- Research
- Open Access
A discrete Hilbert-type inequality in the whole plane
- Dongmei Xin1Email author,
- Bicheng Yang1 and
- Qiang Chen2
Journal of Inequalities and Applications20162016:133
https://doi.org/10.1186/s13660-016-1075-3
© Xin et al. 2016
- Received: 6 March 2016
- Accepted: 19 April 2016
- Published: 5 May 2016
Abstract
By the use of weight coefficients and technique of real analysis, a discrete Hilbert-type inequality in the whole plane with multi-parameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.
Keywords
- Hilbert-type inequality
- weight coefficient
- equivalent form
- operator expression
MSC
- 26D15
- 47A05
1 Introduction
Suppose that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{m},b_{n}\geq 0\), \(a=\{a_{m}\}_{m=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\Vert a\Vert _{p}=(\sum_{m=1}^{\infty}a_{m}^{p})^{\frac{1}{p}}>0\), \(\Vert b\Vert _{q}>0\). We have the following well-known Hardy-Hilbert inequality: where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is the best possible (cf. [1]). Also we have the following Hilbert-type inequality: with the best possible constant factor pq (cf. [2]). Inequalities (1) and (2) are important in analysis and its applications (cf. [2–4]).
$$ \sum_{n=1}^{\infty}\sum _{m=1}^{\infty}\frac{a_{m}b_{n}}{m+n}< \frac {\pi}{\sin(\frac{\pi}{p})}\Vert a \Vert _{p}\Vert b\Vert _{q}, $$
(1)
$$ \sum_{n=1}^{\infty}\sum _{m=1}^{\infty}\frac{a_{m}b_{n}}{\max\{m,n\}}< pq\Vert a\Vert _{p}\Vert b\Vert _{q}, $$
(2)
In 2011, Yang gave an extension of (2) as follows (cf. [5]): If \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(a_{m},b_{n}\geq0\), \(0<\Vert a\Vert _{p,\varphi}=\{\sum_{m=1}^{\infty }m^{p(1-\lambda_{1})-1}a_{m}^{p}\}^{\frac{1}{p}}<\infty \), \(0<\Vert b\Vert _{q,\psi}=\{\sum_{n=1}^{\infty}n^{q(1-\lambda_{2})-1} b_{n}^{q}\}^{\frac{1}{q}}<\infty\), then where the constant factor \(\frac{\lambda}{\lambda_{1}\lambda_{2}}\) is the best possible.
$$ \sum_{n=1}^{\infty}\sum _{m=1}^{\infty}\frac{a_{m}b_{n}}{(\max \{m,n\})^{\lambda}}< \frac{\lambda}{\lambda_{1}\lambda_{2}}\Vert a\Vert _{p,\varphi}\Vert b\Vert _{q,\psi}, $$
(3)
For \(\lambda=1\), \(\lambda_{1}=\frac{1}{q}\), \(\lambda_{2}=\frac{1}{p}\), inequality (3) reduces to (2). Some other results relate to (1)-(3) are provided by [6–23].
In this paper, by the use of weight coefficients and the technique of real analysis, an extension of (3) in the whole plane is given as follows: For \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda _{2}=\lambda\), \(a_{m},b_{n}\geq0\), \(0<\sum_{|m|=1}^{\infty }|m|^{p(1-\lambda _{1})-1}a_{m}^{p}<\infty\), \(0<\sum_{|n|=1}^{\infty}|n|^{q(1-\lambda _{2})-1}b_{n}^{q}<\infty\), we have Moreover, a generation of (4) with multi-parameters and a best possible constant factor is proved. The equivalent forms, the operator expressions and a few particular inequalities are also considered.
$$\begin{aligned} &\sum_{|n|=1}^{\infty}\sum _{|m|=1}^{\infty}\frac{1}{(\max \{|m|,|n|\})^{\lambda}}a_{m}b_{n} \\ &\quad< \frac{2\lambda}{\lambda_{1}\lambda_{2}} \Biggl[ \sum_{|m|=1}^{\infty }|m|^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \Biggl[ \sum_{|n|=1}^{\infty}|n|^{q(1-\lambda_{2})-1}b_{n}^{q} \Biggr] ^{\frac {1}{q}}. \end{aligned}$$
(4)
2 Some lemmas
In the following, we agree that \(\mathbf{N}=\{1,2,\ldots\}\), \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\alpha,\beta\in(0,\pi)\), \(\lambda _{1},\lambda _{2}>-\eta\), \(\lambda_{1}+\lambda_{2}=\lambda\), and for \(|x|,|y|>0\),
$$ k(x,y):=\frac{(\min\{|x|+x\cos\alpha,|y|+y\cos\beta\})^{\eta }}{(\max \{|x|+x\cos\alpha,|y|+y\cos\beta\})^{\lambda+\eta}}. $$
(5)
Lemma 1
(cf. [24])
Suppose that
\(g(t)\) (>0) is decreasing in
\(\mathbf{R}_{+}\)
and strictly decreasing in
\([n_{0},\infty)\) (\(n_{0}\in \mathbf{N}\)), satisfying
\(\int_{0}^{\infty}g(t)\,dt\in\mathbf{R}_{+}\). We have
$$ \int_{1}^{\infty}g(t)\,dt< \sum _{n=1}^{\infty}g(n)< \int_{0}^{\infty}g(t)\,dt. $$
(6)
Definition 1
Define the following weight coefficients:
where \(\sum_{|j|=1}^{\infty}\cdots=\sum_{j=-1}^{-\infty}\cdots +\sum_{j=1}^{\infty}\cdots\) (\(j=m,n\)).
