Open Access

A Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters

Journal of Inequalities and Applications20152015:314

https://doi.org/10.1186/s13660-015-0844-8

Received: 20 March 2015

Accepted: 26 September 2015

Published: 6 October 2015

Abstract

By using the way of real analysis and estimating the weight functions, we build a new Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters. The constant factor related to the beta function is proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular cases.

Keywords

Hilbert-type integral inequalityweight functionequivalent formbeta functionreverse

MSC

26D15

1 Introduction

If \(f(x),g(y)\geq0\), satisfying \(0<\int_{0}^{\infty}f^{2}(x)\, dx<\infty\) and \(0<\int_{0}^{\infty}g^{2}(y)\, dy<\infty\), then we have (cf. [1])
$$ \int_{0}^{\infty}\int_{0}^{\infty} \frac{f(x)g(y)}{x+y}\,dx\,dy< \pi \biggl( \int_{0}^{\infty}f^{2}(x) \,dx\int_{0}^{\infty}g^{2}(y)\,dy \biggr) ^{\frac {1}{2}}, $$
(1)
where the constant factor π is the best possible. Inequality (1) is known as Hilbert’s integral inequality, which is important in analysis and its applications (cf. [1, 2]).

In recent years, by using the way of weight functions, a number of extensions of (1) were given by Yang (cf. [3]). Noticing that inequality (1) is a homogeneous kernel of degree −1, in 2009, A survey of the study of Hilbert-type inequalities with the homogeneous kernels of degree negative numbers and some parameters is given by [4]. Recently, some inequalities with the homogeneous kernels of degree 0 and non-homogeneous kernels have been studied (cf. [510]). All of the above integral inequalities are built in the quarter plane.

In 2007, Yang [11] first gave a Hilbert-type integral inequality in the whole plane as follows:
$$\begin{aligned}& \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{f(x)g(y)}{(1+e^{x+y})^{\lambda}}\,dx\,dy \\& \quad < B\biggl(\frac{\lambda}{2},\frac{\lambda}{2}\biggr) \biggl(\int _{-\infty}^{\infty }e^{-\lambda x}f^{2}(x)\,dx\int _{-\infty}^{\infty}e^{-\lambda y}g^{2}(y)\,dy \biggr)^{\frac{1}{2}}, \end{aligned}$$
(2)
where the constant factor \(B(\frac{\lambda}{2},\frac{\lambda }{2})\) (\(\lambda >0\)) is the best possible, and
$$ B(u,v):=\int_{0}^{\infty}\frac{t^{u+1}}{(1+t)^{u+v}}\, dt \quad (u,v>0) $$
(3)
is the beta function (cf. [12]). He et al. [1324] also provided some Hilbert-type integral inequalities in the whole plane.

In this paper, by using the way of real analysis and estimating the weight functions, we build a new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and a few parameters. The constant factor related to the beta function is proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular cases.

2 Some lemmas

Lemma 1

Suppose that \(0<\alpha_{1}\leq\alpha_{2}<\pi\), \(\mu ,\sigma>0\), \(\mu+\sigma=\lambda\), \(\gamma\in\{\frac{1}{2k+1},2k-1\ (k\in \mathbf{N})\}\), \(\delta\in\{-1,1\}\). We define weight functions \(\omega (\sigma,y)\) (\(y\in\mathbf{R}\)), and \(\varpi(\sigma,x)\) (\(x\in \mathbf{R}\)) as follows:
$$\begin{aligned}& \omega(\sigma,y) : =\int_{-\infty}^{\infty}\min _{i\in\{1,2\}}\frac {|y|^{\sigma}|x|^{\delta\sigma-1}}{[|x^{\delta}y|^{\gamma }+(x^{\delta }y)^{\gamma}\cos\alpha_{i}+1]^{\lambda/\gamma}}\,dx, \end{aligned}$$
(4)
$$\begin{aligned}& \varpi(\sigma,x) : =\int_{-\infty}^{\infty}\min _{i\in\{1,2\}}\frac {|x|^{\delta\sigma}|y|^{\sigma-1}}{[|x^{\delta}y|^{\gamma }+(x^{\delta }y)^{\gamma}\cos\alpha_{i}+1]^{\lambda/\gamma}}\,dy. \end{aligned}$$
(5)
Then for \(y,x\in\mathbf{R}\backslash\{0\}\), we have
$$\begin{aligned} \omega(\sigma,y) =&\varpi(\sigma,x)=K(\sigma) \\ :=&\frac{1}{\gamma2^{\sigma/\gamma}} \biggl[ \biggl(\sec\frac{\alpha _{1}}{2}\biggr)^{\frac{2\sigma}{\gamma}}+ \biggl(\csc\frac{\alpha_{2}}{2}\biggr)^{\frac{2\sigma}{ \gamma}} \biggr] B\biggl( \frac{\mu}{\gamma},\frac{\sigma}{\gamma}\biggr)\in \mathbf{R}_{+}. \end{aligned}$$
(6)

