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A more accurate half-discrete Hardy-Hilbert-type inequality with the logarithmic function
Journal of Inequalities and Applications volume 2017, Article number: 153 (2017)
Abstract
By means of the weight functions, the technique of real analysis and Hermite-Hadamard’s inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given. Moreover, the equivalent forms, the operator expressions, the reverses and some particular cases are also considered.
1 Introduction
Assuming that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f(x),g(y)\geq0\), \(f\in L^{p}( \mathbf{R}_{+})\), \(g\in L^{q}(\mathbf{R}_{+})\), \(\Vert f \Vert _{p} = (\int_{0}^{\infty }f^{p}(x)\,dx)^{\frac{1}{p}}>0\), \(\Vert g \Vert _{q}>0\), we have the following Hardy-Hilbert integral inequality (cf. [1]):
where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is the best possible. If \(a_{m},b_{n}\geq0\), \(a=\{a_{m}\}_{m=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\Vert a \Vert _{p}=(\sum_{m=1}^{\infty }a_{m}^{p})^{\frac{1}{p}}>0\), \(\Vert b \Vert _{q}>0\), then we have the following Hardy-Hilbert inequality with the same best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) (cf. [1]):
Inequalities (1) and (2) are important in analysis and its applications (cf. [1–3]).
Suppose that \(\mu_{i},\nu_{j}>0\) (\(i,j\in\mathbf{N}=\{1,2,\ldots \}\)),
We have the following Hardy-Hilbert-type inequality (cf. Theorem 321 of [1], replacing \(\mu_{m}^{1/q}a_{m}\) and \(\nu_{n}^{1/p}b_{n}\) by \(a_{m}\) and \(b_{n}\)):
For \(\mu_{i}=\nu_{j}=1\) (\(i,j\in\mathbf{N}\)), (4) reduces to (2).
In 1998, by introducing an independent parameter \(\lambda\in(0,1]\), Yang [4] gave an extension of (1) with the kernel \(\frac{1}{(x+y)^{\lambda}}\) for \(p=q=2\). Recently, Yang [3] gave some extensions of (1) and (2) as follows: If \(\lambda _{1},\lambda_{2}\in\mathbf{R}\), \(\lambda_{1}+\lambda_{2}=\lambda \), \(k_{\lambda}(x,y)\) is a non-negative homogeneous function of degree −λ, with \(k(\lambda_{1})=\int_{0}^{\infty}k_{\lambda }(t,1)t^{\lambda_{1}-1}\,dt\in\mathbf{R}_{+}\), \(\phi(x)=x^{p(1-\lambda _{1})-1}\), \(\psi(x)=x^{q(1-\lambda_{2})-1}\), \(f(x), g(y)\geq0\),
\(g\in L_{q,\psi}(\mathbf{R}_{+})\), \(\Vert f \Vert _{p,\phi }, \Vert g \Vert _{q,\psi}>0\), then we have
where the constant factor \(k(\lambda_{1})\) is the best possible. Moreover, if \(k_{\lambda}(x,y)\) keeps a finite value and \(k_{\lambda }(x,y)x^{\lambda _{1}-1} (k_{\lambda}(x,y)y^{\lambda_{2}-1})\) is decreasing with respect to \(x>0\) (\(y>0\)), then, for \(a_{m,}b_{n}\geq0\),
\(b=\{b_{n}\}_{n=1}^{\infty}\in l_{q,\psi}\), \(\Vert a \Vert _{p,\phi}, \Vert b \Vert _{q,\psi }>0\), we have the following inequality with the same best possible constant factor \(k(\lambda_{1})\):
In 2015, Yang [5] gave an extension of (6) for the kernel \(k_{\lambda}(m,n)=\frac{1}{(m+n)^{\lambda}}\) and (4) as follows:
where the constant \(B(\lambda_{1},\lambda_{2})\) is the best possible, and \(B(u,v)\) (\(u,v>0\)) is the beta function. Some other results including multidimensional Hilbert-type inequalities are provided by [6–24].
