Equivalent Statements of Two Multidimensional Hilbert-Type Integral Inequalities with Parameters

Abstract

By means of the weight functions, the idea of introduced parameters and the transfer formulas, two multidimensional Hilbert-type integral inequalities with the general nonhomogeneous kernel as $H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})$ $({\lambda }_1,{\lambda }_2\ne 0)$ are given, which are some extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent conditions of the best value and several parameters related to the new inequalities are provided. Two corollaries regarding the kernel, represented as $k_{\lambda }{(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{{\lambda }_2})}^{}({\lambda }_1,{\lambda }_2\ne 0)$, are given, and a few new inequalities for the particular parameters are obtained.

1. Introduction

If $0<{\displaystyle {\sum }_{m=1}^{\infty }{a}_m^2}<\infty$ and $0<{\displaystyle {\sum }_{n=1}^{\infty }{b}_n^2}<\infty$, then we have the well-known Hilbert’s inequality with the best value $\pi$ as follows (cf. [1], Theorem 315):

$${\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }\frac{a_m{b}_n}{m+n}<\pi {({\displaystyle \sum _{m=1}^{\infty }{a}_m^2}{\displaystyle \sum _{n=1}^{\infty }{b}_n^2})}^{1/2}}}.$$

(1)

Assuming that $0<{\displaystyle {\int }_0^{\infty }{f}^2(x)dx<\infty }$ and $0<{\displaystyle {\int }_0^{\infty }{g}^2(y)}dy<\infty$, we still have the integral analogue of (1) named in Hilbert’s integral inequality as follows (cf. [1], Theorem 316):

$${\displaystyle {\int }_0^{\infty }}{\displaystyle {\int }_0^{\infty }\frac{f(x)g(y)}{x+y}dxdy<\pi ({\displaystyle {\int }_0^{\infty }{f}^2(x)dx{\displaystyle {\int }_0^{\infty }{g}^2(y)}}}dy{)}^{1/2},$$

(2)

where $\pi$ is the best value. (1) and (2), with their extensions, played an important role in real analysis. Among them, the paper [2] studied the generalizations of (1) and (2), and the papers [3,4] considered the properties of m-linear Hilbert-type inequality and two kinds of Hilbert-type inequalities involving differential operators.

A half-discrete Hilbert-type inequality was provided in 1934 as follows: If $K{(x)}^{}(x>0)$ is decreasing, $p>1,\frac{1}{p}+\frac{1}{q}=1,0<\phi (s)={\displaystyle {\int }_0^{\infty }K(x)x^{s-1}}dx<\infty$, $f(x)\ge 0,$ satisfying

$$0<{\displaystyle {\int }_0^{\infty }{f}^p}(x)dx<\infty ,$$

then (cf. [1], Theorem 351)

$${\displaystyle \sum _{n=1}^{\infty }{n}^{p-2}{({\displaystyle {\int }_0^{\infty }K}(nx)f(x)dx)}^p}<{\phi }^p(\frac{1}{q}){\displaystyle {\int }_0^{\infty }{f}^p}(x)dx.$$

(3)

Some new generalizations and applications of (3) were provided by [5,6] in recent years.

In 2006, by means of the summation formula, Krnic et al. [7] gave a generalization of (1) with the kernel as $\frac{1}{{(m+n)}^{\lambda }}(0<\lambda \le 4)$. In 2019, following [7], Adiyasuren et al. [8] gave a generalization of (1) involving two partial sums. In 2016-2017, Hong et al. [9,10] obtained some equivalent statements of the generalizations of (1) and (2) with the best values related to a few parameters. Two similar results were provided by [11,12]. Among them, the paper [11] considered multidimensional Hardy-type inequalities in Hölder spaces, and the paper [12] studied a new form of Hilbert’s integral inequality. To further understand the theory of this field and cite some useful related papers, please see Yang’s book [13]. Recently, Hong et al. [14] gave a new half-discrete multidimensional inequality involving one multiple upper limit function as an application.

In this article, following the idea of [7,8], by means of real analysis, the way of introduced parameters and the transfer formulas, two new multidimensional Hilbert-type integral inequalities with the nonhomogeneous kernel as $H{(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}^{}({\lambda }_1,{\lambda }_2\ne 0)$ are given, which are some new extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent statements of the best possible constant factor and a few parameters related to the new inequalities are provided. Furthermore, two corollaries regard the kernel, represented as $k_{\lambda }(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{{\lambda }_2})$$({\lambda }_1,{\lambda }_2\ne 0)$, are considered, and some new inequalities in a few particular parameters are obtained.

2. Some Lemmas

In what follows, we assume that $i_0,j_0\in \mathrm{N}:=\{1,2,\cdots \},p>1,\frac{1}{p}+\frac{1}{q}=1,{\sigma }_1,\sigma \in \mathrm{R}:=(-\infty ,\infty ),$ $\widehat{\sigma }:=\frac{{\sigma }_1}{p}+\frac{\sigma }{q},$

$$\begin{gathered} {\lambda }_1,{\lambda }_2\ne 0,\alpha ,\beta \in {\mathrm{R}}_{+}:=(0,\infty ), \\ \begin{array}{l}||x|{|}_{\alpha }^{}:={({\displaystyle \sum _{i=1}^{i_0}|x_i{|}^{\alpha }})}^{\frac{1}{\alpha }}(x=(x_1,\cdots ,x_{i_0})\in {\mathrm{R}}^{{\mathrm{i}}_0}),\\ ||y|{|}_{\beta }^{}:={({\displaystyle \sum _{j=1}^{j_0}|y_j{|}^{\beta }})}^{\frac{1}{\beta }}(y=(y_1,\cdots ,y_{j^0})\in {\mathrm{R}}^{{\mathrm{j}}_0}).\end{array} \end{gathered}$$

Two functions $f(x),g(y)\ge 0$, satisfying

$$0<{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\widehat{\sigma })-i_0}{f}^p}(x)dx<0^{} and 0<{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\widehat{\sigma })-j_0} g^q}(y)dy<\infty .$$

We also suppose that $H(u)$ is a nonnegative measurable function in ${\mathrm{R}}_{+}$, such that for any $\eta \in \mathrm{R},$

$$K(\eta ):={\displaystyle {\int }_0^{\infty }H(u)u^{\eta -1}}du>0,$$

which means that there exists a positive constant $T>1$, satisfying ${\displaystyle {\int }_0^{T}H(u)u^{\eta -1}}du>0$.

If $M>0,$ $\psi (u) (u>0)$ is a nonnegative measurable function, then the following transfer formula was provided (cf. [2], (9.3.3)):

$${\displaystyle \int \cdots }{\displaystyle {\int }_{\{x\in R_{+}^{i_0};0<{\displaystyle {\sum }_{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }\le 1\}}}\psi ({\displaystyle \sum _{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }})}d{x}_1\cdots d{x}_{i_0}=\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(\frac{1}{\alpha })}{{\alpha }^{i_0}\mathrm{\Gamma }(\frac{i_0}{\alpha })}{\displaystyle {\int }_0^1\psi (u)u^{\frac{i_0}{\alpha }-1}}du.$$

(4)

In particular, (i) in view of $||x|{|}_{\alpha }=M{[{\displaystyle {\sum }_{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }}]}^{\frac{1}{\alpha }},$ by (4), we have

$${\displaystyle {\int }_{R_{+}^{i_0}}\phi (||x|{|}_{\alpha }})dx=\underset{M\to \infty }{\lim }{\displaystyle \int \cdots }{\displaystyle {\int }_{\{x\in R_{+}^{i_0};0<{\displaystyle {\sum }_{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }\le 1\}}}\phi (M{[{\displaystyle \sum _{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }}]}^{\frac{1}{\alpha }})}d{x}_1\cdots d{x}_{i_0}$$

$$=\underset{M\to \infty }{\lim }\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(\frac{1}{\alpha })}{{\alpha }^{i_0}\mathrm{\Gamma }(\frac{i_0}{\alpha })}{\displaystyle {\int }_0^1\phi (M{u}^{\frac{1}{\alpha }})u^{\frac{i_0}{\alpha }-1}}du\stackrel{v=M{u}^{\frac{1}{\alpha }}}{=}\frac{{\mathrm{\Gamma }}^{i_0}(\frac{1}{\alpha })}{{\alpha }^{i_0-1}\mathrm{\Gamma }(\frac{i_0}{\alpha })}{\displaystyle {\int }_0^{\infty }\phi (v)v^{i_0-1}}dv;$$