Proof

(i) For \(\delta=1\), \(y\in\mathbf{R}\backslash\{0\}\), setting \(u=xy\), we find
$$\begin{aligned} \omega(\sigma,y) =&\int_{-\infty}^{\infty}\min _{i\in\{1,2\}}\frac {1}{(|u|^{\gamma}+u^{\gamma}\cos\alpha_{i}+1)^{\lambda/\gamma }}|u|^{\sigma -1}\,du \\ =&\int_{0}^{\infty}\min_{i\in\{1,2\}} \frac{1}{[u^{\gamma}(1+\cos \alpha _{i})+1]^{\lambda/\gamma}}u^{\sigma-1}\,du \\ &{}+\int_{-\infty}^{0}\min_{i\in\{1,2\}} \frac{1}{[u^{\gamma}(-1+\cos \alpha_{i})+1]^{\lambda/\gamma}}(-u)^{\sigma-1}\,du \\ =&\int_{0}^{\infty}\min_{i\in\{1,2\}} \frac{1}{[u^{\gamma}(1+\cos \alpha _{i})+1]^{\lambda/\gamma}}u^{\sigma-1}\,du \\ &{}+\int_{0}^{\infty}\min_{i\in\{1,2\}} \frac{1}{[v^{\gamma}(1-\cos \alpha _{i})+1]^{\lambda/\gamma}}v^{\sigma-1}\,dv \\ =&\int_{0}^{\infty}\frac{u^{\sigma-1}\,du}{[u^{\gamma}(1+\cos\alpha _{1})+1]^{\lambda/\gamma}}+\int _{0}^{\infty}\frac{v^{\sigma-1}\,dv}{[v^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda/\gamma}}. \end{aligned}$$
(7)
Setting \(t=u^{\gamma}(1+\cos\alpha_{1})\) (\(t=u^{\gamma}(1-\cos \alpha _{2})\)) in the above first (second) integral, by (3), it follows that
$$ \omega(\sigma,y)=\frac{1}{\gamma} \biggl[ \biggl(\frac{\sec^{2}\frac{\alpha _{1}}{2}}{2} \biggr)^{\frac{\sigma}{\gamma}}+\biggl(\frac{\csc^{2}\frac{\alpha _{2}}{2}}{2}\biggr)^{\frac{\sigma}{\gamma}} \biggr] \int_{0}^{\infty}\frac{t^{\frac {\sigma}{\gamma}-1}\,dt}{(t+1)^{\lambda/\gamma}}=K(\sigma). $$

(ii) For \(\delta=-1\), setting \(\frac{y}{x}\), we still can obtain \(\omega (\sigma,y)=K(\sigma)\).

Setting \(u=x^{\delta}y\), we also find
$$ \varpi(\sigma,x)=\int_{-\infty}^{\infty}\min _{i\in\{1,2\}}\frac{1}{ (|u|^{\gamma}+u^{\gamma}\cos\alpha_{i}+1)^{\lambda/\gamma }}|u|^{\sigma -1}\,du=K(\sigma). $$

Hence we have (6). □

Note

If we replace \(\min_{i\in\{1,2\}}\) by \(\max_{i\in\{ 1,2\}}\) in (4) and (5), then we may exchange \(\alpha_{1}\) and \(\alpha_{2}\) in (6).

Lemma 2

Suppose that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0<\alpha _{1}\leq\alpha_{2}<\pi\), \(\mu,\sigma>0\), \(\mu+\sigma=\lambda\), \(\gamma\in \{\frac{1}{2k+1},2k-1\ (k\in\mathbf{N})\}\), \(\delta\in\{-1,1\}\). If \(K(\sigma)\) is indicated by (6), \(f(x)\) is a non-negative measurable function in \((-\infty,\infty)\), then we have
$$\begin{aligned} J :=&\int_{-\infty}^{\infty}|y|^{p\sigma-1} \biggl\{ \int_{-\infty }^{\infty}\min_{i\in\{1,2\}} \frac{1}{[|x^{\delta}y|^{\gamma }+(x^{\delta }y)^{\gamma}\cos\alpha_{i}+1]^{\lambda/\gamma}}f(x)\,dx \biggr\} ^{p}\,dy \\ \leq&K^{p}(\sigma)\int_{-\infty}^{\infty}|x|^{p(1-\delta\sigma )-1}f^{p}(x) \,dx. \end{aligned}$$
(8)