About half-discrete Hilbert-type inequalities with the non-homogeneous kernels, Hardy et al. provided a few results in Theorem 351 of [1]. Yang [25] gave an inequality with the kernel \(\frac {1}{(1+nx)^{\lambda}}\) by introducing an interval variable and proved that the constant factor is the best possible. Zhong et al. [26–28] investigated a few half-discrete Hilbert-type inequalities with the particular kernels. Applying the weight functions, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree \(-\lambda\in\mathbf{R}\) and a best constant factor \(k ( \lambda_{1} ) \) is proved as follows (cf. [29]):
A half-discrete Hilbert-type inequality with a general non-homogeneous kernel and a best constant factor is given by Yang [30].
In this paper, by means of the weight functions, the technique of real analysis and Hermite-Hadamard’s inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given, which is an extension of (8) in a particular kernel of degree 0 similar to (7). The equivalent forms, the operator expressions, the equivalent reverses and some particular cases are also considered.
2 Some lemmas
In the following, we agree that \(\nu_{j}>0\) (\(j\in\mathbf{N}\)), \(V_{n}:=\sum_{j=1}^{n}\nu_{j}\), \(\mu(t)\) is a positive continuous function in \(\mathbf{R}_{+}=(0,\infty)\),
\(0\leq\widetilde{\nu}_{n}\leq\frac{\nu_{n}}{2}\), \(\widetilde {V}_{n}=V_{n}-\widetilde{\nu}_{n}\), \(\nu(t):=\nu_{n}\), \(t\in(n-\frac{1}{2},n+\frac{1}{2} ]\) (\(n\in\mathbf{N}\)), and
\(p\neq0,1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0<\sigma<\gamma\), \(\sigma\leq 1\), \(\delta\in\{-1,1\}\), \(f(x),a_{n}\geq0\) (\(x\in\mathbf{R}_{+}\), \(n\in \mathbf{N}\)), \(\Vert f \Vert _{p,\Phi_{\delta}}=(\int _{0}^{\infty}\Phi_{\delta }(x)f^{p}(x)\,dx)^{\frac{1}{p}}\), \(\Vert a \Vert _{q,\widetilde {\Psi}}=(\sum_{n=1}^{\infty}\widetilde{\Psi}(n)b_{n}^{q})^{\frac{1}{q}}\), where
Example 1
For \(\rho>0\), we set
(i) Setting \(u=\rho t^{-\gamma}\), we find
(ii) We obtain, for \(t>0\), \(h(t)=\ln(1+\frac{\rho}{t^{\gamma}})>0\),
It is evident that, for \(\sigma\leq1\), \(t^{\sigma-1}h(t)>0\), we have
(iii) Since for \(n\in\mathbf{N}\), \(V(y)>0\), \(V^{\prime}(y)=\nu _{n}>0\), \(V^{\prime \prime}(y)=0\) (\(y\in(n-\frac{1}{2},n+\frac{1}{2})\)), then, for \(c>0\), we have
Lemma 1
If \(g(t)>0\), \(g^{\prime}(t)<0\), \(g^{\prime\prime}(t)>0\) (\(t\in (\frac{1}{2},\infty)\)), satisfying \(\int_{\frac{1}{2} }^{\infty}g(t)\,dt\in\mathbf{R}_{+}\), then we have
Proof
For \(n_{0}\in\mathbf{N}\backslash\{1\}\), by the assumptions and Hermite-Hadamard’s inequality (cf. [31]), we have
It follows that
In the same way, we still have
Hence, adding the above two inequalities, we have (10). The lemma is proved. □
Lemma 2
Assuming that \(\rho>0\), we define the following weight functions:
Then we have the following inequalities:
where \(k(\sigma)\) is determined by (9).
Proof
Since
and for \(t\in(n-\frac{1}{2},n+\frac{1}{2})\), \(V^{\prime}(t)=\nu_{n}\), by Examples 1(ii)-(iii), (16), (11) and (10), we have
Setting \(u=U^{\delta}(x)V(t)\) in the above, by (9), we find
Hence, (14) follows.
Setting \(u=\widetilde{V}_{n}U^{\delta}(x)\) in (13), we find \(du=\delta\widetilde{V}_{n}U^{\delta-1}(x)\mu(x)\,dx\) and
If \(\delta=1\), then
if \(\delta=-1\), then
Then by (9), we have (15). The lemma is proved. □
Note
If \(U(\infty)=\infty\), then (15) keeps the form of an equality.