$$\begin{aligned}& \omega(\lambda_{2},m) : =\sum _{|n|=1}^{\infty}k(m,n)\frac{(|m|+m\cos \alpha)^{\lambda_{1}}}{(|n|+n\cos\beta)^{1-\lambda_{2}}},\quad |m|\in \mathbf{N}, \end{aligned}$$
(7)
$$\begin{aligned}& \varpi(\lambda_{1},n) : =\sum _{|m|=1}^{\infty}k(m,n)\frac{(|n|+n\cos \beta)^{\lambda_{2}}}{(|m|+m\cos\alpha)^{1-\lambda_{1}}},\quad |n|\in \mathbf{N}, \end{aligned}$$
(8)
Lemma 2
If
\(\lambda_{2}\leq1-\eta\), then for
\(k_{\beta }(\lambda_{1}):=\frac{2(\lambda+2\eta)\csc^{2}\beta}{(\lambda _{1}+\eta )(\lambda_{2}+\eta)}\), we have
where
$$ k_{\beta}(\lambda_{1}) \bigl(1-\theta( \lambda_{2},m)\bigr)< \omega(\lambda _{2},m)< k_{\beta}( \lambda_{1}),\quad |m|\in\mathbf{N}, $$
(9)
$$\begin{aligned} \theta(\lambda_{2},m) :=&\frac{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}{\lambda+2\eta} \int_{0}^{\frac{1+\cos\beta}{|m|+m\cos\alpha}}\frac{ (\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&O \biggl( \frac{1}{(|m|+m\cos\alpha)^{\eta+\lambda_{2}}} \biggr) \in (0,1),\quad |m|\in\mathbf{N}. \end{aligned}$$
(10)
Proof
For \(|x|>0\), we set from which we have
$$\begin{aligned}& k^{(1)}(x,y) := \frac{[\min\{|x|+x\cos\alpha,y(\cos\beta-1)\} ]^{\eta}}{[\max\{|x|+x\cos\alpha,y(\cos\beta-1)\}]^{\lambda+\eta}},\quad y< 0, \\& k^{(2)}(x,y) := \frac{[\min\{|x|+x\cos\alpha,y(1+\cos\beta)\} ]^{\eta}}{[\max\{|x|+x\cos\alpha,y(1+\cos\beta)\}]^{\lambda+\eta}},\quad y>0, \end{aligned}$$
$$k^{(1)}(x,-y)=\frac{[\min\{|x|+x\cos\alpha,y(1-\cos\beta)\}]^{\eta }}{[\max\{|x|+x\cos\alpha,y(1-\cos\beta)\}]^{\lambda+\eta}},\quad y>0. $$
We obtain
$$\begin{aligned} \omega(\lambda_{2},m) =&\sum _{n=-1}^{-\infty}k^{(1)}(m,n)\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{[n(\cos\beta-1)]^{1-\lambda_{2}}} \\ &{}+\sum_{n=1}^{\infty}k^{(2)}(m,n) \frac{(|m|+m\cos\alpha)^{\lambda _{1}}}{[n(1+\cos\beta)]^{1-\lambda_{2}}} \\ =&\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1-\cos\beta)^{1-\lambda _{2}}}\sum_{n=1}^{\infty} \frac{k^{(1)}(m,-n)}{n^{1-\lambda_{2}}} \\ &{}+\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1+\cos\beta)^{1-\lambda _{2}}}\sum_{n=1}^{\infty} \frac{k^{(2)}(m,n)}{n^{1-\lambda_{2}}}. \end{aligned}$$
(11)
For fixed \(|m|\in\mathbf{N}\), \(\lambda_{2}\leq1-\eta\), we find that is decreasing for \(y>0\) and strictly decreasing for \(y\geq\frac {|m|+m\cos \alpha}{1-\cos\beta}\). Under the same assumptions, it is evident that is decreasing for \(y>0\) and strictly decreasing for \(y\geq\frac {|m|+m\cos \alpha}{1+\cos\beta}\).
$$\begin{aligned} \frac{k^{(1)}(m,-y)}{y^{1-\lambda_{2}}} =&\frac{[\min\{|m|+m\cos \alpha ,y(1-\cos\beta)\}]^{\eta}}{y^{1-\lambda_{2}}[\max\{|m|+m\cos\alpha ,y(1-\cos\beta)\}]^{\lambda+\eta}} \\ =& \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} \frac{(1-\cos\beta)^{\eta}}{(|m|+m\cos\alpha)^{\lambda +\eta}}\frac{1}{y^{1-(\lambda_{2}+\eta)}},&0< y< \frac{|m|+m\cos\alpha}{1-\cos \beta} , \\ \frac{(|m|+m\cos\alpha)^{\eta}}{(1-\cos\beta)^{\lambda +\eta}}\frac{1}{y^{1+(\lambda_{1}+\eta)}},&y\geq\frac{|m|+m\cos\alpha }{1-\cos \beta}\end{array}\displaystyle \right . \end{aligned}$$
$$\begin{aligned} \frac{k^{(2)}(m,y)}{y^{1-\lambda_{2}}} =&\frac{[\min\{|m|+m\cos \alpha ,y(1+\cos\beta)\}]^{\eta}}{y^{1-\lambda_{2}}[\max\{|m|+m\cos\alpha ,y(1+\cos\beta)\}]^{\lambda+\eta}} \\ =& \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} \frac{(1+\cos\beta)^{\eta}}{(|m|+m\cos\alpha)^{\lambda +\eta}}\frac{1}{y^{1-(\lambda_{2}+\eta)}},&0< y< \frac{|m|+m\cos\alpha}{1+\cos \beta} , \\ \frac{(|m|+m\cos\alpha)^{\eta}}{(1+\cos\beta)^{\lambda +\eta}}\frac{1}{y^{1+(\lambda_{1}+\eta)}},&y\geq\frac{|m|+m\cos\alpha }{1+\cos \beta}\end{array}\displaystyle \right . \end{aligned}$$