(5)

(ii) for $\psi (u)=\phi (M{u}^{\frac{1}{\alpha }})=0.u>\frac{1}{M^{\alpha }}$, by (4), we find

$${\displaystyle {\int }_{\{x\in R_{+}^{i_0},||x|{|}_{\alpha }^{}\le 1\}}\phi (||x|{|}_{\alpha }^{}})dx=\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(\frac{1}{\alpha })}{{\alpha }^{i_0}\mathrm{\Gamma }(\frac{i_0}{\alpha })}{\displaystyle {\int }_0^{\frac{1}{M^{\alpha }}}\phi (M{u}^{\frac{1}{\alpha }})u^{\frac{i_0}{\alpha }-1}}du=\frac{{\mathrm{\Gamma }}^{i_0}(\frac{1}{\alpha })}{{\alpha }^{i_0-1}\mathrm{\Gamma }(\frac{i_0}{\alpha })}{\displaystyle {\int }_0^1\phi (v)v^{i_0-1}}dv;$$

(6)

(iii) for $\psi (u)=\phi (M{u}^{\frac{1}{\alpha }})=0.u<\frac{1}{M^{\alpha }}$, by (4), we have

$${\displaystyle {\int }_{\{x\in R_{+}^{i_0},||x|{|}_{\alpha }^{}\ge 1\}}\phi (||x|{|}_{\alpha }^{}})dx=\underset{M\to \infty }{\lim }\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(\frac{1}{\alpha })}{{\alpha }^{i_0}\mathrm{\Gamma }(\frac{i_0}{\alpha })}{\displaystyle {\int }_{\frac{1}{M^{\alpha }}}^1\phi (M{u}^{\frac{1}{\alpha }})u^{\frac{i_0}{\alpha }-1}}du=\frac{{\mathrm{\Gamma }}^{i_0}(\frac{1}{\alpha })}{{\alpha }^{i_0-1}\mathrm{\Gamma }(\frac{i_0}{\alpha })}{\displaystyle {\int }_1^{\infty }\phi (v)v^{i_0-1}}dv.$$

(7)

For given the main results, we obtain the following weight functions:

Lemma 1.

Setting $L_{\alpha }^{(i_0)}:=\frac{{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i_0-1}\mathrm{\Gamma }(i_0/\alpha )}$ and $L_{\beta }^{(j_0)}:=\frac{{\mathrm{\Gamma }}^{j_0}(1/\beta )}{{\beta }^{j_0-1}\mathrm{\Gamma }(j_0/\beta )}$, we have the following expressions of the weight functions:

$$\omega (\sigma ,y):={\displaystyle {\int }_{R_{+}^{i_0}}^{}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}||x|{|}_{\alpha }^{{\lambda }_1\sigma -i_0}dx=L_{\alpha }^{(i_0)}\frac{K(\sigma )}{|{\lambda }_1|}||y|{|}_{\beta }^{-{\lambda }_2\sigma }{}^{}( \mathrm{y}\in {\mathrm{R}}_{+}^{j_0}),$$

(8)

$$\varpi ({\sigma }_1,x):={\displaystyle {\int }_{R_{+}^{j_0}}^{}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}||y|{|}_{\beta }^{{\lambda }_2{\sigma }_1-j_0}dy=L_{\beta }^{(j_0)}\frac{K({\sigma }_1)}{|{\lambda }_2|}||x|{|}_{\alpha }^{-{\lambda }_1{\sigma }_1}{}^{} (\mathrm{x}\in {\mathrm{R}}_{+}^{{\mathrm{i}}_0}).$$

(9)

Proof. 

By (5), for $M>0,$ we have

$$\begin{array}{c}\omega (\sigma ,y)=\underset{M\to \infty }{\lim }{M}^{{\lambda }_1\sigma -i_0}{\displaystyle \int \cdots }{\displaystyle {\int }_{\{x\in R_{+}^{i_0};0<{\displaystyle {\sum }_{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }\le 1\}}}^{}H(M^{{\lambda }_1}{[{\displaystyle \sum _{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }}]}^{\frac{{\lambda }_1}{\alpha }}||y|{|}_{\beta }^{{\lambda }_2})}\\ {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\times {[{\displaystyle \sum _{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }}]}^{\frac{{\lambda }_1\sigma -i_0}{\alpha }}d{x}_1\cdots d{x}_{i_0}\end{array}$$

$$\begin{gathered} {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}=\underset{M\to \infty }{\lim }\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{M}^{{\lambda }_1\sigma -i_0}{\displaystyle {\int }_0^{1}H(M^{{\lambda }_1}{u}^{\frac{{\lambda }_1}{\alpha }}||y|{|}_{\beta }^{{\lambda }_2})}{u}^{\frac{{\lambda }_1\sigma -i_0}{\alpha }}{u}^{\frac{i_0}{\alpha }-1}du \\ =\underset{M\to \infty }{\lim }\frac{{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{M}^{{\lambda }_1\sigma }{\displaystyle {\int }_0^{1}H(M^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2}{u}^{\frac{{\lambda }_1}{\alpha }})}{u}^{\frac{{\lambda }_1\sigma }{\alpha }-1}du. \end{gathered}$$

(10)

Setting $v=M^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }^2}{u}^{\frac{{\lambda }_1}{\alpha }}$ in the above integral, for ${\lambda }_1>0$, we obtain

$$\omega (\sigma ,y)=\frac{{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{|{\lambda }_1|{\alpha }^{i_0-1}\mathrm{\Gamma }(i_0/\alpha )}||y|{|}_{\beta }^{-{\lambda }_2\sigma }{\displaystyle {\int }_0^{\infty }H(v)}{v}^{\sigma -1}dv,$$

namely, (8) follows. For ${\lambda }_1<0$, by (10), we still can obtain (8). In the same way, for ${\lambda }_2\ne 0$, we obtain (9).

This proves the lemma. □

Lemma 2.

For $b\in \mathrm{R},$ we have the expressions as follows:

$$L_1:={\displaystyle {\int }_{\{x\in R_{+}^{i_0},||x|{|}_{\alpha }^{}\le 1\}}||x|{|}_{\alpha }^{b-i_0}}dx=\left\{\begin{array}{c}\frac{1}{b}{L}_{\alpha }^{(i_0)},b>0,\\ \infty ,b\le 0\end{array}\right.,$$

(11)

$$L_2:={\displaystyle {\int }_{\{x\in R_{+}^{i_0},||x|{|}_{\alpha }^{}\ge 1\}}||x|{|}_{\alpha }^{-b-i_0}}dx=\left\{\begin{array}{c}\frac{1}{b}{L}_{\alpha }^{(i_0)}{,}^{}b>0,\\ \infty {,}^{}b\le 0\end{array}\right..$$

(12)

Proof. 

By (6), for $M>0$, we have

$$\begin{array}{l}{L}_1={\displaystyle \int \cdots }{\displaystyle {\int }_{\{x\in R_{+}^{i_0},{\displaystyle {\sum }_{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }}\le \frac{1}{M^{\alpha }}\}}{[{\displaystyle \sum _{i=1}^{i_0}(}\frac{x_i}{M})}^{\alpha }{]}^{\frac{b-i_0}{\alpha }}{M}_{}^{b-i_0}}d{x}_1\cdots d{x}_{i_0}\\ {{\displaystyle }}^{}=\underset{M\to \infty }{\lim }\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{M}_{}^{b-i_0}{\displaystyle {\int }_0^{\frac{1}{M^{\alpha }}}{u}^{\frac{b-i_0}{\alpha }}}{u}^{\frac{i_0}{\alpha }-1}du=\underset{M\to \infty }{\lim }\frac{M^b{\mathrm{\Gamma }}^{i^0}(1/\alpha )}{{\alpha }^{i^0}\mathrm{\Gamma }(i_0/\alpha )}{\displaystyle {\int }_0^{\frac{1}{M^{\alpha }}}{u}^{\frac{b}{\alpha }-1}}du.\end{array}$$

For $b>0$, we find $L_1=\frac{1}{b}{L}_{\alpha }^{(i_0^{})}$; for $b\le 0$, it follows that $L_1=\infty$. Hence, (11) follows.

