# On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function

## Abstract

By introducing some parameters, we establish a generalization of the Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree $-2\lambda$ and the best constant factor which involves the hypergeometric function.

MSC:26D15.

## 1 Introduction

If $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $f\left(x\right),g\left(x\right)\ge 0$ satisfy

$0<{\int }_{0}^{\mathrm{\infty }}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}0<{\int }_{0}^{\mathrm{\infty }}{g}^{q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty },$

then

${\int }_{0}^{\mathrm{\infty }}{\int }_{0}^{\mathrm{\infty }}\frac{f\left(x\right)g\left(y\right)}{x+y}\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy<\frac{\pi }{sin\left(\pi /p\right)}{\left\{{\int }_{0}^{\mathrm{\infty }}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right\}}^{\frac{1}{p}}{\left\{{\int }_{0}^{\mathrm{\infty }}{g}^{q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right\}}^{\frac{1}{q}},$
(1)

where the constant factor $\pi /\left(sin\pi /p\right)$ is the best possible. Inequality (1) is called Hardy-Hilbert’s inequality [1] and is important in analysis and applications [2].

In 2001, Yang gave an extension of (1) involving beta function as (see [3]):

$\begin{array}{r}{\int }_{0}^{\mathrm{\infty }}{\int }_{0}^{\mathrm{\infty }}\frac{f\left(x\right)g\left(y\right)}{{\left(x+y\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}
(2)

where the constant factor $B\left(\frac{p+\lambda -2}{p},\frac{q+\lambda -2}{q}\right)$ ($\lambda >2-min\left\{p,q\right\}$) is the best possible.

Recently, some new Hilbert-type inequalities in the whole plane have been obtained [4, 5]. Xin and Yang in [5] established the following:

If $p>1$, $\frac{1}{p}+\frac{1}{q}=1$, $|\beta |<1$, $0<{\alpha }_{1}<{\alpha }_{2}<\pi$, $f,g\ge 0$, satisfy

$0<{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\beta -1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}0<{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy<\mathrm{\infty },$

then we have

$\begin{array}{r}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{x}^{2}+2xycos{\alpha }_{i}+{y}^{2}}\right\}f\left(x\right)g\left(y\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}
(3)

and

$\begin{array}{r}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{p\left(1-\beta \right)-1}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{x}^{2}+2xycos{\alpha }_{i}+{y}^{2}}\right\}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{p}\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}<{k}^{p}\left(\beta \right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\beta -1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,\end{array}$
(4)

where the constant factors $k\left(\beta \right)=\frac{\pi }{sin\beta \pi }\left(\frac{sin\beta {\alpha }_{1}}{sin{\alpha }_{1}}+\frac{sin\beta \left(\pi -{\alpha }_{2}\right)}{sin{\alpha }_{2}}\right)$ and ${k}^{p}\left(\beta \right)$ are the best possible. Inequalities (3) and (4) are equivalent.

By introducing some parameters, we establish generalizations of inequalities (3) and (4) with the homogeneous kernel of degree $-2\lambda$ and the best constant factor which involves the hypergeometric function.

## 2 Preliminary lemmas

In order to prove our assertions, we need the following lemmas.

Recall that the hypergeometric function $F\left(\alpha ,\beta ;\gamma ;x\right)$ is defined [6] by

$F\left(\alpha ,\beta ;\gamma ;x\right)=\sum _{r=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{r}{\left(\beta \right)}_{r}}{{\left(\gamma \right)}_{r}}\frac{{x}^{r}}{r!},$
(5)

where ${\left(\alpha \right)}_{r}$ is the Pochhammer symbol defined by

${\left(\alpha \right)}_{r}=\alpha \left(\alpha +1\right)\cdots \left(\alpha +r-1\right)=\frac{\mathrm{\Gamma }\left(\alpha +r\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

It is known the series (5) converges for $|x|<1$ and diverges for $|x|>1$. The hypergeometric function satisfies the integral representation

Lemma 2.1 (See [7])

Suppose that $a,c>0$, ${b}^{2}, $0<\alpha <2\lambda$. Then we have

${\int }_{0}^{\mathrm{\infty }}\frac{{x}^{\alpha -1}}{{\left(a{x}^{2}+2bx+c\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx={a}^{-\frac{\alpha }{2}}{c}^{\frac{\alpha }{2}-\lambda }B\left(\alpha ,2\lambda -\alpha \right)F\left(\frac{\alpha }{2},\lambda -\frac{\alpha }{2};\lambda +\frac{1}{2};1-\frac{{b}^{2}}{ac}\right).$

