An Equivalent Form Related to a Hilbert-Type Integral Inequality

Abstract

In the present paper, we establish an equivalent form related to a Hilbert-type integral inequality with a non-homogeneous kernel and a best possible constant factor. We also consider the case of homogeneous kernel as well as certain operator expressions.

1. Introduction

As is well-known, in 1925, Hardy [1] proved the following famous integral inequality:

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

$$0<{\int }_0^{\infty }{f}^p(x)dx<\infty \mathrm{and}0<{\int }_0^{\infty }{g}^q(y)dy<\infty ,$$

then it holds

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{f(x)g(y)}{x+y}dxdy<\frac{\pi }{sin(\pi /p)}{\left({\int }_0^{\infty }{f}^p(x)dx\right)}^{\frac{1}{p}}{\left({\int }_0^{\infty }{g}^q(y)dy\right)}^{\frac{1}{q}},$$

(1)

where the constant factor $\frac{\pi }{sin(\pi /p)}$ is the best possible.

For $p=q=2,$ (1) reduces to the well-known Hilbert integral inequality. Hilbert’s integral inequality and (1) are two very important inequalities, which are well-known for their applicability in various domains of analysis (cf. [2,3]).

In 1934, Hardy et al. presented the following extension of (1):

If $k_1(x,y)$ is a non-negative homogeneous function of degree $-1$,

$$k_p={\int }_0^{\infty }{k}_1(u,1)u^{\frac{-1}{p}}du\in {\mathbf{R}}_{+}=(0,\infty ),$$

then we have

$${\int }_0^{\infty }{\int }_0^{\infty }{k}_1(x,y)f(x)g(y)dxdy<k_p{\left({\int }_0^{\infty }{f}^p(x)dx\right)}^{\frac{1}{p}}{\left({\int }_0^{\infty }{g}^q(y)dy\right)}^{\frac{1}{q}},$$

(2)

where the constant factor $k_p$ is the best possible (cf. [2], Theorem 319). Furthermore, the following Hilbert-type integral inequality with non-homogeneous kernel holds true:

If $h(u)>0,\varphi (\sigma )={\int }_0^{\infty }h(u)u^{\sigma -1}du\in {\mathbf{R}}_{+},$ then

$$\begin{array}{ccc}& & {\int }_0^{\infty }{\int }_0^{\infty }h(xy)f(x)g(y)dxdy\hfill \\ & <& \varphi \left(\frac{1}{p}\right){\left({\int }_0^{\infty }{x}^{p-2}{f}^p(x)dx\right)}^{\left(\frac{1}{p}\right)}{\left({\int }_0^{\infty }{g}^q(y)dy\right)}^{\frac{1}{q}},\hfill \end{array}$$

(3)

where the constant factor $\varphi (\frac{1}{p})$ is the best possible (cf. [2], Theorem 350).

In 1998, by introducing an independent parameter $\lambda >0,$ Yang established an extension of Hilbert’s integral inequality with the kernel $\frac{1}{{(x+y)}^{\lambda }}$ (cf. [4,5] ). In 2004, by introducing two pairs of conjugate exponents $(p,q)$ and $(r,s)$ with an independent parameter $\lambda >0,$ Yang [6] proved the following extension of (1):

If $p,r>1,\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+\frac{1}{s}=1,$ $f(x),g(y)\ge 0,$ such that

$$0<{\int }_0^{\infty }{x}^{p(1-\frac{\lambda }{r})-1}{f}^p(x)dx<\infty \mathrm{and}0<{\int }_0^{\infty }{y}^{q(1-\frac{\lambda }{s})-1}{g}^q(y)dy<\infty ,$$

then

$$\begin{array}{ccc}& & {\int }_0^{\infty }{\int }_0^{\infty }\frac{f(x)g(y)}{x^{\lambda }+y^{\lambda }}dxdy\hfill \\ & <& \frac{\pi }{\lambda sin(\pi /r)}{\left[{\int }_0^{\infty }{x}^{p(1-\frac{\lambda }{r})-1}{f}^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-\frac{\lambda }{s})-1}{g}^q(y)dy\right]}^{\frac{1}{q}},\hfill \end{array}$$

(4)

where the constant factor $\frac{\pi }{\lambda sin(\pi /r)}$ is the best possible.