In the same way, by (7), for $M>0$, we have

$$\begin{array}{l}{L}_2={\displaystyle \int \cdots }{\displaystyle {\int }_{\{x\in R_{+}^{i_0},{\displaystyle {\sum }_{i=1}^{i_0}{(\frac{x_i}{M})}^{\alpha }}\ge \frac{1}{M^{\alpha }}\}}[{\displaystyle \sum _{i=1}^{i_0}(}{\frac{x_i}{M})}^{\alpha }{]}^{\frac{-b-i_0}{\alpha }}{M}_{}^{-b-i_0}}d{x}_1\cdots d{x}_{i_0}\\ {{\displaystyle }}^{}=\underset{M\to \infty }{\lim }\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{M}_{}^{-b-i_0}{\displaystyle {\int }_{\frac{1}{M^{\alpha }}}^1{u}^{\frac{-b-i_0}{\alpha }}}{u}^{\frac{i_0}{\alpha }-1}du=\underset{M\to \infty }{\lim }\frac{M^{-b}{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i^0}\mathrm{\Gamma }(i_0/\alpha )}{\displaystyle {\int }_{\frac{1}{M^{\alpha }}}^1{u}^{\frac{-b}{\alpha }-1}}du.\end{array}$$

For $b>0$, we find $L_2=\frac{1}{b}{L}_{\alpha }^{(i_0^{})}$; for $b\le 0$, it follows that $L_2=\infty$. Hence, we have (12).

This proves the lemma. □

In view of (6) and (7), we give the following expressions:

Lemma 3.

(i) If ${\sigma }_1>\sigma ,$ then for $0<\epsilon <{\sigma }_1-\sigma$, we have

$${\tilde{I}}_{\epsilon }:=\epsilon {\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{{\lambda }_1}\ge 1\}}||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1-\frac{\epsilon }{p})-i_0}}\left[{\displaystyle {\int }_{\{y\in R_{+}^{j_0};||y|{|}_{\beta }^{{\lambda }_2}\le 1\}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})||y|{|}_{\beta }^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}dy}\right]dx=\infty ;$$

(13)

(ii) If ${\sigma }_1<\sigma$, then for $0<\epsilon <\sigma -{\sigma }_1$, we have

$${\widehat{I}}_{\epsilon }:=\epsilon {\displaystyle {\int }_{\{y\in R_{+}^{j_0};||y|{|}_{\beta }^{{\lambda }_2}\ge 1\}}||y|{|}_{\beta }^{{\lambda }_2({\sigma }_1-\frac{\epsilon }{q})-j_0}}\left[{\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{{\lambda }_1}\le 1\}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})||x|{|}_{\alpha }^{{\lambda }_1(\sigma +\frac{\epsilon }{p})-i_0}dy}\right]dx=\infty ;$$

(14)

(iii) If ${\sigma }_1=\sigma$ (in (13)), then

$$\begin{gathered} I_{\epsilon }:=\epsilon {\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{{\lambda }_1}\ge 1\}}||x|{|}_{\alpha }^{{\lambda }_1(\sigma -\frac{\epsilon }{p})-i_0}}\left[{\displaystyle {\int }_{\{y\in R_{+}^{j_0};||y|{|}_{\beta }^{{\lambda }_2}\le 1\}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})||y|{|}_{\beta }^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}dy}\right]dx \\ \ge L_{\alpha }^{(i_0)}{L}_{\beta }^{(j_0)}\frac{K(\sigma )}{|{\lambda }_1{\lambda }_2|}+o{(1)}^{}(\epsilon \to 0^{+}). \end{gathered}$$

(15)

Proof. 

(i) By (6), for $M,{\lambda }_2>0$, we have

$$\begin{array}{ll}h(||x|{|}_{\alpha }^{{\lambda }_1}):& =||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{q})}{\displaystyle {\int }_{\{y\in R_{+}^{j_0};||y|{|}_{\beta }^{{\lambda }_2}\le 1\}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})||y|{|}_{\beta }^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}dy}\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{q})}{\displaystyle \int \cdots }{\displaystyle {\int }_{{\{y\in R_{+}^{j_0};0<{\displaystyle {\sum }_{j=1}^{j_0}(\frac{y_j}{M}})}^{\beta }\le M^{-\beta }\}}H{(||x|{|}_{\alpha }^{{\lambda }_1}{M}^{{\lambda }_2}{[{\displaystyle {\sum }_{j=1}^{j_0}(\frac{y_j}{M}})}^{\beta }]}^{\frac{{\lambda }_2}{\beta }})}\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{} \times M^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}[{{\displaystyle {\sum }_{j=1}^{j_0}(\frac{y_j}{M}})}^{\beta }{]}^{\frac{1}{\beta }[{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0]}d{y}_1\cdots d{y}_{j_0}\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{q})}\underset{M\to \infty }{\lim }\frac{M^{j_0}{\mathrm{\Gamma }}^{j_0}(1/\beta )}{{\beta }^{j_0}\mathrm{\Gamma }(j_0/\beta )}{\displaystyle {\int }_0^{M^{-\beta }}H(||x|{|}_{\alpha }^{{\lambda }_1}{M}^{{\lambda }_2}{u}^{\frac{{\lambda }_2}{\beta }})M^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}{u}^{{}^{\frac{1}{\beta }[{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0]}}}{u}^{\frac{j_0}{\beta }-1}du\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{q})}\underset{M\to \infty }{\lim }\frac{M^{\lambda {(\sigma +\frac{\epsilon }{q})}_2{}_{}}{\mathrm{\Gamma }}^{j_0}(1/\beta )}{{\beta }^{j_0}\mathrm{\Gamma }(j_0/\beta )}{\displaystyle {\int }_0^{M^{-\beta }}H(||x|{|}_{\alpha }^{{\lambda }_1}{M}^{{\lambda }_2}{u}^{\frac{{\lambda }_2}{\beta }})u^{\frac{1}{\beta }{\lambda }_2(\sigma +\frac{\epsilon }{q})-1}}du.\end{array}$$

Setting $v=||x|{|}_{\alpha }^{{\lambda }_1^{}}{M}^{{\lambda }_2}{u}^{\frac{{\lambda }_2}{\beta }}$ in the above expression, in view of ${\lambda }_2>0$, it follows that

$$h(||x|{|}_{\alpha }^{{\lambda }_1})=\frac{1}{{\lambda }_2}{L}_{\beta }^{(j_0)}||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1-\sigma )}{\displaystyle {\int }_0^{||x|{|}_{\alpha }^{{\lambda }_1}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv.$$

(16)

For ${\lambda }_2<0,$ by (7), we obtain

$$\begin{array}{ll}h(||x|{|}_{\alpha }^{{\lambda }_1})& =||x|{|}_{\alpha }^{{\lambda }_1({\sigma }^{+1}\frac{\epsilon }{q})}{\displaystyle {\int }_{\{y\in R_{+}^{j_0};||y|{|}_{\beta }^{}\ge 1\}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})||y|{|}_{\beta }^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}dy}\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{q})}{\displaystyle \int \cdots }{\displaystyle {\int }_{{\{y\in R_{+}^{j_0};{\displaystyle {\sum }_{j=1}^{j_0}(\frac{y_j}{M}})}^{\beta }\ge M^{-\beta }\}}H{(||x|{|}_{\alpha }^{{\lambda }_1}{M}^{{\lambda }_2}{[{\displaystyle {\sum }_{j=1}^{j_0}(\frac{y_j}{M}})}^{\beta }]}^{\frac{{\lambda }_2}{\beta }})}\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{} \times M^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}[{{\displaystyle {\sum }_{j=1}^{j_0}(\frac{y_j}{M}})}^{\beta }{]}^{\frac{1}{\beta }[{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0]}d{y}_1\cdots d{y}_{j_0}\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{q})}\underset{M\to \infty }{\lim }\frac{M^{j_0}{\mathrm{\Gamma }}^{j_0}(1/\beta )}{{\beta }^{j_0}\mathrm{\Gamma }(j_0/\beta )}{\displaystyle {\int }_{M^{-\beta }}^{1}H(||x|{|}_{\alpha }^{{\lambda }_1}{M}^{{\lambda }_2}{u}^{\frac{{\lambda }_2}{\beta }})M^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0}{u}^{{}^{\frac{1}{\beta }[{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0]}}}{u}^{\frac{j_0}{\beta }-1}du\\ & {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{q})}\underset{M\to \infty }{\lim }\frac{M^{\lambda {(\sigma +\frac{\epsilon }{q})}^2{}_{}}{\mathrm{\Gamma }}^{j_0}(1/\beta )}{{\beta }^{j_0}\mathrm{\Gamma }(j_0/\beta )}{\displaystyle {\int }_{M^{-\beta }}^{1}H(||x|{|}_{\alpha }^{{\lambda }_1}{M}^{{\lambda }_2}{u}^{\frac{{\lambda }_2}{\beta }})u^{\frac{1}{\beta }{\lambda }_2(\sigma +\frac{\epsilon }{q})-1}}du.\end{array}$$