By (11) and (6), we have Setting \(u=\frac{y(1-\cos\beta)}{|m|+m\cos\alpha}(\frac{y(1+\cos \beta)}{|m|+m\cos\alpha})\) in the above first (second) integral, by simplifications, we find
$$\begin{aligned} \omega(\lambda_{2},m) < &\frac{(|m|+m\cos\alpha)^{\lambda _{1}}}{(1-\cos \beta)^{1-\lambda_{2}}} \int_{0}^{\infty}\frac {k^{(1)}(m,-y)}{y^{1-\lambda _{2}}}\,dy \\ &{}+\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1+\cos\beta)^{1-\lambda _{2}}} \int_{0}^{\infty}\frac{k^{(2)}(m,y)}{y^{1-\lambda_{2}}}\,dy. \end{aligned}$$
$$\begin{aligned} \omega(\lambda_{2},m) < & \biggl(\frac{1}{1-\cos\beta}+ \frac{1}{1+\cos \beta} \biggr) \int_{0}^{\infty}\frac{(\min\{1,u\})^{\eta}u^{\lambda _{2}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&2\csc^{2}\beta \biggl( \int_{0}^{1}u^{\eta+\lambda _{2}-1}\,du+ \int_{1}^{\infty}\frac{u^{\lambda_{2}-1}}{u^{\lambda+\eta }}\,du \biggr) \\ =&\frac{2(\lambda+2\eta)\csc^{2}\beta}{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}=k_{\beta}(\lambda_{1}). \end{aligned}$$
Still by (11) and (6), we have
$$\begin{aligned} \omega(\lambda_{2},m) >&\frac{(|m|+m\cos\alpha)^{\lambda _{1}}}{(1-\cos \beta)^{1-\lambda_{2}}} \int_{1}^{\infty}\frac {k^{(1)}(m,-y)}{y^{1-\lambda _{2}}}\,dy \\ &{}+\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1+\cos\beta)^{1-\lambda _{2}}} \int_{1}^{\infty}\frac{k^{(2)}(m,y)}{y^{1-\lambda_{2}}}\,dy \\ \geq&\frac{1}{1-\cos\beta} \int_{\frac{1+\cos\beta}{|m|+m\cos \alpha}}^{\infty}\frac{(\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max \{1,u\})^{\lambda+\eta}}\,du \\ &{}+\frac{1}{1+\cos\beta} \int_{\frac{1+\cos\beta}{|m|+m\cos\alpha}}^{\infty}\frac{(\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max \{1,u\})^{\lambda+\eta}}\,du \\ =&k_{\beta}(\lambda_{1}) \bigl(1-\theta( \lambda_{2},m)\bigr)>0. \end{aligned}$$
We obtain for \(|m|+m\cos\alpha\geq1+\cos\beta\)
Then we have (9) and (10). □
$$\begin{aligned} 0 < &\theta(\lambda_{2},m)=\frac{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}{\lambda+2\eta} \int_{0}^{\frac{1+\cos\beta}{|m|+m\cos\alpha}}\frac{ (\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&\frac{(\lambda_{1}+\eta)(\lambda_{2}+\eta)}{\lambda+2\eta} \int _{0}^{\frac{1+\cos\beta}{|m|+m\cos\alpha}}u^{\eta+\lambda_{2}-1}\,du \\ =&\frac{\lambda_{1}+\eta}{\lambda+2\eta} \biggl( \frac{1+\cos\beta }{|m|+m\cos\alpha} \biggr) ^{\eta+\lambda_{2}}. \end{aligned}$$
In the same way, we have the following.
Lemma 3
If
\(\lambda_{1}\leq1-\eta\), then for
\(k_{\alpha }(\lambda_{1})=\frac{2(\lambda+2\eta)\csc^{2}\alpha}{(\lambda _{1}+\eta )(\lambda_{2}+\eta)}\), we have
where
$$ k_{\alpha}(\lambda_{1}) \bigl(1-\vartheta( \lambda_{1},n)\bigr)< \varpi(\lambda _{1},n)< k_{\alpha}( \lambda_{1}),\quad |n|\in\mathbf{N}, $$
(12)
$$\begin{aligned} \vartheta(\lambda_{1},n) :=&\frac{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}{\lambda+2\eta} \int_{0}^{\frac{1+\cos\alpha}{|n|+n\cos\beta }}\frac{(\min\{1,u\})^{\eta}u^{\lambda_{1}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&O \biggl( \frac{1}{(|n|+n\cos\beta)^{\eta+\lambda_{1}}} \biggr) \in (0,1),\quad |n|\in\mathbf{N}. \end{aligned}$$
(13)
Lemma 4
If
\(\theta\in(0,\pi)\), then for
\(\rho>0\), \(H_{\rho }(\theta):=\sum_{|n|=1}^{\infty}\frac{1}{(|n|+n\cos\theta)^{1+\rho}}\), we have
$$ H_{\rho}(\theta)= \biggl[ \frac{1}{(1+\cos\theta)^{1+\rho}}+ \frac{1}{ (1-\cos\theta)^{1+\rho}} \biggr] \frac{1+\rho O(1)}{\rho} \quad\bigl(\rho \rightarrow0^{+}\bigr). $$
(14)
Proof
We have By (6), we find Hence we have (14). □