Proof

We set
$$ k_{\lambda}^{(\delta)}(x,y):=\min_{i\in\{1,2\}} \frac{1}{[|x^{\delta }y|^{\gamma}+(x^{\delta}y)^{\gamma}\cos\alpha_{i}+1]^{\lambda /\gamma}} \quad (x,y\in\mathbf{R}). $$
(9)
By Hölder’s inequality (cf. [25]), we have
$$\begin{aligned}& \biggl( \int_{-\infty}^{\infty}k_{\lambda}^{(\delta )}(x,y)f(x) \,dx \biggr) ^{p} \\& \quad = \biggl\{ \int_{-\infty}^{\infty}k_{\lambda}^{(\delta)}(x,y) \biggl[ \frac{|x|^{(1-\delta\sigma)/q}}{|y|^{(1-\sigma)/p}}f(x) \biggr] \biggl[ \frac{|y|^{(1-\sigma)/p}}{|x|^{(1-\delta\sigma)/q}} \biggr] \,dx \biggr\} ^{p} \\& \quad \leq \int_{-\infty}^{\infty}k_{\lambda}^{(\delta)}(x,y) \frac{|x|^{(1-\delta\sigma)(p-1)}}{|y|^{1-\sigma}}f^{p}(x)\,dx \\& \qquad {}\times \biggl[ \int_{-\infty}^{\infty}k_{\lambda}^{(\delta )}(x,y) \frac{|y|^{(1-\sigma)(q-1)}}{|x|^{1-\delta\sigma}}\,dx \biggr] ^{p-1} \\& \quad =\bigl(\omega(\sigma,y)\bigr)^{p-1}|y|^{-p\sigma+1}\int _{-\infty}^{\infty }k_{\lambda}^{(\delta)}(x,y) \frac{|x|^{(1-\delta\sigma)(p-1)}}{|y|^{(1-\sigma)}}f^{p}(x)\,dx. \end{aligned}$$
(10)
Then by (6) and the Fubini theorem (cf. [26]), it follows that
$$\begin{aligned} J \leq&K^{p-1}(\sigma)\int_{-\infty}^{\infty} \biggl[ \int_{-\infty }^{\infty}k_{\lambda}^{(\delta)}(x,y) \frac{|x|^{(1-\delta\sigma )(p-1)}}{|y|^{(1-\sigma)}}f^{p}(x)\,dx \biggr] \,dy \\ =&K^{p-1}(\sigma)\int_{-\infty}^{\infty}\varpi( \sigma ,x)|x|^{p(1-\delta\sigma)-1}f^{p}(x)\,dx. \end{aligned}$$
Hence, still in view of (6), inequality (8) follows. □

3 Main results and applications

Theorem 1

Suppose that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0<\alpha _{1}\leq\alpha_{2}<\pi\), \(\mu,\sigma>0\), \(\mu+\sigma=\lambda\), \(\gamma\in \{\frac{1}{2k+1},2k-1\ (k\in\mathbf{N})\}\), \(\delta\in\{-1,1\}\). If \(K(\sigma)\) is indicated by (6), \(f(x),g(y)\geq0\), satisfying \(0<\int_{-\infty}^{\infty}|x|^{p(1-\delta\sigma)-1}f^{p}(x)\,dx<\infty\) and \(0<\int_{-\infty}^{\infty}|y|^{q(1-\sigma)-1}g^{q}(y)\,dy<\infty\), then we have the following equivalent inequalities:
$$\begin{aligned}& I : =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \min_{i\in\{ 1,2\}}\frac{1}{[|x^{\delta}y|^{\gamma}+(x^{\delta}y)^{\gamma}\cos\alpha _{i}+1]^{\lambda/\gamma}}f(x)g(y)\,dx\,dy \\& \hphantom{I} < K(\sigma) \biggl[ \int_{-\infty}^{\infty}|x|^{p(1-\delta\sigma )-1}f^{p}(x) \,dx \biggr] ^{\frac{1}{p}} \biggl[ \int_{-\infty}^{\infty }|y|^{q(1-\sigma)-1}g^{q}(y) \,dy \biggr] ^{\frac{1}{q}}, \end{aligned}$$
(11)
$$\begin{aligned}& J : =\int_{-\infty}^{\infty}|y|^{p\sigma-1} \biggl\{ \int _{-\infty }^{\infty}\min_{i\in\{1,2\}} \frac{1}{[|x^{\delta}y|^{\gamma }+(x^{\delta }y)^{\gamma}\cos\alpha_{i}+1]^{\lambda/\gamma}}f(x)\,dx \biggr\} ^{p}\,dy \\& \hphantom{J} < K^{p}(\sigma)\int_{-\infty}^{\infty}|x|^{p(1-\delta\sigma )-1}f^{p}(x) \,dx, \end{aligned}$$
(12)
where the constant factors \(K(\sigma)\) and \(K^{p}(\sigma)\) are the best possible.
In particular, for \(\alpha_{1}=\alpha_{2}=\alpha\in(0,\pi)\), \(\gamma=1\) in (11) and (12), we find
$$ K(\sigma)=k(\sigma):=\frac{1}{2^{\sigma}}\biggl[\biggl(\sec\frac{\alpha}{2}\biggr)^{2\sigma}+\biggl(\csc\frac{\alpha}{2}\biggr)^{2\sigma} \biggr]B(\mu,\sigma), $$
(13)
and the following equivalent inequalities:
$$\begin{aligned}& \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{1}{(|x^{\delta }y|+x^{\delta}y\cos\alpha+1)^{\lambda}}f(x)g(y)\,dx\,dy \\& \quad < k(\sigma) \biggl[ \int_{-\infty}^{\infty}|x|^{p(1-\delta\sigma )-1}f^{p}(x) \,dx \biggr] ^{\frac{1}{p}} \biggl[ \int_{-\infty}^{\infty }|y|^{q(1-\sigma)-1}g^{q}(y) \,dy \biggr] ^{\frac{1}{q}}, \end{aligned}$$
(14)
$$\begin{aligned}& \int_{-\infty}^{\infty}|y|^{p\sigma-1} \biggl[ \int _{-\infty }^{\infty}\frac{1}{(|x^{\delta}y|+x^{\delta}y\cos\alpha+1)^{\lambda }}f(x)\,dx \biggr] ^{p}\,dy \\& \quad < k^{p}(\sigma)\int_{-\infty}^{\infty}|x|^{p(1-\sigma)-1}f^{p}(x) \,dx. \end{aligned}$$
(15)