Lemma 2.2 Let $a,c,\lambda >0$, $b\ge 0$, $1-2\lambda <\beta <1$ and $0<{\alpha }_{1}<{\alpha }_{2}<\pi$ be real parameters such that ${b}^{2}max\left\{{cos}^{2}{\alpha }_{1},{cos}^{2}\left(\pi -{\alpha }_{2}\right)\right\}. Define the weight functions $\omega \left(x\right)$ and $\varpi \left(y\right)$ ($x,y\in \left(-\mathrm{\infty },\mathrm{\infty }\right)$) as follows:

$\begin{array}{c}\omega \left(x\right):={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\frac{{|x|}^{\beta +2\lambda -1}}{{|y|}^{\beta }}\phantom{\rule{0.2em}{0ex}}dy,\hfill \\ \varpi \left(y\right):={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\frac{{|y|}^{1-\beta }}{{|x|}^{-\beta -2\lambda +2}}\phantom{\rule{0.2em}{0ex}}dx.\hfill \end{array}$

Then we have $\omega \left(x\right)=\varpi \left(y\right)={C}_{\lambda }$ ($x,y\ne 0$), where

$\begin{array}{rl}{C}_{\lambda }=& {a}^{\frac{1-\beta }{2}-\lambda }{c}^{\frac{1-\beta }{2}}B\left(1-\beta ,2\lambda +\beta -1\right)\\ ×\left[F\left(\frac{1-\beta }{2},\lambda -\frac{1-\beta }{2};\lambda +\frac{1}{2};1-\frac{{b}^{2}{cos}^{2}{\alpha }_{1}}{ac}\right)\\ +F\left(\frac{1-\beta }{2},\lambda -\frac{1-\beta }{2};\lambda +\frac{1}{2};1-\frac{{b}^{2}{cos}^{2}\left(\pi -{\alpha }_{2}\right)}{ac}\right)\right].\end{array}$

Proof For $x\in \left(-\mathrm{\infty },0\right)$, setting $u=y/x$, $u=-y/x$ in the following two integrals, respectively, and using Lemma 2.1, we get

$\begin{array}{rcl}\omega \left(x\right)& =& {\int }_{-\mathrm{\infty }}^{0}\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{1}+c{y}^{2}\right)}^{\lambda }}\frac{{\left(-x\right)}^{\beta +2\lambda -1}}{{\left(-y\right)}^{\beta }}\phantom{\rule{0.2em}{0ex}}dy\\ +{\int }_{0}^{\mathrm{\infty }}\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{2}+c{y}^{2}\right)}^{\lambda }}\frac{{\left(-x\right)}^{\beta +2\lambda -1}}{{y}^{\beta }}\phantom{\rule{0.2em}{0ex}}dy\\ =& {\int }_{0}^{\mathrm{\infty }}\frac{{u}^{-\beta }}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{0}^{\mathrm{\infty }}\frac{{u}^{-\beta }}{{\left(c{u}^{2}+2bucos\left(\pi -{\alpha }_{2}\right)+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ =& {C}_{\lambda }.\end{array}$

For $x\in \left(0,\mathrm{\infty }\right)$, setting $u=-y/x$, $u=y/x$ in the following two integrals, respectively, and using Lemma 2.1, we get

$\begin{array}{rcl}\omega \left(x\right)& =& {\int }_{-\mathrm{\infty }}^{0}\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{2}+c{y}^{2}\right)}^{\lambda }}\frac{{x}^{\beta +2\lambda -1}}{{\left(-y\right)}^{\beta }}\phantom{\rule{0.2em}{0ex}}dy\\ +{\int }_{0}^{\mathrm{\infty }}\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{1}+c{y}^{2}\right)}^{\lambda }}\frac{{x}^{\beta +2\lambda -1}}{{y}^{\beta }}\phantom{\rule{0.2em}{0ex}}dy\\ =& {\int }_{0}^{\mathrm{\infty }}\frac{{u}^{-\beta }}{{\left(c{u}^{2}+2bucos\left(\pi -{\alpha }_{2}\right)+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{0}^{\mathrm{\infty }}\frac{{u}^{-\beta }}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ =& {C}_{\lambda }.\end{array}$

By the same way, we still can find that $\omega \left(x\right)=\varpi \left(y\right)={C}_{\lambda }$ ($x,y\ne 0$). The lemma is proved. □