$$\begin{aligned} H_{\rho}(\theta) =&\sum_{n=-1}^{-\infty} \frac{1}{[n(\cos\theta -1)]^{1+\rho}}+\sum_{n=1}^{\infty} \frac{1}{[n(\cos\theta +1)]^{1+\rho}} \\ =& \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta )^{1+\rho}} \biggr] \sum _{n=1}^{\infty}\frac{1}{n^{1+\rho}}. \end{aligned}$$
$$\begin{aligned}& \begin{aligned} H_{\rho}(\theta) &= \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+ \frac {1}{(1+\cos\theta)^{1+\rho}} \biggr] \Biggl( 1+\sum_{n=2}^{\infty} \frac {1}{n^{1+\rho}} \Biggr) \\ &< \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta )^{1+\rho}} \biggr] \biggl( 1+ \int_{1}^{\infty}\frac{dy}{y^{1+\rho }} \biggr) \\ &=\frac{1}{\rho} \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta)^{1+\rho}} \biggr] (1+ \rho), \end{aligned} \\& \begin{aligned} H_{\rho}(\theta) &> \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac {1}{(1+\cos\theta)^{1+\rho}} \biggr] \int_{1}^{\infty}\frac {dy}{y^{1+\rho}} \\ &= \frac{1}{\rho} \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta)^{1+\rho}} \biggr] . \end{aligned} \end{aligned}$$
3 Main results
Theorem 1
If
\(\lambda_{1},\lambda_{2}\leq1-\eta\), \(a_{m},b_{n}\geq0\) (\(|m|,|n|\in\mathbf{N}\)),
then we have the following equivalent inequalities:
$$\begin{aligned} &0 < \sum_{|m|=1}^{\infty}\bigl(|m|+m\cos \alpha\bigr)^{p(1-\lambda _{1})-1}a_{m}^{p}< \infty, \\ &0 < \sum_{|n|=1}^{\infty}\bigl(|n|+n\cos \beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q}< \infty, \\ &k_{\alpha,\beta}(\lambda_{1}):=k_{\beta}^{\frac{1}{p}}( \lambda _{1})k_{\alpha}^{\frac{1}{q}}(\lambda_{1})= \frac{2(\lambda+2\eta )\csc^{\frac{2}{p}}\beta\csc^{\frac{2}{q}}\alpha}{(\lambda_{1}+\eta )(\lambda _{2}+\eta)}, \end{aligned}$$
(15)
$$\begin{aligned} &\begin{aligned}[b] I :={}&\sum_{|n|=1}^{\infty} \sum_{|m|=1}^{\infty}k(m,n)a_{m}b_{n} \\ < {}&k_{\alpha,\beta}(\lambda_{1}) \Biggl[ \sum _{|m|=1}^{\infty }\bigl(|m|+m\cos \alpha\bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ &{}\times \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n\cos \beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{q}}, \end{aligned} \end{aligned}$$
(16)
$$\begin{aligned} &\begin{aligned}[b] J :={}& \Biggl[ \sum _{|n|=1}^{\infty}\bigl(|n|+n\cos\beta\bigr)^{p\lambda _{2}-1} \Biggl( \sum _{|m|=1}^{\infty}k(m,n)a_{m} \Biggr) ^{p} \Biggr] ^{\frac{1}{p}} \\ < {}&k_{\alpha,\beta}(\lambda_{1}) \Biggl[ \sum _{|m|=1}^{\infty }\bigl(|m|+m\cos \alpha\bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}}. \end{aligned} \end{aligned}$$
(17)
In particular, for
\(\alpha=\beta=\frac{\pi}{2}\), we have the following equivalent inequalities:
$$\begin{aligned} &\begin{aligned}[b] &\sum_{|n|=1}^{\infty} \sum_{|m|=1}^{\infty}\frac{(\min\{|m|,|n|\} )^{\eta}}{(\max\{|m|,|n|\})^{\lambda+\eta}}a_{m}b_{n} \\ &\quad< \frac{2(\lambda+2\eta)}{(\lambda_{1}+\eta)(\lambda_{2}+\eta )} \Biggl[ \sum_{|m|=1}^{\infty}|m|^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac {1}{p}} \Biggl[ \sum_{|n|=1}^{\infty}|n|^{q(1-\lambda_{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{q}}, \end{aligned} \end{aligned}$$
(18)
$$\begin{aligned} &\begin{aligned}[b] & \Biggl[ \sum_{|n|=1}^{\infty}|n|^{p\lambda_{2}-1} \Biggl( \sum_{|m|=1}^{\infty}\frac{(\min\{|m|,|n|\})^{\eta}}{(\max \{|m|,|n|\})^{\lambda+\eta}}a_{m} \Biggr) ^{p} \Biggr] ^{\frac{1}{p}} \\ &\quad< \frac{2(\lambda+2\eta)}{(\lambda_{1}+\eta)(\lambda_{2}+\eta )} \Biggl[ \sum_{|m|=1}^{\infty}|m|^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac {1}{p}}. \end{aligned} \end{aligned}$$
(19)
Proof
By Hölder’s inequality (cf. [25]) and (8), we have By (12), we have By (9), we have (17).