Setting $v=||x|{|}_{\alpha }^{{\lambda }_1^{}}{M}^{{\lambda }_2}{u}^{\frac{{\lambda }_2}{\beta }}$ in the above expression, in view of ${\lambda }_2<0$, it follows that

$$h(||x|{|}_{\alpha }^{{\lambda }_1})=\frac{1}{-{\lambda }_2}{L}_{\beta }^{(j_0)}||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1-\sigma )}{\displaystyle {\int }_0^{||x|{|}_{\alpha }^{{\lambda }_1}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv.$$

(17)

In view of (16) and (17), we have

$${\tilde{I}}_{\epsilon }=\epsilon {\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{{\lambda }_1}\ge 1\}}||x|{|}_{\alpha }^{-{\lambda }_1\epsilon -i_0}}h(||x|{|}_{\alpha }^{{\lambda }_1})dx$$

$$=\frac{\epsilon }{|{\lambda }_2|}{L}_{\beta }^{(j_0)}{\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{{\lambda }_1}\ge 1\}}||x|{|}_{\alpha }^{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}}[{\displaystyle {\int }_0^{||x|{|}_{\alpha }^{{\lambda }_1}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]dx.$$

For ${\lambda }_1>0$ by (7), we have for $M>0$ that

$$\begin{array}{c}{\tilde{I}}_{\epsilon }=\frac{\epsilon }{|{\lambda }_2|}{L}_{\beta }^{(j_0)}{\displaystyle \int \cdots }{\displaystyle {\int }_{{\{x\in R_{+}^{i_0};{\displaystyle {\sum }_{i=1}^{i_0}(\frac{x_i}{M}})}^{\alpha }\ge M^{-\alpha }\}}{M}^{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}{[{\displaystyle {\sum }_{i=1}^{i_0}(\frac{x_i}{M}})}^{\alpha }{]}^{\frac{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}{\alpha }}}\\ {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\times [{\displaystyle {\int }_0^{M^{{\lambda }_1}{[{\displaystyle {\sum }_{i=1}^{i_0}(\frac{x_i}{M}})}^{\alpha }{]}^{\frac{{\lambda }_1}{\alpha }}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]d{x}_1\cdots d{x}_{i_0}\end{array}$$

$${{\displaystyle }}^{}=\frac{\epsilon }{|{\lambda }_2|}{L}_{\beta }^{(j_0)}\underset{M\to \infty }{\lim }\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{\displaystyle {\int }_{M^{-\alpha }}^1{M}^{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}}{u}^{\frac{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}{\alpha }}[{\displaystyle {\int }_0^{M^{{\lambda }_1}{u}^{\frac{{\lambda }_1}{\alpha }}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]u^{\frac{i_0}{\alpha }-1}du$$

$$\begin{array}{l}{{\displaystyle }}^{}=\frac{\epsilon }{|{\lambda }_2|}{L}_{\beta }^{(j_0)}\underset{M\to \infty }{\lim }\frac{M^{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{\displaystyle {\int }_{M^{-\alpha }}^1{u}^{\frac{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )}{\alpha }-1}}[{\displaystyle {\int }_0^{M^{{\lambda }_1}{u}^{\frac{{\lambda }_1}{\alpha }}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]du\\ {{\displaystyle }}^{}\stackrel{t=M^{{\lambda }_1}{u}^{\frac{{\lambda }_1}{\alpha }}}{=}\frac{\epsilon }{|{\lambda }_1{\lambda }_2|}{L}_{\beta }^{(j_0)}{L}_{\alpha }^{(i_0)}{\displaystyle {\int }_1^{\infty }{t}^{-(\sigma -{\sigma }_1+\epsilon )-1}}[{\displaystyle {\int }_0^{t}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]dt.\end{array}$$

For ${\lambda }_1<0,M>0$, by (7), we still have

$${\tilde{I}}_{\epsilon }=\frac{\epsilon }{|{\lambda }_2|}{L}_{\beta }^{(j_0)}{\displaystyle \int \cdots }{\displaystyle {\int }_{{\{x\in R_{+}^{i_0};0<{\displaystyle {\sum }_{i=1}^{i_0}(\frac{x_i}{M}})}^{\alpha }\le M^{-\alpha }\}}{M}^{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}{[{\displaystyle {\sum }_{i=1}^{i_0}(\frac{x_i}{M}})}^{\alpha }{]}^{\frac{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}{\alpha }}}$$

$${{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\times [{\displaystyle {\int }_0^{M^{{\lambda }_1}{[{\displaystyle {\sum }_{i=1}^{i_0}(\frac{x_i}{M}})}^{\alpha }{]}^{\frac{{\lambda }_1}{\alpha }}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]d{x}_1\cdots d{x}_{i_0}$$

$${{\displaystyle }}^{}=\frac{\epsilon }{|{\lambda }_2|}{L}_{\beta }^{(j_0)}\underset{M\to \infty }{\lim }\frac{M^{i_0}{\mathrm{\Gamma }}^{i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{\displaystyle {\int }_0^{M^{-\alpha }}{M}^{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}}{u}^{\frac{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}{\alpha }}[{\displaystyle {\int }_0^{M^{{\lambda }_1}{u}^{\frac{{\lambda }_1}{\alpha }}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]u^{\frac{i_0}{\alpha }-1}du$$

$$\begin{array}{l}{{\displaystyle }}^{}=\frac{\epsilon }{|{\lambda }_2|}{L}_{\beta }^{(j_0)}\underset{M\to \infty }{\lim }\frac{M^{-{\lambda }_1(\sigma -{\sigma }_1+\epsilon )-i_0}(1/\alpha )}{{\alpha }^{i_0}\mathrm{\Gamma }(i_0/\alpha )}{\displaystyle {\int }_0^{M^{-\alpha }}{u}^{\frac{-{\lambda }_1\epsilon }{\alpha }-1}}[{\displaystyle {\int }_0^{M^{{\lambda }_1}{u}^{\frac{{\lambda }_1}{\alpha }}}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]du\\ {{\displaystyle }}^{}\stackrel{t=M^{{\lambda }_1}{u}^{\frac{{\lambda }_1}{\alpha }}}{=}\frac{\epsilon }{|{\lambda }_1{\lambda }_2|}{L}_{\beta }^{(j_0)}{L}_{\alpha }^{(i_0)}{\displaystyle {\int }_1^{\infty }{t}^{-(\sigma -{\sigma }_1+\epsilon )-1}}[{\displaystyle {\int }_0^{t}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]dt.\end{array}$$

Hence, we have

$${\tilde{I}}_{\epsilon }\ge \frac{\epsilon }{|{\lambda }_1{\lambda }_2|}{L}_{\beta }^{(j_0)}{L}_{\alpha }^{(i_0)}{\displaystyle {\int }_T^{\infty }{t}^{-(\sigma -{\sigma }_1+\epsilon )-1}}dt{\displaystyle {\int }_0^{T}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}d{v}^{}(T>1),$$

(18)

Satisfying ${\displaystyle {\int }_0^{T}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv>0$. For $\sigma -{\sigma }_1+\epsilon <0$, we have ${\displaystyle {\int }_T^{\infty }{t}^{-(\sigma -{\sigma }_1+\epsilon )-1}}dt=\infty$, in view of (18), we have ${\tilde{I}}_{\epsilon }=\infty$. Hence, we have (13).