Lemma 3
If \(\rho>0\), there exists a \(n_{0}\in\mathbf{N}\), such that \(\nu_{n}\geq\nu_{n+1}\) (\(n\in \{n_{0},n_{0}+1,\ldots\}\)), and \(V_{\infty}=\infty\), then: (i) for \(x\in \mathbf{R}_{+}\), we have
where
(ii) for any \(b>0\), we have
Proof
(i) Since for \(t\in(n,n+1)\) (\(n\geq n_{0}\)), \(\nu _{n}\geq \nu_{n+1}=V^{\prime}(t+\frac{1}{2})\), by Examples 1(iii) and (11), we have
Setting \(u=U^{\delta}(x)V(t+\frac{1}{2})\) in the above, in view of \(V_{\infty}=\infty\), by (9), we find
Since \(F(u)=u^{\frac{\sigma}{2}}\ln(1+\frac{\rho}{u^{\gamma}})\) is continuous in \((0,\infty)\) satisfying \(F(u)\rightarrow0\) (\(u\rightarrow 0^{+}\) or \(u\rightarrow\infty\)), there exists a constant \(L>0\), such that \(F(u)\leq L\), namely,
Hence we find
and then (18) follows.
(ii) For \(b>0\), by (11), we find
Hence we have (19). The lemma is proved. □
Note
For example, \(\nu_{n}=\frac{1}{n^{\beta}}\) (\(n\in \mathbf{N}\); \(0\leq\beta\leq1\)) satisfies the conditions of Lemma 3 (for \(n_{0}=1\)).
3 Main results and operator expressions
Theorem 1
If \(\rho>0\), \(k(\sigma)\) is determined by (9), then, for \(p>1\), \(0< \Vert f \Vert _{p,\Phi _{\delta}}\), \(\Vert a \Vert _{q,\widetilde{\Psi}}<\infty\), we have the following equivalent inequalities:
Proof
By Hölder’s inequality with weight (cf. [31]), we have
In view of (15) and Lebesgue term by term integration theorem (cf. [32]), we find
Then by (14), we have (21). By Hölder’s inequality (cf. [31]), we have
In view of (21), we have (20). On the other hand, assuming that (20) is valid, we set
Then we find \(J_{1}^{p}= \Vert a \Vert _{q,\widetilde{\Psi }}^{q}\). If \(J_{1}=0\), then (21) is trivially valid; if \(J_{1}=\infty\), then (21) remains impossible. Suppose that \(0< J_{1}<\infty\). By (20), we have
and then (21) follows, which is equivalent to (20).
Still by Hölder’s inequality with weight (cf. [31]), we have
Then by (14) and Lebesgue term by term integration theorem (cf. [32]), it follows that
In view of (15), we have (22). By Hölder’s inequality (cf. [31]), we have
Then by (22), we have (20). On the other hand, assuming that (22) is valid, we set
Then we find \(J_{2}^{q}= \Vert f \Vert _{p,\Phi_{\delta }}^{p}\). If \(J_{2}=0\), then (22) is trivially valid; if \(J_{2}=\infty\), then (22) keeps impossible. Suppose that \(0< J_{2}<\infty\). By (20), we have
and then (22) follows, which is equivalent to (20).
Therefore, inequalities (20), (21) and (22) are equivalent. The theorem is proved. □
Theorem 2
As regards the assumptions of Theorem 1, if there exists a \(n_{0}\in\mathbf{N}\), such that \(\nu_{n}\geq\nu_{n+1}\) (\(n\in \{n_{0},n_{0}+1,\ldots\}\)), and \(U(\infty)=V_{\infty}=\infty\), then the constant factor \(k(\sigma)\) in (20), (21) and (22) is the best possible.
