{2} - 1}\)) is strictly decreasing and strictly convex with respect to \(x > 0\) (resp. \(y > 0\)), such that
By Theorem 3, it follows that \(\lambda _{1} + \lambda _{2} = \lambda \) if and only if
5 Conclusions
In this paper, by means of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane is obtained in Lemma 4, which is an extension of (1). An equivalent form is given in Theorem 1. The equivalent statements of the best possible constant factor related to several parameters are considered in Theorem 2. The operator expressions and some particular cases are provided in Theorem 3, Remark 1 and Examples 1–2. The lemmas and theorems provide an extensive account of this type of inequalities.
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References
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Yang, B.C.: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778–785 (1998)
Yang, B.C.: A note on Hilbert’s integral inequality. Chin. Q. J. Math. 13(4), 83–86 (1998)
Yang, B.C.: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing, (2009)
Yang, B.C.: Hilbert-Type Integral Inequalities. Bentham Science, The United Arab Emirates (2009)
Krnić, M., Pečarić, J.: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8, 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., Krnić, M.: Hilbert-type inequalities involving differential operators, the best constants and applications. Math. Inequal. Appl. 18(1), 111–124 (2015)
Rassias, M.T., 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.T., Yang, B.C.: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263–277 (2013)
Rassias, M.T., 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.Q.: 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)
Huang, Z.X., Yang, B.C.: Equivalent property of a half-discrete Hilbert’s inequality with parameters. J. Inequal. Appl. 2018, 333 (2018)
He, L.P., Liu, H.Y., Yang, B.C.: Parametric Mulholland-type inequalities. J. Appl. Anal. Comput. 9(5), 1973–1986 (2019)
Yang, B.C., Hauang, M.F., Zhong, Y.R.: On an extended Hardy–Hilbert’s inequality in the whole plane. J. Appl. Anal. Comput. 9(6), 2124–2136 (2019)
Yang, B.C., Wu, S., Wang, A.: On a reverse half-discrete Hardy–Hilbert’s inequality with parameters. Mathematics 7, 1054 (2019)
Wang, A.Z., Yang, B.C., Chen, Q.: Equivalent properties of a reverse half-discrete Hilbert’s inequality. J. Inequal. Appl. 2019, 279 (2019)
Yang, B.C., Wu, S.H., Liao, J.Q.: On a new extended Hardy–Hilbert’s inequality with parameters. Mathematics 8, 73 (2020). https://doi.org/10.3390/math8010073
Mo, H.M., Yang, B.C.: On a new Hilbert-type integral inequality involving the upper limit functions. J. Inequal. Appl. 2020, 5 (2020)
Huang, X.S., Luo, R.C., Yang, B.C.: On a new extended half-discrete Hilbert’s inequality involving partial sums. J. Inequal. Appl. 2020, 16 (2020)
Yang, B.C., Wu, S.H., Chen, Q.: On an extended Hardy–Littlewood–Polya’s inequality. AIMS Math. 5(2), 1550–1561 (2020)
Liao, J.Q., Wu, S.H., Yang, B.C.: On a new half-discrete Hilbert-type inequality involving the variable upper limit integral and the partial sum. Mathematics 8, 229 (2020). https://doi.org/10.3390/math8020229
Yang, B.C., Huang, M.F., Zhong, Y.R.: Equivalent statements of a more accurate extended Mulholland’s inequality with a best possible constant factor. Math. Inequal. Appl. 23(1), 231–244 (2020)
Kuang, J.C.: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004)
Kuang, J.C.: Real Analysis and Functional Analysis (Continuation), 2nd edn. Higher Education Press, Beijing (2015)
Acknowledgements
The authors thank the referee for his useful suggestions in reforming the paper.
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This work is supported by the National Natural Science Foundation (Nos. 11961021, 11561019), Hechi University Research Fund for Advanced Talents (No. 2019GCC005) and Science and Technology Planning Project Item of Guangzhou City (No. 201707010229). 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 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., Yang, B. On a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel. J Inequal Appl 2021, 10 (2021). https://doi.org/10.1186/s13660-020-02542-2
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DOI: https://doi.org/10.1186/s13660-020-02542-2