(ii) In the same way, by the symmetry, we have (14).

(iii) If ${\sigma }_1=\sigma$, then in view of (18), by Fubini theorem and Fatou lemma (cf. [15]), we obtain

$$\begin{array}{l}\underset{\epsilon \to 0^{+}}{\lim }{I}_{\epsilon }=\underset{\epsilon \to 0^{+}}{\lim }\frac{\epsilon }{|{\lambda }_1{\lambda }_2|}{L}_{\beta }^{(j_0)}{L}_{\alpha }^{(i_0)}\\ {{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}\times \{{\displaystyle {\int }_1^{\infty }{t}^{-\epsilon -1}}[{\displaystyle {\int }_0^{1}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]dt+{\displaystyle {\int }_1^{\infty }{t}^{-\epsilon -1}}[{\displaystyle {\int }_1^{t}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv]dt\}\end{array}$$

$$\begin{array}{l}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=\underset{\epsilon \to 0^{+}}{\lim }\frac{\epsilon }{|{\lambda }_1{\lambda }_2|}{L}_{\beta }^{(j_0)}{L}_{\alpha }^{(i_0)}[\frac{1}{\epsilon }{\displaystyle {\int }_0^{1}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv+{\displaystyle {\int }_1^{\infty }({\displaystyle {\int }_v^{\infty }{t}^{-\epsilon -1}dt}})H(v)v^{(\sigma +\frac{\epsilon }{q})-1}dv]\\ {{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}=\frac{1}{|{\lambda }_1{\lambda }_2|}{L}_{\beta }^{(j_0)}{L}_{\alpha }^{(i_0)}\underset{\epsilon \to 0^{+}}{\lim }[{\displaystyle {\int }_0^{1}H(v)v^{(\sigma +\frac{\epsilon }{q})-1}}dv+{\displaystyle {\int }_1^{\infty }H(v)v^{(\sigma -\frac{\epsilon }{p})-1}}dv]\end{array}$$

$${{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{}{}\ge \frac{1}{|{\lambda }_1{\lambda }_2|}{L}_{\beta }^{(j_0)}{L}_{\alpha }^{(i_0)}({\displaystyle {\int }_0^1\underset{\epsilon \to 0^{+}}{\underset{¯}{\lim }}H(v)v^{\sigma +\frac{\epsilon }{q}-1}}dv+{\displaystyle {\int }_1^{\infty }\underset{\epsilon \to 0^{+}}{\underset{¯}{\lim }}H(v)v^{\sigma -\frac{\epsilon }{p}-1}}dv)=L_{\alpha }^{(i_0)}{L}_{\beta }^{(j_0)}\frac{K(\sigma )}{|{\lambda }_1{\lambda }_2|},$$

namely, (15) follows.

The lemma is proved. □

By Lemma 1, we obtain the following main inequality:

Lemma 4.

If $K(\eta )\in {\mathrm{R}}_{+}{}^{}(\eta \in \{{\sigma }_1,\sigma \})$, then we have the following inequality

$$\begin{array}{r}I:={\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}}f(x)g(y)dxdy<{(L_{\beta }^{(j_0)}\frac{K({\sigma }_1)}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{K(\sigma )}{|{\lambda }_1|})}^{\frac{1}{q}}\\ \times {[{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\widehat{\sigma })-i_0}{f}^p}(x)dx]}^{\frac{1}{p}}{[{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\widehat{\sigma })-j_0}{g}^q}(y)dy]}^{\frac{1}{q}}.\end{array}$$

(19)

Proof. 

By Hölder’s inequality (cf. [16]), we have

$$\begin{array}{l}I={\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})[}}\frac{||y|{|}_{\beta }^{({\lambda }_2{\sigma }_1-j_0)/p}}{||x|{|}_{\alpha }^{({\lambda }_1\sigma -i_0)/q}}f(x)][\frac{||x|{|}_{\alpha }^{({\lambda }_1\sigma -i_0)/q}}{||y|{|}_{\beta }^{({\lambda }_2{\sigma }_1-j_0)/p}}g(y)]dxdy\\ {{\displaystyle }}^{}\le \{{\displaystyle {\int }_{R_{+}^{i_0}}[{\displaystyle {\int }_{R_{+}^{j_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}}\frac{||y|{|}_{\beta }^{{\lambda }_2{\sigma }_1-j_0}}{||x|{|}_{\alpha }^{({\lambda }_1\sigma -i_0)(p-1)}}dy]f^p(x)dx{\}}^{\frac{1}{p}}\end{array}$$

$$\begin{array}{l} {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\times \{{\displaystyle {\int }_{R_{+}^{j_0}}[{\displaystyle {\int }_{R_{+}^{i_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}}\frac{||x|{|}_{\alpha }^{{\lambda }_1\sigma -i_0}}{||y|{|}_{\beta }^{({\lambda }_2{\sigma }_1-j_0)(q-1)}}dx]g^q(y)dy{\}}^{\frac{1}{q}}\\ {{\displaystyle }}^{}={[{\displaystyle {\int }_{R_{+}^{i_0}}\varpi ({\sigma }_1,x)}||x|{|}_{\alpha }^{(p-1)(i_0-{\lambda }_1\sigma )}{f}^p(x)dx]}^{\frac{1}{p}}\end{array}$$

$$\times {[{\displaystyle {\int }_{R_{+}^{j_0}}\omega (\sigma ,y)||y|{|}_{\beta }^{(q-1)(j_0-{\lambda }_2{\sigma }_1)}}{g}^q(y)dy]}^{\frac{1}{q}}.$$

(20)

If (20) pertain to the form of equality, then (cf. [16]), there exist constants $A$ and $B$, satisfying they are not both zero, and

$$A\frac{||y|{|}_{\beta }^{{\lambda }_2{\sigma }_1-j_0}}{||x|{|}_{\alpha }^{({\lambda }_1\sigma -i_0)(p-1)}}{f}^p(x)=B\frac{||x|{|}_{\alpha }^{{\lambda }_1\sigma -i_0}}{||y|{|}_{\beta }^{({\lambda }_2{\sigma }_1-j_0)(q-1)}}{g}^q{(y)}^{}a.e{.}^{}i{n}^{}{\mathrm{R}}_{+}^{{\mathrm{i}}_0}\times {\mathrm{R}}_{+}^{{\mathrm{j}}_0}.$$

Assuming that $A\ne 0$, there exists a $y\in {\mathrm{R}}_{+}^{{\mathrm{j}}_0},$ such that

$$||x|{|}_{\alpha }^{p(i-0{\lambda }_1\widehat{\sigma })-i_0}{f}^p(x)=\frac{B{g}^q(y)}{A||y|{|}_{\beta }^{q({\lambda }_2{\sigma }_1-j_0)}}||x|{|}_{\alpha }^{{\lambda }_1(\sigma -{\sigma }_1)-i_0}{}^{}a.e{.}^{}i{n}^{}{\mathrm{R}}_{+}^{{\mathrm{i}}_0},$$

which contradicts that

$$0<{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p[i_0-{\lambda }_1\widehat{\sigma }]-i_0}{f}^p}(x)dx<\infty .$$

In fact, by (11) and (12), for $b={\lambda }_1(\sigma -{\sigma }_1)\in \mathrm{R}$, we have

$${\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{b-i_0}}dx={\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }\le 1\}}||x|{|}_{\alpha }^{b-i_0}}dx+{\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }\ge 1\}}||x|{|}_{\alpha }^{b-i_0}}dx=\infty .$$

By (8) and (9), we obtain (19).