Lemma 2.3 Let p and q be conjugate parameters with $p>1$, and let $a,c,\lambda >0$, $b\ge 0$, $1-2\lambda <\beta <1$, $0<{\alpha }_{1}<{\alpha }_{2}<\pi$ and ${b}^{2}max\left\{{cos}^{2}{\alpha }_{1},{cos}^{2}\left(\pi -{\alpha }_{2}\right)\right\}, and $f\left(x\right)$ be a nonnegative measurable function in $\left(-\mathrm{\infty },\mathrm{\infty }\right)$, then we have

$\begin{array}{rl}J& :={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{p\left(1-\beta \right)-1}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{p}\phantom{\rule{0.2em}{0ex}}dy\\ \le {C}_{\lambda }^{p}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(6)

Proof By Lemma 2.2 and Hölder’s inequality [8], we have

$\begin{array}{r}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{p}\\ \phantom{\rule{1em}{0ex}}=\left[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\\ \phantom{\rule{2em}{0ex}}×{\left(\frac{{|x|}^{\left(-\beta -2\lambda +2\right)/q}}{{|y|}^{\beta /p}}f\left(x\right)\right)\left(\frac{{|y|}^{\beta /p}}{{|x|}^{\left(-\beta -2\lambda +2\right)/q}}\right)\phantom{\rule{0.2em}{0ex}}dx\right]}^{p}\\ \phantom{\rule{1em}{0ex}}\le {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\frac{{|x|}^{\left(1-p\right)\left(\beta +2\lambda -2\right)}}{{|y|}^{\beta }}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\frac{{|y|}^{\left(q-1\right)\beta }}{{|x|}^{\left(-\beta -2\lambda +2\right)}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{p-1}\\ \phantom{\rule{1em}{0ex}}={C}_{\lambda }^{p-1}{|y|}^{p\left(\beta -1\right)+1}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\frac{{|x|}^{\left(1-p\right)\left(\beta +2\lambda -2\right)}}{{|y|}^{\beta }}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(7)

Then by the Fubini theorem, it follows that

$\begin{array}{rl}J& \le {C}_{\lambda }^{p-1}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\frac{{|x|}^{\left(1-p\right)\left(\beta +2\lambda -2\right)}}{{|y|}^{\beta }}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right]\phantom{\rule{0.2em}{0ex}}dy\\ ={C}_{\lambda }^{p-1}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\omega \left(x\right){|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ ={C}_{\lambda }^{p}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$

The lemma is proved. □

## 3 Main results

Theorem 3.1 Let p and q be conjugate parameters with $p>1$, and let $a,c,\lambda >0$, $b\ge 0$, $1-2\lambda <\beta <1$, $0<{\alpha }_{1}<{\alpha }_{2}<\pi$ and ${b}^{2}max\left\{{cos}^{2}{\alpha }_{1},{cos}^{2}\left(\pi -{\alpha }_{2}\right)\right\}, and $f,g\ge 0$, satisfy $0<{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }$ and $0<{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy<\mathrm{\infty }$. Then

$\begin{array}{rl}I& :={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}f\left(x\right)g\left(y\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy\\ <{C}_{\lambda }{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q},\end{array}$
(8)

and

$\begin{array}{rl}J& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{p\left(1-\beta \right)-1}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{p}\phantom{\rule{0.2em}{0ex}}dy\\ <{C}_{\lambda }^{p}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,\end{array}$
(9)

where the constant factors ${C}_{\lambda }$ and ${C}_{\lambda }^{p}$ are the best possible and ${C}_{\lambda }$ is defined in Lemma  2.2. Inequalities (8) and (9) are equivalent.

Proof If (7) takes the form of the equality for a $y\in \left(-\mathrm{\infty },0\right)\cup \left(0,\mathrm{\infty }\right)$, then there exist constants A and B such that they are not all zero, and

Hence, there exists a constant K such that

We suppose $A\ne 0$ (otherwise $B=A=0$). Then ${|x|}^{p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)=K/\left(A|x|\right)$ a.e. in $\left(-\mathrm{\infty },\mathrm{\infty }\right)$, which contradicts the fact that $0<{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }$. Hence, (7) takes the form of a strict inequality, so does (6), and we have (9).