Proof

If (10) takes the form of equality for \(y\in(-\infty,0)\cup(0,\infty)\), then there exist constants A and B, such that they are not all zero, and
$$ A\frac{|x|^{(1-\delta\sigma)(p-1)}}{|y|^{(1-\sigma)}}f^{p}(x)=B\frac{ |y|^{(1-\sigma)(q-1)}}{|x|^{(1-\delta\sigma)}}\quad \mbox{a.e. in } (-\infty,\infty). $$
We suppose \(A\neq0\) (otherwise \(B=A=0\)). Then it follows that
$$ |x|^{p(1-\delta\sigma)-1}f^{p}(x)=|y|^{q(1-\sigma)}\frac{B}{A|x|}\quad \mbox{a.e. in }(-\infty,\infty), $$
which contradicts the fact that \(0<\int_{-\infty}^{\infty }|x|^{p(1-\delta \sigma)-1}f^{p}(x)\,dx<\infty\). Hence (10) takes the form of a strict inequality. So does (9), and we have (12).
By Hölder’s inequality (cf. [25]), we find
$$\begin{aligned} I =&\int_{-\infty}^{\infty} \biggl( |y|^{\sigma-\frac{1}{p}}\int _{-\infty }^{\infty}k_{\lambda}^{(\delta)}(x,y)f(x) \,dx \biggr) \bigl(|y|^{\frac{1}{p}-\sigma}g(y)\,dy\bigr) \\ \leq&J^{\frac{1}{p}} \biggl[ \int_{-\infty}^{\infty}|y|^{q(1-\sigma )-1}g^{q}(y) \,dy \biggr] ^{\frac{1}{q}}. \end{aligned}$$
(16)
Then by (12), we have (11). On the other hand, suppose that (11) is valid. Setting
$$ g(y):=|y|^{p\sigma-1} \biggl( \int_{-\infty}^{\infty}k_{\lambda }^{(\delta )}(x,y)f(x) \,dx \biggr) ^{p-1} ,\quad y\in\mathbf{R}, $$
then it follows that \(J=\int_{-\infty}^{\infty}|y|^{q(1-\sigma )-1}g^{q}(y)\,dy\). By (9), we have \(J<\infty\). If \(J=0\), then (12) is obviously of value; if \(0< J<\infty\), then by (11), we obtain
$$\begin{aligned}& \int_{-\infty}^{\infty}|y|^{q(1-\sigma)-1}g^{q}(y) \,dy \\& \quad =J=I < K(\sigma) \biggl[ \int_{-\infty}^{\infty}|x|^{p(1-\delta\sigma )-1}f^{p}(x) \,dx \biggr] ^{\frac{1}{p}} \biggl[ \int_{-\infty}^{\infty }|y|^{q(1-\sigma)-1}g^{q}(y) \,dy \biggr] ^{\frac{1}{q}}, \end{aligned}$$
(17)
$$\begin{aligned}& J^{\frac{1}{p}}= \biggl[ \int_{-\infty}^{\infty}|y|^{q(1-\sigma )-1}g^{q}(y) \,dy \biggr] ^{\frac{1}{p}} \\& \hphantom{J^{\frac{1}{p}}}< K(\sigma) \biggl[ \int_{-\infty }^{\infty}|x|^{p(1-\delta\sigma)-1}f^{p}(x) \,dx \biggr] ^{\frac{1}{p}}. \end{aligned}$$
(18)
Hence we have (12), which is equivalent to (11).
We set \(E_{\delta}:=\{x\in\mathbf{R};|x|^{\delta}\geq1\}\), and \(E_{\delta}^{+}:=E_{\delta}\cap\mathbf{R}_{+}=\{x\in\mathbf{R}_{+};x^{\delta}\geq1\}\). For \(\varepsilon>0\), we define functions \(\tilde{f}(x)\), \(\tilde{g}(y)\) as follows:
$$\begin{aligned}& \tilde{f}(x) :=\left \{ \textstyle\begin{array}{l@{\quad}l} |x|^{\delta(\sigma-\frac{2\varepsilon}{p})-1}, & x\in E_{\delta} , \\ 0, & x\in\mathbf{R}\backslash E_{\delta}, \end{array}\displaystyle \right . \\& \tilde{g}(y) :=\left \{ \textstyle\begin{array}{l@{\quad}l} 0, & y\in(-\infty,-1)\cup(1,\infty), \\ |y|^{\sigma+\frac{2\varepsilon}{q}-1}, & y\in[-1,1].\end{array}\displaystyle \right . \end{aligned}$$
Then we obtain
$$\begin{aligned} \tilde{L} :=& \biggl[ \int_{-\infty}^{\infty}|x|^{p(1-\delta\sigma )-1}\tilde{f}^{p}(x)\,dx \biggr] ^{\frac{1}{p}} \biggl[ \int _{-\infty}^{\infty }|y|^{q(1-\sigma)-1}\tilde{g}^{q}(y) \,dy \biggr] ^{\frac{1}{q}} \\ =&2 \biggl( \int_{E_{\delta}^{+}}x^{-2\delta\varepsilon-1}\,dx \biggr) ^{ \frac{1}{p}} \biggl( \int_{0}^{1}y^{2\varepsilon-1} \,dy \biggr) ^{\frac {1}{q}} \\ =&\frac{1}{\varepsilon}. \end{aligned}$$