This proves the lemma. □

Remark 1 

(i) In particular, for $\sigma =1\sigma$ in (19), we have $\widehat{\sigma }=\sigma$,

$$0<{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\sigma )-i_0}{f}^p}(x)dx<\infty ,0<{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\sigma )-j_0}{g}^q}(y)dy<\infty ,$$

and the following:

$$\begin{gathered} I={\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}}f(x)g(y)dxdy<{(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma ) \\ \times {[{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\sigma )-i_0}{f}^p}(x)dx]}^{\frac{1}{p}}{[{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\sigma )-j_0}{g}^q}(y)dy]}^{\frac{1}{q}}. \end{gathered}$$

(21)

(ii) By Hölder’s inequality (cf. [16]), we still have

$$\begin{array}{l}0<K(\widehat{\sigma })=K(\frac{{\sigma }_1}{p}+\frac{\sigma }{q})={\displaystyle {\int }_0^{\infty }H(u)u^{\frac{{\sigma }_1}{p}+\frac{\sigma }{q}-1}}du\\ \stackrel{}{}={\displaystyle {\int }_0^{\infty }H(u)(u^{\frac{{\sigma }_1-1}{p}})(u^{\frac{\sigma -1}{q}})}du\end{array}$$

$$\le {({\displaystyle {\int }_0^{\infty }H(u)u^{{\sigma }_1-1}}du)}^{\frac{1}{p}}{({\displaystyle {\int }_0^{\infty }H(u)u^{\sigma -1}}du)}^{\frac{1}{q}}={(K({\sigma }_1))}^{\frac{1}{p}}{(K(\sigma ))}^{\frac{1}{q}}<\infty .$$

(22)

Now, we use Lemma 2 and Lemma 3 to show the best value in the key inequality (21).

Lemma 5.

For $K(\sigma )\in {\mathrm{R}}_{+}$, ${(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma )$ in (21) is the best value.

Proof. 

For any $\epsilon >0$, we set

$$f_{\epsilon }(x):=\left\{\begin{array}{c}0,||x|{|}_{\alpha }^{{\lambda }_1}<1,\\ ||x|{|}_{\alpha }^{{\lambda }_1(\sigma -\frac{\epsilon }{p})-i_0},||x|{|}_{\alpha }^{{\lambda }_1}\ge 1,\end{array}\right.g_{\epsilon }(y):=\left\{\begin{array}{c}||y|{|}_{\beta }^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0},||y|{|}_{\beta }^{{\lambda }_2}\le 1,\\ 0,||y|{|}_{\beta }^{{\lambda }_2}>1.\end{array}\right.$$

By (11) and (12), we have

$$\begin{array}{ll}& {{\displaystyle }}^{}{\displaystyle {\int }_{x\in R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\sigma )-i_0}{f}_{\epsilon }^p}(x)dx={\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{{\lambda }_1}\ge 1\}}||x|{|}_{\alpha }^{-{\lambda }_1\epsilon -i_0}}dx\\ =& \left\{\begin{array}{c}{\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{}\le 1\}}||x|{|}_{\alpha }^{|{\lambda }_1|\epsilon -i_0}}dx,{\lambda }_1<0\\ {\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\delta }^{}\ge 1\}}||x|{|}_{\alpha }^{-|{\lambda }_1|\epsilon -i_0}}dx,{\lambda }_1>0\end{array}\right.=\frac{1}{|\lambda |1\epsilon }{L}_{\alpha }^{(i_0)},\end{array}$$

$${\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-\sigma )-j_0}{g}_{\epsilon }^q}(y)dy={\displaystyle {\int }_{\{y\in R_{+}^{j_0};||y|{|}_{\beta }^{{\lambda }_2}\le 1\}}||y|{|}_{\beta }^{{\lambda }_2\epsilon -j_0}}dy=\frac{1}{|{\lambda }_2|\epsilon }{L}_{\beta }^{(j_0)}.$$

(23)

If there exists a positive constant

$$M\le {(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma ),$$

such that (21) is valid as we replace ${(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma )$ by $M$, then in particular, by (15), we have

$$L_{\alpha }^{(i_0)}{L}_{\beta }^{(j_0)}\frac{K(\sigma )}{|{\lambda }_1{\lambda }_2|}+o(1)\le I_{\epsilon }=\epsilon {\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}}{f}_{\epsilon }(x)g(y)\epsilon dxdy$$

$$\begin{array}{l}<\epsilon M{[{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\sigma )-i_0}{f}_{\epsilon }{}^p}(x)dx]}^{\frac{1}{p}}{[{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\sigma )-j_0}{g}_{\epsilon }{}^q}(y)dy]}^{\frac{1}{q}}\\ =M{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{p}}{(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{q}}.\end{array}$$

For $\epsilon \to 0^{+}$, it follows that

$$L_{\alpha }^{(i_0)}{L}_{\beta }^{(j_0)}\frac{K(\sigma )}{|{\lambda }_1{\lambda }_2|}\le M{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{p}}{(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{q}}.$$

We find that ${(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma )$$\le M,$ which follows that

$$M={(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(j_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma )$$

is the best possible constant factor of (21).

This proves the lemma. □

3. Main Results and Two Corollaries

Theorem 1.

For $p>1,\frac{1}{p}+\frac{1}{q}=1,$ if there exists $M(\ge 0)$, satisfying the following inequality holds:

$$\begin{array}{l} I={\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}}f(x)g(y)dxdy\\ \le M{[{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1{\sigma }_1)-i_0}{f}^p}(x)dx]}^{\frac{1}{p}}{[{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\sigma )-j_0}{g}^q}(y)dy]}^{\frac{1}{q}},\end{array}$$

(24)

then we have ${\sigma }_1=\sigma$ and $M>0$. Hence, $K(\sigma )\in {\mathrm{R}}_{+}$ and

$${(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K{(\sigma )}^{}(\in (0,M])$$

is the best value of (24) (for ${\sigma }_1=\sigma$).

Proof. 

If ${\sigma }_1<\sigma$, then for any $\epsilon >0$, we set

$$\begin{array}{l}{\tilde{f}}_{\epsilon }(x):=\left\{\begin{array}{c}0,||x|{|}_{\alpha }^{{\lambda }_1}<1,\\ ||x|{|}_{\alpha }^{\lambda {({\sigma }_1-\frac{\epsilon }{p})}^1-i_0},||x|{|}_{\alpha }^{{\lambda }_1}\ge 1,\end{array}\right.\\ {\tilde{g}}_{\epsilon }(y):=\left\{\begin{array}{c}||y|{|}_{\beta }^{{\lambda }_2(\sigma +\frac{\epsilon }{q})-j_0},||y|{|}_{\beta }^{{\lambda }_2}\le 1,\\ 0,||y|{|}_{\beta }^{{\lambda }_2}>1.\end{array}\right.\end{array}$$

By (8), (19) and (18), we have

$$\infty ={\tilde{I}}_{\epsilon }=\epsilon {\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}}{\tilde{f}}_{\epsilon }(x){\tilde{g}}_{\epsilon }(y)dxdy$$

$$\begin{array}{l}\le \epsilon M{[{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1{\sigma }_1)-i_0}{\tilde{f}}_{\epsilon }^p}(x)dx]}^{\frac{1}{p}}{[{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\sigma )-j_0}{\tilde{g}}_{\epsilon }^q}(y)dy]}^{\frac{1}{q}}\\ =\epsilon M{({\displaystyle {\int }_{\{x\in R_{+}^{i_0};||x|{|}_{\alpha }^{{\lambda }_1}\ge 1\}}||x|{|}_{\alpha }^{-{\lambda }_1\epsilon -i_0}}dx)}^{\frac{1}{p}}{({\displaystyle {\int }_{\{y\in R_{+}^{j_0};||y|{|}_{\beta }^{{\lambda }_2}\le 1\}}||y|{|}_{\beta }^{{\lambda }_2\epsilon -j_0}}dy)}^{\frac{1}{q}}<\infty ,\end{array}$$

which is a contradiction.

If ${\sigma }_1<\sigma$, then for any $\epsilon >0$, we set

$$\begin{array}{l}{\widehat{f}}_{\epsilon }(x):=\left\{\begin{array}{c}||x|{|}_{\alpha }^{{\lambda }_1({\sigma }_1+\frac{\epsilon }{p})-i_0}{,}^{}||x|{|}_{\alpha }^{{\lambda }_1}\le 1,\\ 0{,}^{}||x|{|}_{\alpha }^{{\lambda }_1}>1,\end{array}\right.\\ {\widehat{g}}_{\epsilon }(y):=\left\{\begin{array}{c}0{,}^{}||y|{|}_{\beta }^{{\lambda }_2}<1,\\ ||y|{|}_{\beta }^{{\lambda }_2(\sigma -\frac{\epsilon }{q})-j_0}{,}^{}||y|{|}_{\beta }^{{\lambda }_2}\ge 1.\end{array}\right.\end{array}$$

By (9), (19) and (18), in the same way, we still obtain a contradiction.