$$\begin{aligned} \Biggl( \sum_{|m|=1}^{\infty}k(m,n)a_{m} \Biggr) ^{p} =& \Biggl[ \sum_{|m|=1}^{\infty}k(m,n) \frac{(|m|+m\cos\alpha )^{(1-\lambda _{1})/q}a_{m}}{(|n|+n\cos\beta)^{(1-\lambda_{2})/p}}\frac{(|n|+n\cos \beta)^{(1-\lambda_{2})/p}}{(|m|+m\cos\alpha)^{(1-\lambda_{1})/q}} \Biggr] ^{p} \\ \leq& \sum_{|m|=1}^{\infty}k(m,n) \frac{(|m|+m\cos\alpha)^{(1-\lambda _{1})p/q}}{(|n|+n\cos\beta)^{1-\lambda_{2}}}a_{m}^{p} \\ &{} \times \Biggl[ \sum_{|m|=1}^{\infty}k(m,n) \frac{(|n|+n\cos\beta )^{(1-\lambda_{2})q/p}}{(|m|+m\cos\alpha)^{1-\lambda_{1}}} \Biggr] ^{p-1} \\ =&\frac{(\varpi(\lambda_{1},n))^{p-1}}{(|n|+n\cos\beta)^{p\lambda _{2}-1}}\sum_{|m|=1}^{\infty}k(m,n) \frac{(|m|+m\cos\alpha)^{(1-\lambda _{1})p/q}}{(|n|+n\cos\beta)^{1-\lambda_{2}}}a_{m}^{p}. \end{aligned}$$
$$\begin{aligned} J < &k_{\alpha}^{\frac{1}{q}}(\lambda_{1}) \Biggl[ \sum_{|n|=1}^{\infty }\sum _{|m|=1}^{\infty}k(m,n)\frac{(|m|+m\cos\alpha)^{(1-\lambda _{1})(p-1)}}{(|n|+n\cos\beta)^{1-\lambda_{2}}}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ =&k_{\alpha}^{\frac{1}{q}}(\lambda_{1}) \Biggl[ \sum _{|m|=1}^{\infty }\sum_{|n|=1}^{\infty}k(m,n) \frac{(|m|+m\cos\alpha)^{(1-\lambda _{1})(p-1)}}{(|n|+n\cos\beta)^{1-\lambda_{2}}}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ =&k_{\alpha}^{\frac{1}{q}}(\lambda_{1}) \Biggl[ \sum _{|m|=1}^{\infty }\omega (\lambda_{2},m) \bigl(|m|+m\cos \alpha\bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}}. \end{aligned}$$
(20)
By Hölder’s inequality (cf. [25]), we have Then by (17), we have (16).
$$\begin{aligned} I =&\sum_{|n|=1}^{\infty} \Biggl[ \bigl(|n|+n\cos\beta\bigr)^{\lambda_{2}-\frac {1}{p}}\sum_{|m|=1}^{\infty}k(m,n)a_{m} \Biggr] \bigl(|n|+n\cos\beta\bigr)^{\frac {1}{p}-\lambda_{2}}b_{n} \\ \leq&J \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n\cos \beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{q}}. \end{aligned}$$
(21)
On the other hand, assuming that (16) is valid, we set Then it follows that By (20), we find \(J<\infty\). If \(J=0\), then (17) is evidently valid; if \(J>0\), then by (16), we have namely, (17) follows, which is equivalent to (16). □
$$b_{n}:=\bigl(|n|+n\cos\beta\bigr)^{p\lambda_{2}-1} \Biggl( \sum _{|m|=1}^{\infty }k(m,n)a_{m} \Biggr) ^{p-1},\quad |n|\in\mathbf{N}. $$
$$J= \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n\cos \beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{p}}. $$
$$\begin{aligned}& \begin{aligned} 0 < {}&\sum_{|n|=1}^{\infty}\bigl(|n|+n \cos\beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q}=J^{p}=I \\ < {}&k_{\alpha,\beta}(\lambda_{1}) \Biggl[ \sum _{|m|=1}^{\infty }\bigl(|m|+m\cos \alpha\bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ &{}\times \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n\cos \beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{q}}, \end{aligned} \\& \begin{aligned} J ={}& \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n \cos\beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{p}} \\ < {}&k_{\alpha,\beta}(\lambda_{1}) \Biggl[ \sum _{|m|=1}^{\infty }\bigl(|m|+m\cos \alpha\bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}}, \end{aligned} \end{aligned}$$
Theorem 2
As regards the assumptions of Theorem 1, the constant factor \(k_{\alpha,\beta}(\lambda_{1})\) in (16) and (17) is the best possible.
Proof
For any \(\varepsilon\in(0,q(\lambda_{2}+\eta))\), we set \(\widetilde{\lambda}_{1}=\lambda_{1}+\frac{\varepsilon}{q}\) (\(>-\eta\)), \(\widetilde{\lambda}_{2}=\lambda_{2}-\frac{\varepsilon}{q}\) (\(\in(-\eta ,1-\eta)\)), and Then by (14) and (9), we find
$$\begin{aligned}& \widetilde{a}_{m} : =\bigl(|m|+m\cos\alpha\bigr)^{(\lambda_{1}-\frac {\varepsilon}{p})-1}=\bigl(|m|+m\cos \alpha\bigr)^{\widetilde{\lambda}_{1}-\varepsilon -1}\quad\bigl(|m|\in \mathbf{N}\bigr), \\& \widetilde{b}_{n} : =\bigl(|n|+n\cos\beta\bigr)^{(\lambda_{2}-\frac {\varepsilon}{q})-1}=\bigl(|n|+n\cos \beta\bigr)^{\widetilde{\lambda}_{2}-1}\quad\bigl(|n|\in\mathbf{N}\bigr). \end{aligned}$$