We find
$$ h(x):=\int_{-1}^{1}\min_{i\in\{1,2\}} \frac{|y|^{\sigma+\frac {2\varepsilon }{q}-1}}{[|x^{\delta}y|^{\gamma}+(x^{\delta}y)^{\gamma}\cos\alpha _{i}+1]^{\lambda/\gamma}}\,dy=h(-x). $$
In fact, setting \(Y=-y\), we obtain
$$\begin{aligned} h(-x) =&\int_{-1}^{1}\min_{i\in\{1,2\}} \frac{|y|^{\sigma+\frac{2\varepsilon}{q}-1}}{[|-x^{\delta}y|^{\gamma}+(-x^{\delta}y)^{\gamma }\cos\alpha_{i}+1]^{\lambda/\gamma}}\,dy \\ =&\int_{-1}^{1}\min_{i\in\{1,2\}} \frac{|Y|^{\sigma+\frac {2\varepsilon}{q}-1}}{[|x^{\delta}Y|^{\gamma}+(x^{\delta}Y)^{\gamma}\cos\alpha _{i}+1]^{\lambda/\gamma}}\, dY=h(x). \end{aligned}$$
It follows that
$$\begin{aligned} \tilde{I} =&\int_{-\infty}^{\infty}\int _{-\infty}^{\infty }k_{\lambda }^{(\delta)}(x,y) \tilde{f}(x)\tilde{g}(y)\,dx\,dy \\ =&\int_{E_{\delta}}|x|^{\delta(\sigma-\frac{2\varepsilon}{p})-1}h(x)\,dx=2\int _{E_{\delta}^{+}}x^{\delta(\sigma-\frac{2\varepsilon }{p})-1}h(x)\,dx \\ \stackrel{u=x^{\delta}y}{=}&2\int_{E_{\delta}^{+}}x^{-2\delta \varepsilon -1} \biggl\{ \int_{-x^{\delta}}^{x^{\delta}}\min_{i\in\{1,2\}} \frac{|u|^{\sigma+\frac{2\varepsilon}{q}-1}}{[|u|^{\gamma}+u^{\gamma}\cos \alpha_{i}+1]^{\lambda/\gamma}}\,du \biggr\} \,dx. \end{aligned}$$
Setting \(v=x^{\delta}\) in the above integral, by the Fubini theorem (cf. [26]), we find
$$\begin{aligned} \tilde{I} =&2\int_{1}^{\infty}v^{-2\varepsilon-1} \biggl\{ \int_{-v}^{v}\min_{i\in\{1,2\}} \frac{|u|^{\sigma+\frac{2\varepsilon }{q}-1}}{[|u|^{\gamma}+u^{\gamma}\cos\alpha_{i}+1]^{\lambda/\gamma}}\,du \biggr\} \,dv \\ =& 2\int_{1}^{\infty}v^{-2\varepsilon-1}\biggl\{ \int _{0}^{v}\biggl[ \min_{i\in\{1,2\}} \frac{1}{[u^{\gamma}(1+\cos\alpha_{i})+1]^{\lambda /\gamma}} \\ &{} +\min_{i\in\{1,2\}}\frac{1}{[u^{\gamma}(1-\cos\alpha _{i})+1]^{\lambda/\gamma}}\biggr] u^{\sigma+\frac{2\varepsilon}{q}-1}\,du \biggr\} \,dv \\ =& 2\int_{1}^{\infty}v^{-2\varepsilon-1}\biggl\{ \int _{0}^{v}\biggl[ \frac{1}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}} +\frac{1}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}}\biggr] u^{\sigma+\frac{2\varepsilon}{q}-1}\,du\biggr\} \,dv \\ =&2\int_{1}^{\infty}v^{-2\varepsilon-1} \biggl\{ \int _{0}^{1}\biggl[\frac {u^{\sigma +\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\frac{ \lambda}{\gamma}}}+ \frac{u^{\sigma+\frac{2\varepsilon }{q}-1}}{[u^{\gamma }(1-\cos\alpha_{2})+1]^{\frac{\lambda}{\gamma}}}\biggr]\,du \biggr\} \,dv \\ &{}+2\int_{1}^{\infty}v^{-2\varepsilon-1} \biggl\{ \int _{1}^{v}\biggl[\frac {u^{\sigma +\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\frac{ \lambda}{\gamma}}}+ \frac{u^{\sigma+\frac{2\varepsilon }{q}-1}}{[u^{\gamma }(1-\cos\alpha_{2})+1]^{\frac{\lambda}{\gamma}}}\biggr]\,du \biggr\} \,dv \\ =& \frac{1}{\varepsilon}\int_{0}^{1} \biggl\{ \frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda /\gamma}}+\frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1-\cos\alpha _{2})+1]^{\lambda/\gamma}} \biggr\} \,du \\ &{}+2\int_{1}^{\infty}\biggl(\int _{u}^{\infty}v^{-2\varepsilon-1}\,dv\biggr) \\ &{}\times \biggl\{ \frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma }(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+\frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du \\ =& \frac{1}{\varepsilon}\biggl\{ \int_{0}^{1} \biggl[ \frac{u^{\sigma +\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda /\gamma}}+\frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1-\cos\alpha _{2})+1]^{\lambda/\gamma}} \biggr] \,du \\ &{} +\int_{1}^{\infty} \biggl[ \frac{u^{\sigma-\frac{2\varepsilon }{p}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+ \frac {u^{\sigma -\frac{2\varepsilon}{p}-1}}{[u^{\gamma}(1-\cos\alpha _{2})+1]^{\lambda /\gamma}} \biggr] \,du\biggr\} . \end{aligned}$$
If the constant factor \(K(\sigma)\) in (11) is not the best possible, then there exists a positive number k, with \(K(\sigma)< k\), such that (11) is valid when replacing \(K(\sigma)\) by k. Then we have \(\varepsilon \tilde{I}<\varepsilon k\tilde{L}\), and
$$\begin{aligned} \begin{aligned}[b] &\int_{0}^{1} \biggl\{ \frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+ \frac{u^{\sigma +\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du \\ &\qquad {} +\int_{1}^{\infty} \biggl\{ \frac{u^{\sigma-\frac{2\varepsilon }{p}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+\frac{u^{\sigma -\frac{2\varepsilon}{p}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du \\ &\quad = \varepsilon\tilde{I}< \varepsilon k\tilde{L}=k. \end{aligned} \end{aligned}$$
(19)
By (7) and the Levi theorem (cf. [26]), we have
$$\begin{aligned} K(\sigma) =&\int_{0}^{\infty}\frac{u^{\sigma-1}\,du}{[u^{\gamma }(1+\cos \alpha_{1})+1]^{\lambda/\gamma}}+\int _{0}^{\infty}\frac{u^{\sigma -1}\,du}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda/\gamma}} \\ =&\int_{0}^{1}\lim_{\varepsilon\rightarrow0^{+}} \biggl\{ \frac {u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda /\gamma}}+\frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma }(1-\cos \alpha_{2})+1]^{\lambda/\gamma}} \biggr\} \,du \\ &{}+\int_{1}^{\infty}\lim_{\varepsilon\rightarrow0^{+}} \biggl\{ \frac{u^{\sigma-\frac{2\varepsilon}{p}-1}}{[u^{\gamma}(1+\cos\alpha _{1})+1]^{\lambda/\gamma}}+\frac{u^{\sigma-\frac{2\varepsilon }{p}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda/\gamma}} \biggr\} \,du \\ =&\lim_{\varepsilon\rightarrow0^{+}}\biggl\{ \int_{0}^{1} \biggl[ \frac{ u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha _{1})+1]^{\lambda/\gamma}}+\frac{u^{\sigma+\frac{2\varepsilon }{q}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda/\gamma}} \biggr] \,du \\ &{} +\int_{1}^{\infty} \biggl[ \frac{u^{\sigma-\frac{2\varepsilon }{p}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+ \frac {u^{\sigma-\frac{2\varepsilon}{p}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr] \,du\biggr\} \leq k, \end{aligned}$$
which contradicts the fact that \(k< K(\sigma)\). Hence the constant factor \(K(\sigma)\) in (11) is the best possible.