Hence, we have ${\sigma }_1=\sigma$.

For ${\sigma }_1=\sigma$ in (19), replacing $f{(x)}^{}(resp{.}^{}g(y))$ by $f_{\epsilon }{(x)}^{}(resp{.}^{}{g}_{\epsilon }(y))$ in Lemma 5 and following the proof of Lemma 5, for $K(\sigma )>0$, we still find

$$0<{(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma )\le M<\infty ,$$

which follows that

$$0<K(\sigma )\le M{(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{-\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{-\frac{1}{q}}<\infty .$$

By Lemma 5, ${(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma )$$(\in (0,M])$ is the best possible constant of (24) (for ${\sigma }_1=\sigma$).

This proves the theorem. □

Theorem 2.

For $K(\eta )\in {\mathrm{R}}_{+}(\eta \in \{\sigma ,{\sigma }_1\})$, we have the following equivalent statements:

(i) Both ${(K({\sigma }_1))}^{\frac{1}{p}}{(K(\sigma ))}^{\frac{1}{q}}$ and $K(\frac{{\sigma }_1}{p}+\frac{\sigma }{q})$ are independent of $p,q$;

$$(\mathrm{ii}) {(K({\sigma }_1))}^{\frac{1}{p}}{(K(\sigma ))}^{\frac{1}{q}}\le K(\frac{{\sigma }_1}{p}+\frac{\sigma }{q});$$

(25)

(iii) ${\sigma }_1=\sigma$;

(iv) The constant factor ${(L_{\beta }^{(j_0)}\frac{K({\sigma }_1)}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{K(\sigma )}{|{\lambda }_1|})}^{\frac{1}{q}}$ in (19) is the best value;

(v) there exists a constant $M$, such that (24) holds.

Proof. 

(ii)$\Rightarrow$(iii). By (25), it follows that (22) protains the form of equality. Then, there exist $A$ and $B$, satisfying they are not both zero and $A{u}^{{\sigma }_1-1}=B{u}^{\sigma -1}{}^{}a.e{.}^{} i{n}^{} {\mathrm{R}}_{+}$ (cf. [16]). Supposing that $A\ne 0$, we have $u^{{\sigma }_1-\sigma }={\frac{B}{A}}^{}a.e{.}^{} i{n}^{} {\mathrm{R}}_{+}$, and then ${\sigma }_1-\sigma =0$, namely, ${\sigma }_1=\sigma$.

(iii)$\Rightarrow$(iv). In view of Lemma 5, we obtain (iv).

(iv)$\Rightarrow$(ii). If the constant factor ${(L_{\beta }^{(j_0)}\frac{K({\sigma }_1)}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{K(\sigma )}{|{\lambda }_1|})}^{\frac{1}{q}}$ in (19) is the best value, then by (21) (for $\sigma =\widehat{\sigma }$), we have

$${(L_{\beta }^{(j_0)}\frac{K({\sigma }_1)}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{K(\sigma )}{|{\lambda }_1|})}^{\frac{1}{q}}\le {(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\widehat{\sigma })\in {\mathrm{R}}_{+},$$

namely, (25) follows. Hence, it follows that (ii)$\iff$(iii)$\iff$(iv).

(i)$\Rightarrow$(ii). By (i), we find

$${(K({\sigma }_1))}^{\frac{1}{p}}{(K(\sigma ))}^{\frac{1}{q}}=\underset{p\to \infty }{\lim }\underset{q\to 1}{\lim }{(K({\sigma }_1))}^{\frac{1}{p}}{(K(\sigma ))}^{\frac{1}{q}}=K(\sigma ),$$

and then, in view of Fatou lemma (cf. [15]), we have

$${(K({\sigma }_1))}^{\frac{1}{p}}{(K(\sigma ))}^{\frac{1}{q}}=K(\sigma )=K(\underset{p\to \infty }{\underset{¯}{\lim }}\frac{{\sigma }_1-\sigma }{p}+\sigma )\le \underset{p\to \infty }{\underset{¯}{\lim }}K(\frac{{\sigma }_1-\sigma }{p}+\sigma )=K(\frac{{\sigma }_1}{p}+\frac{\sigma }{q}),$$

namely, (25) follows.

(iii)$\Rightarrow$(i). For ${\sigma }_1=\sigma$, both ${(K({\sigma }_1))}^{\frac{1}{p}}{(K(\sigma ))}^{\frac{1}{q}}$ and $K(\frac{{\sigma }_1}{p}+\frac{\sigma }{q})$ equal $K(\sigma )$, which are independent of $p,q$. Hence, we have (i)$\iff$(ii)$\iff$(iii)$\iff$(iv).

(v)$\Rightarrow$(iii). By Theorem 1, for $K(\eta )\in {\mathrm{R}}_{+}(\eta =\sigma ,\sigma )1$, we still have ${\sigma }_1=\sigma$.

(iii)$\Rightarrow$(v). If ${\sigma }_1=\sigma$, then by Lemma 5, we set $M\ge {(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}K(\sigma ),$ and then (24) holds. Hence, we have (iii)$\iff$(v).

Therefore, we have (i)$\iff$(ii)$\iff$(iii)$\iff$(iv)$\iff$(v).

This proved that theorem. □

Replacing ${\lambda }_1$ to $-{\lambda }_1$ in Theorems 1 and 2, setting $H(v)=k_{\lambda }(1,v),$ where $k_{\lambda }(u,v)$ is a homogeneous function of degree $-\lambda$, such that $k_{\lambda }(tu,tv)=t^{-\lambda }{k}_{\lambda }{(u,v)}^{}(t,u,v>0)$, and

$$K_{\lambda }(\eta ):={\displaystyle {\int }_0^{\infty }{k}_{\lambda }(1,u)u^{\eta -1}}du>0^{}(\eta =\lambda -\mu ,\sigma ).$$

For $\mu =\lambda -{\sigma }_1,\widehat{\sigma }=\frac{\lambda -\mu }{p}+\frac{\sigma }{q},\widehat{\mu }=\lambda -\widehat{\sigma }=\frac{\lambda -\sigma }{q}+\frac{\mu }{p},$ replacing $||x|{|}_{\alpha }^{{\lambda }_1\lambda }f(x)$ to $f(x)$, by calculation, we have

Corollary 1.

If there exists $M(\ge 0)$, such that the following inequality holds:

$$\begin{array}{l} {\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}{k}_{\lambda }(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{{\lambda }_2})}}f(x)g(y)dxdy\\ \le M{[{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\mu )-i_0}{f}^p}(x)dx]}^{\frac{1}{p}}{[{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\sigma )-j_0}{g}^q}(y)dy]}^{\frac{1}{q}},\end{array}$$

(26)

then we have $\mu +\sigma =\lambda$, and ${(L_{\beta }^{(j_0)}\frac{1}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{1}{|{\lambda }_1|})}^{\frac{1}{q}}{K}_{\lambda }{(\sigma )}^{}(\in (0,M])$ is the best possible constant in (26) (for $\mu +\sigma =\lambda$).

Corollary 2.