Proof
For \(\varepsilon\in(0,q\sigma)\), we set \(\widetilde {\sigma }=\sigma-\frac{\varepsilon}{q}\) (\(<\min\{1,\gamma\}\)), and \(\widetilde{f}=\widetilde{f}(x)\), \(x\in\mathbf{R}_{+}\), \(\widetilde{a}=\{\widetilde{a}_{n}\}_{n=1}^{\infty}\),
Then, for \(\delta=\pm1\), since \(U(\infty)=\infty\), we obtain
By (31), (19) and (17), we find
If there exists a positive constant \(K\leq k(\sigma)\), such that (20) is valid when replacing \(k(\sigma)\) to K, then in particular, by Lebesgue term by term integration theorem, we have \(\varepsilon \widetilde{I}<\varepsilon K \Vert \widetilde{f} \Vert _{p,\Phi_{\delta }} \Vert \widetilde{a} \Vert _{q,\widetilde{\Psi}}\), namely,
It follows that \(k(\sigma)\leq K\) (\(\varepsilon\rightarrow0^{+}\)). Hence, \(K=k(\sigma)\) is the best possible constant factor of (20).
The constant factor \(k(\sigma)\) in (21) ((22)) is still the best possible. Otherwise, we would reach a contradiction by (25) ((28)) that the constant factor in (20) is not the best possible. The theorem is proved. □
For \(p>1\), we find \(\widetilde{\Psi}^{1-p}(n)=\frac{\nu _{n}}{\widetilde{V}_{n}^{1-p\sigma}}\) (\(n\in\mathbf{N}\)), \(\Phi_{\delta}^{1-q}(x)=\frac {\mu (x)}{U^{1-q\delta\sigma}(x)}\) (\(x\in\mathbf{R}_{+}\)), and define the following real normed spaces:
Assuming that \(f\in L_{p,\Phi_{\delta}}(\mathbf{R}_{+})\), setting
we can rewrite (21) as \(\Vert c \Vert _{p,\widetilde {\Psi}^{1-p}}< k(\sigma ) \Vert f \Vert _{p,\Phi_{\delta}}<\infty\), namely, \(c\in l_{p,\widetilde{\Psi}^{1-p}}\).
Definition 1
Define a half-discrete Hardy-Hilbert-type operator \(T_{1}:L_{p,\Phi_{\delta}}(\mathbf{R}_{+})\rightarrow l_{p,\widetilde {\Psi}^{1-p}}\) as follows: For any \(f\in L_{p,\Phi_{\delta}}(\mathbf{R}_{+})\), there exists a unique representation \(T_{1}f=c\in l_{p,\widetilde{\Psi} ^{1-p}}\). Define the formal inner product of \(T_{1}f\) and \(a=\{a_{n}\}_{n=1}^{\infty}\in l_{q,\widetilde{\Psi}}\) as follows:
Then we can rewrite (20) and (21) as follows:
Define the norm of operator \(T_{1}\) as follows:
Then by (35), it follows that \(\Vert T_{1} \Vert \leq k(\sigma)\). Since by Theorem 2, the constant factor in (35) is the best possible, we have
Assuming that \(a=\{a_{n}\}_{n=1}^{\infty}\in l_{q,\widetilde{\Psi}}\), setting
we can rewrite (22) as \(\Vert h \Vert _{q,\Phi _{\delta}^{1-q}}< k(\sigma ) \Vert a \Vert _{q,\widetilde{\Psi}}<\infty\), namely, \(h\in L_{q,\Phi_{\delta }^{1-q}}(\mathbf{R}_{+})\).
Definition 2
Define a half-discrete Hardy-Hilbert-type operator \(T_{2}:l_{q,\widetilde{\Psi}}\rightarrow L_{q,\Phi_{\delta }^{1-q}}(\mathbf{R}_{+})\) as follows: For any \(a=\{a_{n}\}_{n=1}^{\infty}\in l_{q,\widetilde{\Psi}}\), there exists a unique representation \(T_{2}a=h\in L_{q,\Phi _{\delta}^{1-q}}(\mathbf{R}_{+})\). Define the formal inner product of \(T_{2}a\) and \(f\in L_{p,\Phi_{\delta}}(\mathbf{R}_{+})\) as follows:
Then we can rewrite (20) and (22) as follows:
Define the norm of operator \(T_{2}\) as follows:
Then by (39), we find \(\Vert T_{2} \Vert \leq k(\sigma)\). Since, by Theorem 2, the constant factor in (39) is the best possible, we have
4 Some equivalent reverses
In the following, we also set
For \(0< p<1\) or \(p<0\), we still use the formal symbols \(\Vert f \Vert _{p,\Phi _{\delta}}\), \(\Vert f \Vert _{p,\widetilde{\Phi}_{\delta }}\) and \(\Vert a \Vert _{q,\widetilde{\Psi}}\).