If the constant factor in (12) is not the best possible, then by (16), we may get a contradiction: that the constant factor in (11) is not the best possible. □

Theorem 2

As the assumptions of Theorem  1, replacing \(p>1\) by \(0< p<1\), we have the equivalent reverses of (11) and (12) with the same best constant factors.

Proof

By the reverse Hölder’s inequality (cf. [25]), we have the reverses of (9) and (16). It is easy to obtain the reverse of (12). In view of the reverses of (12) and (16), we obtain the reverse of (11). On the other hand, suppose that the reverse of (11) is valid. Setting the same \(g(y)\) as Theorem 1, by the reverse of (9), we have \(J>0\). If \(J=\infty\), then the reverse of (12) is obviously value; if \(J<\infty\), then by the reverse of (11), we obtain the reverses of (17) and (18). Hence we have the reverse of (12), which is equivalent to the reverse of (11).

If the constant factor \(K(\sigma)\) in the reverse of (11) is not the best possible, then there exists a positive constant k, with \(k>K(\sigma)\), such that the reverse of (11) is still valid when replacing \(K(\sigma)\) by k. By the reverse of (19), we have
$$\begin{aligned}& \int_{0}^{1} \biggl\{ \frac{u^{\sigma+\frac{2\varepsilon }{q}-1}}{[u^{\gamma }(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+ \frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du \\& \quad {}+\int_{1}^{\infty} \biggl\{ \frac{u^{\sigma-\frac{2\varepsilon}{p}-1}}{ [u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+\frac{u^{\sigma -\frac{2\varepsilon}{p}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du>k. \end{aligned}$$
(20)
For \(\varepsilon\rightarrow0^{+}\), by the Levi theorem (cf. [26]), we find that
$$\begin{aligned}& \int_{1}^{\infty} \biggl\{ \frac{u^{\sigma-\frac{2\varepsilon}{p}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+ \frac{u^{\sigma -\frac{2\varepsilon}{p}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du \\& \quad \rightarrow\int_{1}^{\infty} \biggl\{ \frac{u^{\sigma-1}}{[u^{\gamma}(1+\cos \alpha _{1})+1]^{\lambda/\gamma}}+\frac{u^{\sigma-1}}{[u^{\gamma}(1-\cos \alpha _{2})+1]^{\lambda/\gamma}} \biggr\} \,du. \end{aligned}$$
(21)
There exists a constant \(\delta_{0}>0\), such that \(\sigma-\frac{1}{2}\delta_{0}>0\), and then \(K(\sigma-\frac{\delta_{0}}{2})<\infty\). For \(0<\varepsilon<\frac{\delta_{0}|q|}{4}\) (\(q<0\)), since \(u^{\sigma+\frac{2\varepsilon}{q}-1}\leq u^{\sigma-\frac{\delta_{0}}{2}-1}\), \(u\in(0,1]\), and
$$\begin{aligned} 0 < &\int_{0}^{1} \biggl\{ \frac{u^{\sigma-\frac{\delta_{0}}{2}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+ \frac{u^{\sigma -\frac{\delta_{0}}{2}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du \\ \leq& K\biggl(\sigma-\frac{\delta_{0}}{2}\biggr)< \infty, \end{aligned}$$
then by the Lebesgue control convergence theorem (cf. [26]), for \(\varepsilon\rightarrow0^{+}\), we have
$$\begin{aligned}& \int_{0}^{1} \biggl\{ \frac{u^{\sigma+\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1+\cos\alpha_{1})+1]^{\lambda/\gamma}}+ \frac{u^{\sigma +\frac{2\varepsilon}{q}-1}}{[u^{\gamma}(1-\cos\alpha_{2})+1]^{\lambda /\gamma}} \biggr\} \,du \\& \quad \rightarrow \int_{0}^{1} \biggl\{ \frac{u^{\sigma-1}}{[u^{\gamma }(1+\cos \alpha_{1})+1]^{\lambda/\gamma}}+\frac{u^{\sigma-1}}{[u^{\gamma }(1-\cos \alpha_{2})+1]^{\lambda/\gamma}} \biggr\} \,du. \end{aligned}$$
(22)
By (20), (21), and (22), for \(\varepsilon\rightarrow 0^{+}\), we find \(K(\sigma)\geq k\), which contradicts the fact that \(k>K(\sigma)\). Hence, the constant factor \(K(\sigma)\) in the reverse of (11) is the best possible.

If the constant factor in reverse of (12) is not the best possible, then by the reverse of (16), we may get a contradiction that the constant factor in the reverse of (11) is not the best possible. □

Remarks

For \(\delta=-1\) in (11) and (12), replacing \(|x|^{\lambda}f(x)\) by \(f(x)\), we obtain the following equivalent inequalities with the homogeneous kernel and the best possible constant factors:
$$\begin{aligned}& \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \min_{i\in\{1,2\} }\frac{1}{(|y|^{\gamma}+\operatorname{sgn}(x)y^{\gamma}\cos\alpha_{i}+|x|^{\gamma })^{\lambda /\gamma}}f(x)g(y)\,dx\,dy \\& \quad < K(\sigma) \biggl[ \int_{-\infty}^{\infty}|x|^{p(1-\mu)-1}f^{p}(x) \,dx \biggr] ^{\frac{1}{p}} \biggl[ \int_{-\infty}^{\infty}|y|^{q(1-\sigma )-1}g^{q}(y) \,dy \biggr] ^{\frac{1}{q}}, \end{aligned}$$
(23)
$$\begin{aligned}& \int_{-\infty}^{\infty}|y|^{p\sigma-1} \biggl[ \int _{-\infty }^{\infty }\min_{i\in\{1,2\}} \frac{1}{(|y|^{\gamma}+\operatorname{sgn}(x)y^{\gamma}\cos\alpha _{i}+|x|^{\gamma})^{\lambda/\gamma}}f(x)\,dx \biggr] ^{p}\,dy \\& \quad < K^{p}(\sigma)\int_{-\infty}^{\infty}|x|^{p(1-\mu)-1}f^{p}(x) \,dx. \end{aligned}$$
(24)
In particular, for \(\alpha_{1}=\alpha_{2}=\alpha\in(0,\pi)\), \(\gamma=1\) in (23) and (24), we obtain the following equivalent inequalities:
$$\begin{aligned}& \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{1}{(|y|+\operatorname{sgn}(x)y\cos\alpha+|x|)^{\lambda}}f(x)g(y)\,dx\,dy \\& \quad < k(\sigma) \biggl[ \int_{-\infty}^{\infty}|x|^{p(1-\mu)-1}f^{p}(x) \,dx \biggr] ^{\frac{1}{p}} \biggl[ \int_{-\infty}^{\infty}|y|^{q(1-\sigma )-1}g^{q}(y) \,dy \biggr] ^{\frac{1}{q}}, \end{aligned}$$
(25)
$$\begin{aligned}& \int_{-\infty}^{\infty}|y|^{p\sigma-1} \biggl[ \int _{-\infty }^{\infty}\frac{1}{(|y|+\operatorname{sgn}(x)y\cos\alpha+|x|)^{\lambda}}f(x)\,dx \biggr] ^{p}\,dy \\& \quad < k^{p}(\sigma)\int_{-\infty}^{\infty}|x|^{p(1-\mu)-1}f^{p}(x) \,dx, \end{aligned}$$
(26)
where \(k(\sigma)\) is indicated by (13).