For $K_{\lambda }(\eta )\in {\mathrm{R}}_{+}(\eta \in \{\sigma ,\lambda -\mu \})$, the following statements are equivalent:

(I) Both ${(K_{\lambda }(\lambda -\mu ))}^{\frac{1}{p}}{(K_{\lambda }(\sigma ))}^{\frac{1}{q}}$ and $K_{\lambda }(\frac{\lambda -\mu }{p}+\frac{\sigma }{q})$ are independent of $p,q$;

$$(\mathrm{II}) {(K_{\lambda }(\lambda -\mu ))}^{\frac{1}{p}}{(K_{\lambda }(\sigma ))}^{\frac{1}{q}}\le K_{\lambda }(\frac{\lambda -\mu }{p}+\frac{\sigma }{q});$$

(27)

(III) $\mu +\sigma =\lambda$;

(IV) the constant factor ${(L_{\beta }^{(j_0)}\frac{K_{\lambda }(\lambda -\mu )}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{K_{\lambda }(\sigma )}{|{\lambda }_1|})}^{\frac{1}{q}}$ in the following inequality

$$\begin{array}{l} {\displaystyle {\int }_{R_{+}^{j_0}}{\displaystyle {\int }_{R_{+}^{i_0}}{k}_{\lambda }(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{{\lambda }_2})}}f(x)g(y)dxdy\\ <{(L_{\beta }^{(j_0)}\frac{K_{\lambda }(\lambda -\mu )}{|{\lambda }_2|})}^{\frac{1}{p}}{(L_{\alpha }^{(i_0)}\frac{K_{\lambda }(\sigma )}{|{\lambda }_1|})}^{\frac{1}{q}}\\ \times {[{\displaystyle {\int }_{R_{+}^{i_0}}||x|{|}_{\alpha }^{p(i_0-{\lambda }_1\widehat{\mu })-i_0}{f}^p}(x)dx]}^{\frac{1}{p}}{[{\displaystyle {\int }_{R_{+}^{j_0}}||y|{|}_{\beta }^{q(j_0-{\lambda }_2\widehat{\sigma })-j_0}{g}^q}(y)dy]}^{\frac{1}{q}},\end{array}$$

(28)

is the best possible;

(V) there exists a constant $M$, such that inequality (26) holds.

Example 1.

Setting $h(u)=k(1,u)\lambda =\frac{1}{{(1+u)}^{\lambda }}(\lambda >0;u>0)$, we find

$$\begin{array}{l}K(\eta )=K(\eta )\lambda ={\displaystyle {\int }_0^{\infty }\frac{u^{\eta -1}}{{(1+u)}^{\lambda }}du=B(\eta ,\lambda -\eta )\in }{\mathrm{R}}_{+}{}^{}(0<\eta <\lambda )\\ H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})=\frac{1}{{(1+||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})}^{\lambda }},k_{\lambda }(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{{\lambda }_2})=\frac{1}{{(||x|{|}_{\alpha }^{{\lambda }_1}+||y|{|}_{\beta }^{{\lambda }_2})}^{\lambda }}.\end{array}$$

Example 2.

(i) For $\lambda ,\gamma >0,$ we set $H(u)=k_{\lambda }^{}(1,u)={\frac{1-u^{\gamma }}{1-u^{\lambda +\gamma }}}^{}(u>0)$. We find

$$H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})=\frac{1-||x|{|}_{\alpha }^{\gamma {\lambda }_1}||y|{|}_{\beta }^{\gamma {\lambda }_2}}{1-||x|{|}_{\alpha }^{(\lambda +\gamma ){\lambda }_1}||y|{|}_{\beta }^{(\lambda +\gamma ){\lambda }_2}},k_{\lambda }(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{{\lambda }_2})=\frac{||x|{|}_{\alpha }^{\gamma {\lambda }_1}-||y|{|}_{\beta }^{\gamma {\lambda }_2}}{||x|{|}_{\alpha }^{(\lambda +\gamma ){\lambda }_1}-||y|{|}_{\beta }^{(\lambda +\gamma ){\lambda }_2}}.$$

In particular, for $\gamma =\lambda$, we have

$$H(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{{\lambda }_2})=\frac{1}{1+||x|{|}_{\alpha }^{\lambda {\lambda }_1}||y|{|}_{\beta }^{\lambda {\lambda }_2}},k_{\lambda }(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{{\lambda }_2})=\frac{1}{||x|{|}_{\alpha }^{\lambda {\lambda }_1}+||y|{|}_{\beta }^{\lambda {\lambda }_2}}.$$

(ii) In view of (cf. [17]):

$$\cot x=\frac{1}{x}+{\displaystyle \sum _{k=1}^{\infty }{(\frac{1}{x-\pi k}+\frac{1}{x+\pi k})}^{}(x\in (0,\pi )),}$$

for $b\in (0,1)$, by Lebesgue term by term theorem (cf. [14], we find

$$\begin{array}{l}{A}_b:={\displaystyle {\int }_0^{\infty }\frac{u^{b-1}}{1-u}}du={\displaystyle {\int }_0^1\frac{u^{b-1}}{1-u}}du+{\displaystyle {\int }_1^{\infty }\frac{u^{b-1}}{1-u}}du\\ {{\displaystyle }}^{}={\displaystyle {\int }_0^1\frac{u^{b-1}}{1-u}}du-{\displaystyle {\int }_0^1\frac{v^{-b}}{1-v}}dv={\displaystyle {\int }_0^1\frac{u^{b-1}-u^{-b}}{1-u}}du\\ {{\displaystyle }}^{}={\displaystyle {\int }_0^1{\displaystyle \sum _{k=0}^{\infty }(u^{k+b-1}-u^{k-b}}})du={\displaystyle \sum _{k=0}^{\infty }{\displaystyle {\int }_0^1(}{u}^{k+b-1}-u^{k-b}})du\\ {{\displaystyle }}^{}={\displaystyle \sum _{k=0}^{\infty }(\frac{1}{k+b}-\frac{1}{k+1-b})=}\pi [\frac{1}{\pi b}+{\displaystyle \sum _{k=1}^{\infty }(\frac{1}{\pi b-\pi k}+\frac{1}{\pi b+\pi k})]}\\ {{\displaystyle }}^{}=\pi \cot \pi b\in \mathrm{R}:=(-\infty ,\infty ).\end{array}$$

Note .

For $b\in (0,\frac{1}{2}),A_b>0;$ for $b\in (\frac{1}{2},1),A_b<0;A_{1/2}=0.$

(iii) For $\eta \in (0,\lambda ),$ by (ii), we obtain (cf. [18])

$$\begin{array}{l}K(\eta )=K_{\lambda }(\eta ):={\displaystyle {\int }_0^{\infty }H}(u)u^{\eta -1}du={\displaystyle {\int }_0^{\infty }\frac{1-u^{\gamma }}{1-u^{\lambda +\gamma }}}{u}^{\eta -1}du\\ {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}\stackrel{v=u^{\lambda +\gamma }}{=}\frac{1}{\lambda +\gamma }\left({\displaystyle {\int }_0^{\infty }\frac{v^{\frac{\eta }{\lambda +\gamma }-1}}{1-v}dv-{\displaystyle {\int }_0^{\infty }\frac{v^{\frac{\eta +\gamma }{\lambda +\gamma }-1}}{1-v}dv}}\right)\\ {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}=\frac{\pi }{\lambda +\gamma }\left[\cot (\frac{\pi \eta }{\lambda +\gamma })-\cot (\frac{\pi (\eta +\gamma )}{\lambda +\gamma })\right]\\ {{\displaystyle }}^{}{{\displaystyle }}^{}{{\displaystyle }}^{}=\frac{\pi }{\lambda +\gamma }\left[\cot (\frac{\pi \eta }{\lambda +\gamma })+\cot (\frac{\pi (\lambda -\eta )}{\lambda +\gamma })\right]\in {\mathrm{R}}_{+}.\end{array}$$

In particular, for $\gamma =\lambda$, we obtain

$$K(\eta )=K_{\lambda }(\eta )=\frac{\pi }{2\lambda }\left[\cot (\frac{\pi \eta }{2\lambda })+\cot (\frac{\pi (\lambda -\eta )}{2\lambda })\right]=\frac{\pi }{\lambda \sin (\frac{\pi \eta }{\lambda })}.$$

We can use Examples 1 and 2 as the particular kernels to Theorems 1 and 2 and Corollaries 1 and 2.

4. Conclusions

In this article, following the idea of [7,8], by means of the technique of real analysis, the way of introduced parameters, and a few useful formulas, two new multidimensional Hilbert-type integral inequalities with the nonhomogeneous kernel as

$$H{(||x|{|}_{\alpha }^{{\lambda }_1}||y|{|}_{\beta }^{\lambda 2})}^{}({\lambda }_1,\lambda \ne 20)$$

are given in (19) and (24), which are some new extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent statements related to the two inequalities, the best value and several parameters are provided in Theorem 2. Two corollaries about the homogeneous kernel as $k_{\lambda }{(||x|{|}_{\alpha }^{{\lambda }_1},||y|{|}_{\beta }^{\lambda 2})}^{}({\lambda }_1,\lambda \ne 20)$ are given in Corollaries 1 and 2, and some new inequalities in particular parameters are obtained in Examples 1 and 2.