$$\begin{aligned}& \begin{aligned} \widetilde{I}_{1} :={}& \Biggl[ \sum _{|m|=1}^{\infty}\bigl(|m|+m\cos\alpha \bigr)^{p(1-\lambda_{1})-1} \widetilde{a}_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ &{}\times \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n\cos \beta\bigr)^{q(1-\lambda _{2})-1}\widetilde{b}_{n}^{q} \Biggr] ^{\frac{1}{q}} \\ ={}& \Biggl[ \sum_{|m|=1}^{\infty} \frac{1}{(|m|+m\cos\alpha )^{1+\varepsilon}} \Biggr] ^{\frac{1}{p}} \Biggl[ \sum _{|n|=1}^{\infty}\frac{1}{(|n|+n\cos \beta)^{1+\varepsilon}} \Biggr] ^{\frac{1}{q}} \\ ={}&\frac{1}{\varepsilon} \biggl[ \frac{1}{(1+\cos\alpha )^{1+\varepsilon}}+\frac{1}{(1-\cos\alpha)^{1+\varepsilon}} \biggr] ^{\frac{1}{p}}\bigl(1+\varepsilon O_{1}(1)\bigr)^{\frac{1}{p}} \\ &{}\times \biggl[ \frac{1}{(1+\cos\beta)^{1+\varepsilon}}+\frac {1}{(1-\cos \beta)^{1+\varepsilon}} \biggr] ^{\frac{1}{q}} \bigl(1+\varepsilon O_{2}(1)\bigr)^{\frac{1}{q}}, \end{aligned} \\& \begin{aligned} \widetilde{I} :={}&\sum_{|n|=1}^{\infty} \sum_{|m|=1}^{\infty}k(m,n)\widetilde{a}_{m}\widetilde{b}_{n} \\ ={}&\sum_{|m|=1}^{\infty}\sum _{|m|=1}^{\infty}k(m,n)\frac{(|m|+m\cos \alpha )^{\widetilde{\lambda}_{1}-\varepsilon-1}}{(|n|+n\cos\beta)^{1-\widetilde{\lambda}_{2}}} \\ ={}&\sum_{|m|=1}^{\infty}\frac{\omega(\widetilde{\lambda}_{2},m)}{(|m|+m\cos\alpha)^{\varepsilon+1}}\geq k_{\beta}(\widetilde{\lambda }_{1})\sum _{|m|=1}^{\infty}\frac{1-\theta(\widetilde{\lambda}_{2},m)}{ (|m|+m\cos\alpha)^{\varepsilon+1}} \\ ={}&k_{\beta}(\widetilde{\lambda}_{1}) \Biggl[ \sum _{|m|=1}^{\infty }\frac{1}{(|m|+m\cos\alpha)^{\varepsilon+1}}-\sum _{|m|=1}^{\infty}\frac{1}{O((|m|+m\cos\alpha)^{(\frac{\varepsilon}{p}+\lambda_{2}+\eta)+1})} \Biggr] \\ ={}&\frac{k_{\beta}(\widetilde{\lambda}_{1})}{\varepsilon}\biggl\{ \biggl[\frac{1}{ (1+\cos\alpha)^{1+\varepsilon}}+\frac{1}{(1-\cos\alpha )^{1+\varepsilon}} \biggr]\bigl(1+\varepsilon O_{1}(1)\bigr)-\varepsilon O(1)\biggr\} . \end{aligned} \end{aligned}$$
If there exists a constant \(k\leq k_{\alpha,\beta}(\lambda_{1})\), such that (16) is valid when replacing \(k_{\alpha,\beta}(\lambda_{1})\) by k, then in particular, we have \(\varepsilon\widetilde {I}<\varepsilon k\widetilde{I}_{1}\), namely, It follows that namely, Hence, \(k=k_{\alpha,\beta}(\lambda_{1})\) is the best possible constant factor of (16).
$$\begin{aligned}& k_{\beta}(\widetilde{\lambda}_{1})\biggl\{ \biggl[ \frac{1}{(1+\cos\alpha )^{1+\varepsilon}}+\frac{1}{(1-\cos\alpha)^{1+\varepsilon}}\biggr]\bigl(1+\varepsilon O_{1}(1)\bigr)-\varepsilon O(1)\biggr\} \\& \quad < k \biggl[ \frac{1}{(1+\cos\alpha)^{1+\varepsilon}}+\frac{1}{(1-\cos \alpha)^{1+\varepsilon}} \biggr] ^{\frac{1}{p}} \bigl(1+\varepsilon O_{1}(1)\bigr)^{\frac{1}{p}} \\& \qquad{} \times \biggl[ \frac{1}{(1+\cos\beta)^{1+\varepsilon}}+\frac {1}{(1-\cos \beta)^{1+\varepsilon}} \biggr] ^{\frac{1}{q}}\bigl(1+\varepsilon O_{2}(1)\bigr)^{\frac{1}{q}}. \end{aligned}$$
$$\frac{4(\lambda+2\eta)}{(\lambda_{1}+\eta)(\lambda_{2}+\eta)}\csc ^{2}\beta\csc^{2}\alpha\leq2k \csc^{\frac{2}{p}}\alpha\csc^{\frac {2}{q}}\beta \quad\bigl( \varepsilon\rightarrow0^{+}\bigr), $$
$$k_{\alpha,\beta}(\lambda_{1})=\frac{2(\lambda+2\eta)\csc^{\frac {2}{p}}\beta\csc^{\frac{2}{q}}\alpha}{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}\leq k. $$
The constant factor \(k_{\alpha,\beta}(\lambda_{1})\) in (17) is still the best possible. Otherwise, we would reach a contradiction by (21) that the constant factor in (16) is not the best possible. □
4 Operator expressions
We set functions \(\Phi(m)\) and \(\Psi(n)\) as follows: from which we have We also set the following weight normed spaces: Then for \(a=\{a_{m}\}_{|m|=1}^{\infty}\in l_{p,\Phi }\), \(c=\{c_{n}\}_{|n|=1}^{\infty}\), \(c_{n}=\sum_{|m|=1}^{\infty }k(m,n)a_{m}\), in view of (17), we have \(\Vert c\Vert _{p,\Psi^{1-p}}< k_{\alpha,\beta }(\lambda_{1})\Vert a\Vert _{p,\Phi}<\infty\), namely, \(c\in l_{p,\Psi^{1-p}}\).