For $\lambda =1,r=q,s=p,$ (4) reduces to (1). In 2005, the work [7] also provided an extension of (1) with the kernel $\frac{1}{{(x+y)}^{\lambda }}$ and two pairs of conjugate exponents. In papers [8,9,10,11,12], the authors proved some interesting extensions and particular cases of (1)–(3) with parameters.

In 2009, Yang presented the following extension of (2) and (5) (cf. [13,14]):

If ${\lambda }_1+{\lambda }_2=\lambda \in \mathbf{R}=(-\infty ,\infty ),$$k_{\lambda }(x,y)$ is a non-negative homogeneous function of degree $-\lambda$, satisfying

$$k_{\lambda }(ux,uy)=u^{-\lambda }{k}_{\lambda }(x,y)(u,x,y>0),$$

$$k({\lambda }_1)={\int }_0^{\infty }{k}_{\lambda }(u,1)u^{{\lambda }_1-1}du\in {\mathbf{R}}_{+},$$

then we have

$$\begin{array}{ccc}& & {\int }_0^{\infty }{\int }_0^{\infty }{k}_{\lambda }(x,y)f(x)g(y)dxdy\hfill \\ & <& k({\lambda }_1){\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-{\lambda }_2)-1}{g}^q(y)dy\right]}^{\frac{1}{q}},\hfill \end{array}$$

(5)

where the constant factor $k({\lambda }_1)$ is the best possible.

For $\lambda =1,{\lambda }_1=\frac{1}{q},{\lambda }_2=\frac{1}{p},$ (5) reduces to (2), whereas for $\lambda >0,$${\lambda }_1=\frac{\lambda }{r},$${\lambda }_2=\frac{\lambda }{s},$$k_{\lambda }(x,y)=\frac{1}{x^{\lambda }+y^{\lambda }},$ (5) reduces to (4).

Additionally, the extension below of (3) has been established:

$$\begin{array}{ccc}& & {\int }_0^{\infty }{\int }_0^{\infty }h(xy)f(x)g(y)dxdy\hfill \\ & <& \varphi (\sigma ){\left[{\int }_0^{\infty }{x}^{p(1-\sigma )-1}{f}^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-\sigma )-1}{g}^q(y)dy\right]}^{\frac{1}{q}},\hfill \end{array}$$

(6)

where the constant factor $\varphi (\sigma )$ is the best possible (cf. [15]).

Some equivalent inequalities of (5) and (6) were constructed in [14]. In 2013, Yang [15] also studied the equivalency of (5) and (6). In 2017, Hong [16] investigated an equivalent condition between (5) and a few parameters. Since 2018, in the papers [17,18,19,20,21,22,23,24,25,26], the authors proved some novel extensions of the above Hilbert-type inequalities.

In the present paper, we establish an equivalent form related to a Hilbert-type integral inequality with the non-homogeneous kernel

$$|lnxy|\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}$$

and a best possible constant factor. We also consider the case of homogeneous kernel and operator expressions.

2. An Example and a Lemma

In the following, we assume that $s,s_0\in \mathbf{N}=\{1,2,\cdots \},$ $0<c_1\le \cdots \le c_s<\infty ,$$s_0\le s, 0=c_0\le c_{s_0}\le 1<c_{s_0+1}\le c_{s+1}=\infty , {\lambda }_1,{\lambda }_2>-\alpha ,$${\lambda }_1+{\lambda }_2=\lambda .$

Example 1.

We consider the following function:

$$h(u):=|lnu|\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}(u\in {\mathbf{R}}_{+}),$$

(7)

and define

$$k({\lambda }_1):={\int }_0^{\infty }h(u)u^{{\lambda }_1-1}du={\int }_0^{\infty }|lnu|\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-1}du.$$

(8)

Note. For $0<a\le b\le 1,\eta \ne 0,$ we have

$$\begin{array}{ccc}& & {\int }_a^b{x}^{\eta -1}|lnx|dx=\frac{1}{\eta }{\int }_a^b(-lnx)d{x}^{\eta }=\frac{1}{\eta }[(-lnx)x^{\eta }{|}_a^b+{\int }_a^b{x}^{\eta -1}dx]\hfill \\ & =& \frac{1}{{\eta }^2}[\eta (-lnx)x^{\eta }{|}_a^b+(b^{\eta }-a^{\eta })].\hfill \end{array}$$

(9)

Since we have

$$\begin{array}{ccc}& & {\int }_a^b{x}^{-1}|lnx|dx={\int }_a^b(-lnx)dlnx\hfill \\ & =& -\frac{1}{2}{ln}^2{x|}_a^b=\underset{\eta \to 0^{+}}{lim}\frac{1}{{\eta }^2}[\eta (-lnx)x^{\eta }{|}_a^b+(b^{\eta }-a^{\eta })],\hfill \end{array}$$

we still denote this as (9) for $\eta =0$.