Theorem 3
If \(\rho>0\), \(k(\sigma)\) is determined by (9), there exists a \(n_{0}\in\mathbf{N}\), such that \(\nu_{n}\geq\nu_{n+1}\) (\(n\in\{n_{0},n_{0}+1,\ldots \}\)), and \(U(\infty)=V_{\infty}=\infty\), then, for \(p<0\), \(0< \Vert f \Vert _{p,\Phi _{\delta}}, \Vert a \Vert _{q,\widetilde{\Psi}}<\infty\), we have the following equivalent inequalities with the best possible constant factor \(k(\sigma)\):
Proof
By the reverse Hölder inequality with weight (cf. [31]), since \(p<0\), in a similar way to obtaining (23) and (24), we have
Then by the note of Lemma 2 and the Lebesgue term by term integration theorem, it follows that
In view of (14), we have (42). By the reverse Hölder inequality (cf. [31]), we have
Then by (42), we have (41). On the other hand, assuming that (41) is valid, we set \(a_{n}\) as in Theorem 1. Then we find \(J_{1}^{p}= \Vert a \Vert _{q,\widetilde{\Psi}}^{q}\). If \(J_{1}=\infty\), then (42) is trivially valid; if \(J_{1}=0\), then (42) remains impossible. Suppose that \(0< J_{1}<\infty\). By (41), it follows that
and then (42) follows, which is equivalent to (41).
Still by the reverse Hölder inequality with weight (cf. [31]), since \(0< q<1\), in a similar way to obtaining (26) and (27), we have
Then by (14) and the Lebesgue term by term integration theorem (cf. [32]), it follows that
In view of the note of Lemma 2, we have (43). By the reverse Hölder inequality, we have
Then by (43), we have (41). On the other hand, assuming that (43) is valid, we set \(f(x)\) as in Theorem 1. Then we find \(J_{2}^{q}= \Vert f \Vert _{p,\Phi_{\delta}}^{p}\). If \(J_{2}=\infty\), then (43) is trivially valid; if \(J_{2}=0\), then (43) remains impossible. Suppose that \(0< J_{2}<\infty\). By (41), it follows that
and then (43) follows, which is equivalent to (41).
Therefore, inequalities (41), (42) and (43) are equivalent.
For \(\varepsilon\in(0,q\sigma)\), we set \(\widetilde{\sigma}=\sigma- \frac{\varepsilon}{q}\), and \(\widetilde{f}=\widetilde{f}(x)\), \(x\in \mathbf{R}_{+}\), \(\widetilde{a}=\{\widetilde{a}_{n}\}_{n=1}^{\infty}\),
By (19), (31) and (14), we obtain
If there exists a positive constant \(K\geq k(\sigma)\), such that (41) is valid when replacing \(k(\sigma)\) to K, then in particular, we have \(\varepsilon\widetilde{I}>\varepsilon K \Vert \widetilde{f} \Vert _{p,\Phi_{\delta }} \Vert \widetilde{a} \Vert _{q,\widetilde{\Psi}}\), namely,
It follows that \(k(\sigma)\geq K\) (\(\varepsilon\rightarrow0^{+}\)). Hence, \(K=k(\sigma)\) is the best possible constant factor of (41). The constant factor \(k(\sigma)\) in (42) ((43)) is still the best possible. Otherwise, we would reach the contradiction by (44) ((45)) that the constant factor in (41) is not the best possible. The theorem is proved. □
Theorem 4
As regards the assumptions of Theorem 3, if \(0< p<1\), \(0< \Vert f \Vert _{p,\Phi_{\delta}}\), \(\Vert a \Vert _{q,\widetilde{\Psi}}<\infty\), then we have the following equivalent inequalities with the best possible constant factor \(k(\sigma)\):
Proof
By the reverse Hölder inequality with weight (cf. [31]), since \(0< p<1\), in a similar way to obtaining (23) and (24), we have
In view of the note of Lemma 2 and the Lebesgue term by term integration theorem (cf. [32]), we find
Then by (17), we have (47). By the reverse Hölder inequality, we have
Then by (47), we have (46). On the other hand, assuming that (46) is valid, we set \(a_{n}\) as in Theorem 1. Then we find \(J_{1}^{p}= \Vert a \Vert _{q,\widetilde{\Psi}}^{q}\). If \(J_{1}=\infty\), then (47) is trivially valid; if \(J_{1}=0\), then (47) remains impossible. Suppose that \(0< J_{1}<\infty\). By (46), it follows that
and then (47) follows, which is equivalent to (46).