$$\begin{aligned}& \Phi(m) : =\bigl(|m|+m\cos\alpha\bigr)^{p(1-\lambda_{1})-1} \quad\bigl(|m|\in\mathbf {N}\bigr), \\& \Psi(n) : =\bigl(|n|+n\cos\beta\bigr)^{q(1-\lambda_{2})-1} \quad\bigl(|n|\in\mathbf{N}\bigr), \end{aligned}$$
$$\Psi^{1-p}(n)=\bigl(|n|+n\cos\beta\bigr)^{p\lambda_{2}-1} \quad\bigl(|n|\in\mathbf{N}\bigr). $$
$$\begin{aligned}& l_{p,\Phi} : =\Biggl\{ a=\{a_{m}\}_{|m|=1}^{\infty}; \Vert a\Vert _{p,\Phi}= \Biggl\{ \sum_{|m|=1}^{\infty} \Phi(m)|a_{m}|^{p}\Biggr\} ^{\frac{1}{p}}< \infty \Biggr\} , \\& l_{q,\Psi} : =\Biggl\{ b=\{b_{n}\}_{|n|=1}^{\infty}; \Vert b\Vert _{q,\Psi}= \Biggl\{ \sum_{|n|=1}^{\infty} \Psi(n)|b_{n}|^{q}\Biggr\} ^{\frac{1}{q}}< \infty \Biggr\} , \\& l_{p,\Psi^{1-p}} : =\Biggl\{ c=\{c_{n}\}_{|n|=1}^{\infty}; \Vert c\Vert _{p,\Psi ^{1-p}}= \Biggl\{ \sum_{|n|=1}^{\infty} \Psi^{1-p}(n)|c_{n}|^{p}\Biggr\} ^{\frac {1}{p}}< \infty \Biggr\} . \end{aligned}$$
Definition 2
Define a Hilbert-type operator \(T:l_{p,\Phi }\rightarrow l_{p,\Psi^{1-p}}\) as follows: For any \(a=\{a_{m}\} _{|m|=1}^{\infty}\in l_{p,\Phi}\), there exists a unique representation \(c=Ta\in l_{p,\Psi^{1-p}}\). We also define the formal inner product of Ta and \(b=\{b_{n}\}_{|n|=1}^{\infty}\in l_{q,\Psi}\) (\(b_{n}\geq0\)) as follows:
$$ (Ta,b):=\sum_{|n|=1}^{\infty}\sum _{|m|=1}^{\infty}k(m,n)a_{m}b_{n}. $$
(22)
Then for \(a_{m}\geq0\) (\(|m|\in\mathbf{N}\)), we may rewrite (16) and (17) as follows:
$$\begin{aligned}& (Ta,b) < k_{\alpha,\beta}(\lambda_{1})\Vert a\Vert _{p,\Phi}\Vert b\Vert _{q,\Psi}, \end{aligned}$$
(23)
$$\begin{aligned}& \Vert Ta\Vert _{p,\Psi^{1-p}} < k_{\alpha,\beta}( \lambda_{1})\Vert a\Vert _{p,\Phi}. \end{aligned}$$
(24)
We define the norm of operator T as follows: Then \(\Vert Ta\Vert _{p,\Psi^{1-p}}\leq\Vert T\Vert \cdot\Vert a\Vert _{p,\Phi}\). Since by Theorem 2, the constant factor \(k_{\alpha,\beta}(\lambda_{1})\) in (24) is the best possible, we have
$$ \Vert T\Vert :=\sup_{a\ (\neq\theta)\in l_{p,\Phi}}\frac{\Vert Ta\Vert _{p,\Psi ^{1-p}}}{\Vert a\Vert _{p,\Phi}}. $$
(25)
$$ \Vert T\Vert =k_{\alpha,\beta}(\lambda_{1})= \frac{2(\lambda+2\eta)\csc ^{\frac{2}{p}}\beta\csc^{\frac{2}{q}}\alpha}{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}. $$
(26)
Remark 1
(i) For \(\eta=0\), (16) reduces to the following inequality: In particular, for \(\alpha=\beta=\frac{\pi}{2}\), (27) reduces to (4). If \(a_{-m}=a_{m}\), \(b_{-n}=b_{n}\) (\(m,n\in\mathbf{N}\)), then (4) reduces to (3). Hence, (16) is an extension of (4) with multi-parameters.
$$\begin{aligned} & \sum_{|n|=1}^{\infty}\sum _{|m|=1}^{\infty}\frac{1}{(\max\{|m|+m\cos \alpha,|n|+n\cos\beta\})^{\lambda}}a_{m}b_{n} \\ & \quad< \frac{2\lambda}{\lambda_{1}\lambda_{2}}\csc^{\frac{2}{p}}\beta \csc^{\frac{2}{q}}\alpha \Biggl[ \sum_{|m|=1}^{\infty}\bigl(|m|+m\cos\alpha \bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ & \qquad{} \times \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n \cos\beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{q}}. \end{aligned}$$
(27)
(ii) For \(\eta=-\lambda\), \(-1\leq\lambda_{1}\), \(\lambda_{2}<0\) in (16), we have In particular, for \(\alpha=\beta=\frac{\pi}{2}\), we have
$$\begin{aligned} & \sum_{|n|=1}^{\infty}\sum _{|m|=1}^{\infty}\frac{1}{(\min\{|m|+m\cos \alpha,|n|+n\cos\beta\})^{\lambda}}a_{m}b_{n} \\ &\quad < \frac{2(-\lambda)}{\lambda_{1}\lambda_{2}}\csc^{\frac{2}{p}}\beta \csc^{\frac{2}{q}}\alpha \Biggl[ \sum_{|m|=1}^{\infty}\bigl(|m|+m\cos\alpha \bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ & \qquad{} \times \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n \cos\beta\bigr)^{q(1-\lambda _{2})-1}b_{n}^{q} \Biggr] ^{\frac{1}{q}}. \end{aligned}$$
(28)
$$\begin{aligned} &\sum_{|n|=1}^{\infty}\sum _{|m|=1}^{\infty}\frac{1}{(\min \{|m|,|n|\})^{\lambda}}a_{m}b_{n} \\ &\quad< \frac{2(-\lambda)}{\lambda_{1}\lambda_{2}} \Biggl[ \sum_{|m|=1}^{\infty }|m|^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \Biggl[ \sum_{|n|=1}^{\infty}|n|^{q(1-\lambda_{2})-1}b_{n}^{q} \Biggr] ^{\frac {1}{q}}. \end{aligned}$$
(29)
(iii) For \(\lambda=0\) in (16), we have \(\lambda_{2}=-\lambda _{1}\), \(|\lambda_{1}|<\eta\) (\(\eta>0\)) and In particular, for \(\alpha=\beta=\frac{\pi}{2}\), we have
$$ \begin{aligned}[b] &\sum_{|n|=1}^{\infty}\sum _{|m|=1}^{\infty} \biggl( \frac{\min\{ |m|+m\cos \alpha,|n|+n\cos\beta\}}{\max\{|m|+m\cos\alpha,|n|+n\cos\beta\}} \biggr) ^{\eta}a_{m}b_{n} \\ &\quad< \frac{4\eta}{\eta^{2}-\lambda_{1}^{2}}\csc^{\frac{2}{p}}\beta \csc^{\frac{2}{q}}\alpha \Biggl[ \sum_{|m|=1}^{\infty}\bigl(|m|+m\cos\alpha \bigr)^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \\ &\qquad{}\times \Biggl[ \sum_{|n|=1}^{\infty}\bigl(|n|+n \cos\beta\bigr)^{q(1+\lambda _{1})-1}b_{n}^{q} \Biggr] ^{\frac{1}{q}}. \end{aligned} $$