Declarations

Acknowledgements

The authors wish to express their thanks to the referees for their careful reading of the manuscript and for their valuable suggestions. This work is supported by the National Natural Science Foundation (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2013KJCX0140).

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)
School of Economics and Trade, Guangdong University of Foreign Studies
(2)
Department of Mathematics, Guangdong University of Education

References

  1. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934) Google Scholar
  2. Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991) MATHView ArticleGoogle Scholar
  3. Yang, B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009) Google Scholar
  4. Yang, B: A survey of the study of Hilbert-type inequalities with parameters. Adv. Math. 38(3), 257-268 (2009) MathSciNetGoogle Scholar
  5. Yang, B: On the norm of an integral operator and applications. J. Math. Anal. Appl. 321, 182-192 (2006) MATHMathSciNetView ArticleGoogle Scholar
  6. Xu, J: Hardy-Hilbert’s inequalities with two parameters. Adv. Math. 36(2), 63-76 (2007) Google Scholar
  7. Yang, B: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 325, 529-541 (2007) MATHMathSciNetView ArticleGoogle Scholar
  8. Xin, D: A Hilbert-type integral inequality with the homogeneous kernel of zero degree. Math. Theory Appl. 30(2), 70-74 (2010) MathSciNetGoogle Scholar
  9. Yang, B: A Hilbert-type integral inequality with the homogeneous kernel of degree 0. J. Shandong Univ. Nat. Sci. 45(2), 103-106 (2010) MathSciNetGoogle Scholar
  10. Debnath, L, Yang, B: Recent developments of Hilbert-type discrete and integral inequalities with applications. Int. J. Math. Math. Sci. 2012, Article ID 871845 (2012) MathSciNetGoogle Scholar
  11. Yang, B: A new Hilbert-type integral inequality. Soochow J. Math. 33(4), 849-859 (2007) MATHMathSciNetGoogle Scholar
  12. Wang, Z, Guo, D: Introduction to Special Functions. Science Press, Beijing (1979) Google Scholar
  13. He, B, Yang, B: On a Hilbert-type integral inequality with the homogeneous kernel of 0-degree and the hypergeometric function. Math. Pract. Theory 40(18), 105-211 (2010) MathSciNetGoogle Scholar
  14. Yang, B: A new Hilbert-type integral inequality with some parameters. J. Jilin Univ. Sci. Ed. 46(6), 1085-1090 (2008) Google Scholar
  15. Yang, B: A Hilbert-type integral inequality with a non-homogeneous kernel. J. Xiamen Univ. Nat. Sci. 48(2), 165-169 (2008) Google Scholar
  16. Zeng, Z, Xie, Z: On a new Hilbert-type integral inequality with the homogeneous kernel of degree 0 and the integral in whole plane. J. Inequal. Appl. 2010, Article ID 256796 (2010) MathSciNetView ArticleGoogle Scholar
  17. Yang, B: A reverse Hilbert-type integral inequality with some parameters. J. Xinxiang Univ. Nat. Sci. Ed. 27(6), 1-4 (2010) MATHGoogle Scholar
  18. Wang, A, Yang, B: A new Hilbert-type integral inequality in whole plane with the non-homogeneous kernel. J. Inequal. Appl. 2011, 123 (2011) View ArticleGoogle Scholar
  19. Xin, D, Yang, B: A Hilbert-type integral inequality in whole plane with the homogeneous kernel of degree −2. J. Inequal. Appl. 2011, Article ID 401428 (2011) MathSciNetView ArticleGoogle Scholar
  20. He, B, Yang, B: On an inequality concerning a non-homogeneous kernel and the hypergeometric function. Tamsui Oxf. J. Inf. Math. Sci. 27(1), 75-88 (2011) MATHMathSciNetGoogle Scholar
  21. Yang, B: A reverse Hilbert-type integral inequality with a non-homogeneous kernel. J. Jilin Univ. Sci. Ed. 49(3), 437-441 (2011) Google Scholar
  22. Xie, Z, Zeng, Z, Sun, Y: A new Hilbert-type inequality with the homogeneous kernel of degree −2. Adv. Appl. Math. Sci. 12(7), 391-401 (2013) MATHMathSciNetGoogle Scholar
  23. Zeng, Z, Raja Rama Gandhi, K, Xie, Z: 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) Google Scholar
  24. Yang, B, Chen, Q: Two kinds of Hilbert-type integral inequalities in the whole plane. J. Inequal. Appl. 2015, 21 (2015) View ArticleGoogle Scholar
  25. Kuang, J: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004) Google Scholar
  26. Kuang, J: Introduction to Real Analysis. Hunan Education Press, Changsha (1996) Google Scholar

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