By Hölder’s inequality [8], we have

$\begin{array}{rl}I=& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({|y|}^{1/q-\beta }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)\\ ×\left({|y|}^{\beta -1/q}g\left(y\right)\right)\phantom{\rule{0.2em}{0ex}}dy\\ \le & {J}^{1/p}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}.\end{array}$
(10)

By (9), we have (8). On the other hand, suppose that (8) is valid. Set

$g\left(y\right)={|y|}^{p\left(1-\beta \right)-1}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{p-1},$

then it follows $J={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$. By (6), we have $J<\mathrm{\infty }$. If $J=0$, then (9) is obviously valid. If $0, then by (8), we obtain

$\begin{array}{rcl}0& <& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy=J=I\\ <& {C}_{\lambda }{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q},\end{array}$

and

$\begin{array}{rcl}{J}^{1/p}& =& {\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ <& {C}_{\lambda }{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}.\end{array}$

Hence, we have (9), which is equivalent to (8).

For $\epsilon >0$, define functions $\stackrel{˜}{f}\left(x\right)$, $\stackrel{˜}{g}\left(y\right)$ as follows:

$\begin{array}{rcl}\stackrel{˜}{f}\left(x\right)& :=& \left\{\begin{array}{ll}{x}^{\left(\beta +2\lambda -2\right)-2\epsilon /p},& x\in \left(1,\mathrm{\infty }\right),\\ 0,& x\in \left[-1,1\right],\\ {\left(-x\right)}^{\left(\beta +2\lambda -2\right)-2\epsilon /p},& x\in \left(-\mathrm{\infty },-1\right),\end{array}\\ \stackrel{˜}{g}\left(y\right)& :=& \left\{\begin{array}{ll}{y}^{-\beta -2\epsilon /q},& y\in \left(1,\mathrm{\infty }\right),\\ 0,& y\in \left[-1,1\right],\\ {\left(-y\right)}^{-\beta -2\epsilon /q},& y\in \left(-\mathrm{\infty },-1\right).\end{array}\end{array}$

Then

$\stackrel{˜}{L}:={\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta +2\lambda -2\right)-1}{\stackrel{˜}{f}}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{\stackrel{˜}{g}}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}=\frac{1}{\epsilon },$

and

$\begin{array}{rcl}\stackrel{˜}{I}& :=& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{{\left(a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}\right)}^{\lambda }}\right\}\stackrel{˜}{f}\left(x\right)\stackrel{˜}{g}\left(y\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy\\ =& {I}_{1}+{I}_{2}+{I}_{3}+{I}_{4},\end{array}$

where

$\begin{array}{rcl}{I}_{1}& :=& {\int }_{-\mathrm{\infty }}^{-1}{\left(-x\right)}^{\left(\beta +2\lambda -2\right)-2\epsilon /p}\left[{\int }_{-\mathrm{\infty }}^{-1}\frac{{\left(-y\right)}^{-\beta -2\epsilon /q}}{{\left(a{x}^{2}+2bxycos{\alpha }_{1}+c{y}^{2}\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dy\right]\phantom{\rule{0.2em}{0ex}}dx,\\ {I}_{2}& :=& {\int }_{-\mathrm{\infty }}^{-1}{\left(-x\right)}^{\left(\beta +2\lambda -2\right)-2\epsilon /p}\left[{\int }_{1}^{\mathrm{\infty }}\frac{{y}^{-\beta -2\epsilon /q}}{{\left(a{x}^{2}+2bxycos{\alpha }_{2}+c{y}^{2}\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dy\right]\phantom{\rule{0.2em}{0ex}}dx,\\ {I}_{3}& :=& {\int }_{1}^{\mathrm{\infty }}{x}^{\left(\beta +2\lambda -2\right)-2\epsilon /p}\left[{\int }_{-\mathrm{\infty }}^{-1}\frac{{\left(-y\right)}^{-\beta -2\epsilon /q}}{{\left(a{x}^{2}+2bxycos{\alpha }_{2}+c{y}^{2}\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dy\right]\phantom{\rule{0.2em}{0ex}}dx,\end{array}$

and

${I}_{4}:={\int }_{1}^{\mathrm{\infty }}{x}^{\left(\beta +2\lambda -2\right)-2\epsilon /p}\left[{\int }_{1}^{\mathrm{\infty }}\frac{{y}^{-\beta -2\epsilon /q}}{{\left(a{x}^{2}+2bxycos{\alpha }_{1}+c{y}^{2}\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dy\right]\phantom{\rule{0.2em}{0ex}}dx.$