Still by the reverse Hölder inequality with weight (cf. [31]), since \(q<0\), we have
Then by (17) and the Lebesgue term by term integration theorem, it follows that
Then by the note of Lemma 2, we have (48). By the reverse Hölder inequality (cf. [31]), we have
Then by (48), we have (46). On the other hand, assuming that (46) is valid, we set \(f(x)\) as in Theorem 1. Then we find \(J^{q}= \Vert f \Vert _{p,\widetilde{\Phi}_{\delta}}^{p}\). If \(J=\infty\), then (48) is trivially valid; if \(J=0\), then (48) remains impossible. Suppose that \(0< J<\infty\). By (46), it follows that
and then (48) follows, which is equivalent to (46).
Therefore, inequalities (46), (47) and (48) are equivalent.
For \(\varepsilon\in(0,p\sigma)\), we set \(\widetilde{\sigma}=\sigma+ \frac{\varepsilon}{p}\), and \(\widetilde{f}=\widetilde{f}(x)\), \(x\in \mathbf{R}_{+}\), \(\widetilde{a}=\{\widetilde{a}_{n}\}_{n=1}^{\infty}\),
By (19), (31) and the note of Lemma 2, we obtain
If there exists a positive constant \(K\geq k(\sigma)\), such that (41) is valid when replacing \(k(\sigma)\) to K, then in particular we have \(\varepsilon\widetilde{I}>\varepsilon K \Vert \widetilde{f} \Vert _{p,\widetilde{\Phi}_{\delta}} \Vert \widetilde{a} \Vert _{q,\widetilde{\Psi}}\), namely,
It follows that \(k(\sigma)\geq K\) (\(\varepsilon\rightarrow0^{+}\)). Hence, \(K=k(\sigma)\) is the best possible constant factor of (46). The constant factor \(k(\sigma)\) in (47) ((48)) is still the best possible. Otherwise, we would reach the contradiction by (49) ((50)) that the constant factor in (46) is not the best possible. The theorem is proved. □
Remark
(i) For \(\delta=-1\) in (20), we obtain the following inequality with the homogeneous kernel of degree 0:
(ii) For \(\delta=1\) in (20), we obtain the following inequality with the non-homogeneous kernel:
(iii) For \(\widetilde{\mu}_{n}=0\) (\(n\in\mathbf{N}\)) in (20), we have the following inequality:
where the constant factor \(\frac{\rho^{\sigma/\gamma}\pi}{\sigma \sin(\frac{\pi\sigma}{\gamma})}\) is still the best possible. Hence, inequality (20) is a more accurate form of (53) (for \(0<\widetilde {\mu}_{n}\leq\frac{\mu_{n}}{2}\), \(n\in\mathbf{N}\)).
(iv) For \(\mu(x)=\mu_{n}=1\) (\(x\in\mathbf{R}_{+}\), \(n\in\mathbf{N}\)), \(\delta=-1\) in (53), we have the following inequality:
which is a particular case of (8) for \(\lambda=0\), \(\lambda _{1}=-\sigma\), \(\lambda_{2}=\sigma\) and \(k_{\lambda}(x,n)=\ln [ 1+\rho ( \frac{x}{n} ) ^{\gamma} ] \).
We still can obtain some inequalities with the best possible constant factors in Theorems 1-4, by using some particular parameters.
5 Conclusions
In this paper, by means of the weight functions, the technique of real analysis and Hermite-Hadamard’s inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given by Theorems 1-2. Moreover, the equivalent forms and the operator expressions are considered. We also obtain the reverses and some particular cases in Theorems 3-4. The method of weight functions is very important, which is the key to help us proving the main inequalities with the best possible constant factor. The lemmas and theorems provide an extensive account of this type of inequalities.