(30)
$$\begin{aligned} &\sum_{|n|=1}^{\infty}\sum _{|m|=1}^{\infty} \biggl( \frac{\min\{ |m|,|n|\}}{\max\{|m|,|n|\}} \biggr) ^{\eta}a_{m}b_{n} \\ &\quad< \frac{4\eta}{\eta^{2}-\lambda_{1}^{2}} \Biggl[ \sum_{|m|=1}^{\infty }|m|^{p(1-\lambda_{1})-1}a_{m}^{p} \Biggr] ^{\frac{1}{p}} \Biggl[ \sum_{|n|=1}^{\infty}|n|^{q(1-\lambda_{2})-1}b_{n}^{q} \Biggr] ^{\frac {1}{q}}. \end{aligned}$$
(31)
The above particular inequalities are all with the best possible constant factors.
Declarations
Acknowledgements
This work is supported by the Science and Technology Planning Project of Guangdong Province (No. 2013A011403002), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong, 510303, P.R. China
(2)
Department of Computer Science, Guangdong University of Education, Guangzhou, Guangdong, 510303, P.R. China
References
- Hardy, GH: Note on a theorem of Hilbert concerning series of positive terms. Proc. Lond. Math. Soc. 23(2), 45-46 (1925) Google Scholar
- Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934) MATHGoogle Scholar
- Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht (1991) View ArticleMATHGoogle Scholar
- Yang, BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009) Google Scholar
- Yang, BC: Discrete Hilbert-Type Inequalities. Bentham Science Publishers, Sharjah (2011) Google Scholar
- Gao, MZ, Yang, BC: On extended Hilbert’s inequality. Proc. Am. Math. Soc. 126(3), 751-759 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Yang, BC, Debnath, L: On a new generalization of Hardy-Hilbert’s inequality and its applications. J. Math. Anal. Appl. 233, 484-497 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Yang, BC: On new generalizations of Hilbert’s inequality. J. Math. Anal. Appl. 248, 29-40 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Yang, BC, Debnath, L: On the extended Hardy-Hilbert’s inequality. J. Math. Anal. Appl. 272, 187-199 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Yang, BC: On a generalization of Hilbert’s double series theorem. Math. Inequal. Appl. 5(2), 197-204 (2002) MathSciNetMATHGoogle Scholar
- Yang, BC, Rasias, TM: On the way of weight coefficient and research for the Hilbert-type inequalities. Math. Inequal. Appl. 6(4), 625-658 (2003) MathSciNetGoogle Scholar
- Yang, BC: On new extension of Hilbert’s inequality. Acta Math. Hung. 104(4), 291-299 (2004) View ArticleMATHGoogle Scholar
- Yang, BC: On a new Hardy-Hilbert’s type inequality. Math. Inequal. Appl. 7(3), 355-363 (2004) MathSciNetMATHGoogle Scholar
- Krnić, M, Pečarić, JE: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 67(3-4), 315-331 (2005) MATHGoogle Scholar
- Yang, BC: A new Hilbert-type inequality. Bull. Belg. Math. Soc. Simon Stevin 13(3), 479-487 (2006) MathSciNetMATHGoogle Scholar
- Yang, BC: On the norm of a self-adjoint operator and applications to Hilbert’s type inequalities. Bull. Belg. Math. Soc. Simon Stevin 13, 577-584 (2006) MathSciNetMATHGoogle Scholar
- Yang, BC: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 325, 529-541 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Li, YJ, He, B: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 76(1), 1-13 (2007) View ArticleMATHGoogle Scholar
- Yang, BC, Krnić, M: On the norm of a multi-dimensional Hilbert-type operator. Sarajevo J. Math. 7(20), 223-243 (2011) MathSciNetMATHGoogle Scholar
- Krnić, M, Vuković, P: On a multidimensional version of the Hilbert-type inequality. Anal. Math. 38, 291-303 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Chen, Q, Yang, BC: A survey on the study of Hilbert-type inequalities. J. Inequal. Appl. 2015, 302 (2015) MathSciNetView ArticleGoogle Scholar
- Shi, YP, Yang, BC: On a multidimensional Hilbert-type inequality with parameters. J. Inequal. Appl. 2015, 371 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Shi, YP, Yang, BC: A new Hardy-Hilbert-type inequality with multiparameters and a best possible constant factor. J. Inequal. Appl. 2015, 380 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Yang, BC, Chen, Q: On a Hardy-Hilbert-type inequality with parameters. J. Inequal. Appl. 2015, 339 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Kuang, JC: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004) Google Scholar
Copyright
© Xin et al. 2016