Taking $u=y/x$, by the Fubini theorem, we obtain

$\begin{array}{rl}{I}_{1}=& {I}_{4}={\int }_{1}^{\mathrm{\infty }}{x}^{-1-2\epsilon }{\int }_{1/x}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\phantom{\rule{0.2em}{0ex}}dx\\ =& {\int }_{1}^{\mathrm{\infty }}{x}^{-1-2\epsilon }\left({\int }_{1/x}^{1}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\right)\phantom{\rule{0.2em}{0ex}}dx\\ =& {\int }_{0}^{1}\left({\int }_{1/u}^{\mathrm{\infty }}{x}^{-1-2\epsilon }\phantom{\rule{0.2em}{0ex}}dx\right)\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ +\frac{1}{2\epsilon }{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ =& \frac{1}{2\epsilon }\left({\int }_{0}^{1}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\right),\end{array}$

and

${I}_{2}={I}_{3}=\frac{1}{2\epsilon }\left({\int }_{0}^{1}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\right).$

In view of the above results, if the constant factor ${C}_{\lambda }$ in (8) is not the best possible, then there exists a positive number $\stackrel{˜}{C}$ with $\stackrel{˜}{C}<{C}_{\lambda }$ such that

$\begin{array}{r}{\int }_{0}^{1}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ \phantom{\rule{2em}{0ex}}+{\int }_{0}^{1}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ \phantom{\rule{1em}{0ex}}=\epsilon \stackrel{˜}{I}<\epsilon \stackrel{˜}{C}\cdot \stackrel{˜}{L}=\stackrel{˜}{C}.\end{array}$
(11)

By the Fatou lemma and (11), we have

$\begin{array}{rcl}{C}_{\lambda }& =& {\int }_{0}^{\mathrm{\infty }}\frac{{u}^{-\beta }}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{0}^{\mathrm{\infty }}\frac{{u}^{-\beta }}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ =& {\int }_{0}^{1}\underset{\epsilon \to {0}^{+}}{lim}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\underset{\epsilon \to {0}^{+}}{lim}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ +{\int }_{0}^{1}\underset{\epsilon \to {0}^{+}}{lim}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\underset{\epsilon \to {0}^{+}}{lim}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ \le & \underset{\epsilon \to {0}^{+}}{lim}\left[{\int }_{0}^{1}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}+2bucos{\alpha }_{1}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\\ +{\int }_{0}^{1}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du+{\int }_{1}^{\mathrm{\infty }}\frac{{u}^{-\beta -2\epsilon /q}}{{\left(c{u}^{2}-2bucos{\alpha }_{2}+a\right)}^{\lambda }}\phantom{\rule{0.2em}{0ex}}du\right]\\ \le & \stackrel{˜}{C},\end{array}$

which contradicts the fact that $\stackrel{˜}{C}<{C}_{\lambda }$. Hence, the constant factor ${C}_{\lambda }$ in (8) is the best possible.

If the constant factor in (9) is not the best possible, then by (10), we may get a contradiction that the constant factor in (8) is not the best possible. Thus the theorem is proved. □

Remark 1 Setting $\lambda =a=b=c=1$ in Theorem 3.1, we have (3) and (4).

Remark 2 Setting $\lambda =1/2$ in Theorem 3.1, we have the following particular results:

$\begin{array}{rl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{\sqrt{a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}}}\right\}f\left(x\right)g\left(y\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy\\ <& {C}_{1/2}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta -1\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|y|}^{q\beta -1}{g}^{q}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q},\end{array}$

and

$\begin{array}{rl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}& {|y|}^{p\left(1-\beta \right)-1}{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{i\in \left\{1,2\right\}}{min}\left\{\frac{1}{\sqrt{a{x}^{2}+2bxycos{\alpha }_{i}+c{y}^{2}}}\right\}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{p}\phantom{\rule{0.2em}{0ex}}dy\\ <& {C}_{1/2}^{p}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x|}^{-p\left(\beta -1\right)-1}{f}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$

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Correspondence to Tserendorj Batbold.

### Competing interests

The authors declare that they have no competing interests.

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Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

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Adiyasuren, V., Batbold, T. On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function. J Inequal Appl 2013, 189 (2013). https://doi.org/10.1186/1029-242X-2013-189

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• DOI: https://doi.org/10.1186/1029-242X-2013-189

### Keywords

• Hilbert’s inequality
• Hölder’s inequality
• homogeneous kernel
• weight function
• equivalent form