References
Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934)
Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
Yang, BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijin (2009)
Yang, BC: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778-785 (1998)
Yang, BC: An extension of a Hardy-Hilbert-type inequality. J. Guangdong Univ. Educ. 35(3), 1-8 (2015)
Yang, BC, Brnetić, I, Krnić, M, Pečarić, JE: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 8(2), 259-272 (2005)
Krnić, M, Pečarić, JE: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 67(3-4), 315-331 (2005)
Krnić, M, Pečarić, JE: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8(1), 29-51 (2005)
Yang, BC, Rassias, T: On a Hilbert-type integral inequality in the subinterval and its operator expression. Banach J. Math. Anal. 4(2), 100-110 (2010)
Azar, L: On some extensions of Hardy-Hilbert’s inequality and applications. J. Inequal. Appl. 2008, 546829 (2009)
Arpad, B, Choonghong, O: Best constant for certain multilinear integral operator. J. Inequal. Appl. 2006, 28582 (2006)
Kuang, JC, Debnath, L: On Hilbert’s type inequalities on the weighted Orlicz spaces. Pac. J. Appl. Math. 1(1), 95-103 (2007)
Zhong, WY: The Hilbert-type integral inequality with a homogeneous kernel of Lambda-degree. J. Inequal. Appl. 2008, 917392 (2008)
Hong, Y: On Hardy-Hilbert integral inequalities with some parameters. J. Inequal. Pure Appl. Math. 6(4), 92 (2005)
Zhong, WY, Yang, BC: On multiple Hardy-Hilbert’s integral inequality with kernel. J. Inequal. Appl. 2007, 27962 (2007)
Krnić, M, Pečarić, JE, Vuković, P: On some higher-dimensional Hilbert’s and Hardy-Hilbert’s type integral inequalities with parameters. Math. Inequal. Appl. 11, 701-716 (2008)
Krnić, M, Vuković, P: On a multidimensional version of the Hilbert-type inequality. Anal. Math. 38, 291-303 (2012)
Rassias, M, Yang, BC: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75-93 (2013)
Rassias, M, Yang, BC: On a multidimensional Hilbert-type integral inequality associated to the gamma function. Appl. Math. Comput. 249, 408-418 (2014)
Huang, QL: A new extension of Hardy-Hilbert-type inequality. J. Inequal. Appl. 2015, 397 (2015)
Perić, I, Vuković, P: Multiple Hilbert’s type inequalities with a homogeneous kernel. Banach J. Math. Anal. 5(2), 33-43 (2011)
Vandanjav, A, Tserendorj, B, Krnić, M: Multiple Hilbert-type inequalities involving some differential operators. Banach J. Math. Anal. 10(2), 320-337 (2016)
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)
Wang, AZ, Huang, QL, Yang, BC: A strengthened Mulholland-type inequality with parameters. J. Inequal. Appl. 2015, 329 (2015)
Yang, BC: A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl. Math. 20(3), 319-328 (2005)
Zhong, WY: A mixed Hilbert-type inequality and its equivalent forms. J. Guangdong Univ. Educ. 31(5), 18-22 (2011)
Zhong, WY: A half discrete Hilbert-type inequality and its equivalent forms. J. Guangdong Univ. Educ. 32(5), 8-12 (2012)
Wang, AZ, Yang, BC: A more accurate reverse half-discrete Hilbert-type inequality. J. Inequal. Appl. 2015, 85 (2015)
Yang, BC, Debnath, L: Half-Discrete Hilbert-Type Inequalities. World Scientific, Singapore (2014)
Yang, BC: A half-discrete Hilbert-type inequality with a non-homogeneous kernel and two variables. Mediterr. J. Math. 10, 677-692 (2013)
Kuang, JC: Applied Inequalities. Shangdong Science Technic Press, Jinan (2004)
Kuang, JC: Real Analysis and Functional Analysis (Continuation), vol. 2. Higher Education Press, Beijing (2015)
Acknowledgements
This work is supported by the National Natural Science Foundation (No. 61370186), and Science and Technology Planning Project 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. AW participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Wang, A., Yang, B. A more accurate half-discrete Hardy-Hilbert-type inequality with the logarithmic function. J Inequal Appl 2017, 153 (2017). https://doi.org/10.1186/s13660-017-1408-x
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DOI: https://doi.org/10.1186/s13660-017-1408-x