For $0<a\le b,$ we also use the above viewpoint in the following.

By the above Note, indicating

$$\prod _{k=1}^0{c}_k^{\frac{\alpha }{s}}=\prod _{k=s+1}^s{c}_k^{\frac{\lambda +\alpha }{s}}=1$$

we obtain

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& {\int }_{c_0}^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-1}du\hfill \\ & & +{\int }_1^{c_{s+1}}lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-1}du\hfill \\ & =& \sum _{i=0}^{s_0-1}{\int }_{c_i}^{c_{i+1}}(-lnu)\left(\prod _{k=1}^i\frac{c{}_k^{\frac{\alpha }{s}}}{u^{\frac{\lambda +\alpha }{s}}}\right)\left(\prod _{k=i+1}^s\frac{u^{\frac{\alpha }{s}}}{c{}_k^{\frac{\lambda +\alpha }{s}}}\right)u^{{\lambda }_1-1}du\hfill \\ & & +{\int }_{c_{s_0}}^1(-lnu)\left(\prod _{k=1}^{s_0}\frac{c_k^{\frac{\alpha }{s}}}{u^{\frac{\lambda +\alpha }{s}}}\right)\left(\prod _{k=s_0+1}^s\frac{u^{\frac{\alpha }{s}}}{c_k^{\frac{\lambda +\alpha }{s}}}\right)u^{{\lambda }_1-1}du\hfill \\ & & +{\int }_1^{c_{s_0+1}}lnu\left(\prod _{k=1}^{s_0}\frac{c_k^{\frac{\alpha }{s}}}{u^{\frac{\lambda +\alpha }{s}}}\right)\left(\prod _{k=s_0+1}^s\frac{u^{\frac{\alpha }{s}}}{c_k^{\frac{\lambda +\alpha }{s}}}\right)u^{{\lambda }_1-1}du\hfill \\ & & +\sum _{i=s_0+1}^s{\int }_{c_i}^{c_{i+1}}lnu\left(\prod _{k=1}^i\frac{c_k^{\frac{\alpha }{s}}}{u^{\frac{\lambda +\alpha }{s}}}\right)\left(\prod _{k=i+1}^s\frac{u^{\frac{\alpha }{s}}}{c_k^{\frac{\lambda +\alpha }{s}}}\right)u^{{\lambda }_1-1}du\hfill \\ & =& \sum _{i=0}^{s_0-1}\left[{\int }_{c_i}^{c_{i+1}}(-lnu)u^{{\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i-1}du\right]\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \\ & & +\left[{\int }_{c_{s_0}}^1(-lnu)u^{{\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0-1}du\right]\frac{{\prod }_{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \\ & & +\left[{\int }_1^{c_{s_0+1}}(lnu)u^{{\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0-1}du\right]\frac{{\prod }_{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \\ & & +\sum _{i=s_0+1}^s\left[{\int }_{c_i}^{c_{i+1}}(lnu)u^{{\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i-1}du\right]\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}.\hfill \end{array}$$

Hence, we find that

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& \sum _{i=0}^{s_0-1}\frac{1}{{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}^2}\left\{[1-({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_{i+1}]c_{i+1}^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right.\hfill \\ & & -\left.[1-({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_i]c_i^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right\}\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \\ & & +\frac{1-[1-({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0)ln{c}_{s_0}]c_{s_0}^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0)}}{{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0)}^2}\frac{{\prod }_{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \\ & & +\frac{[({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0)ln{c}_{s_0+1}-1]c_{s_0+1}^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0)}+1}{{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}{s}_0)}^2}\frac{{\prod }_{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \end{array}$$

$$\begin{array}{ccc}& & +\sum _{i=s_0+1}^s\frac{1}{{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}^2}\left\{[({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_{i+1}-1]c_{i+1}^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right.\hfill \\ & & -\left.[({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_i-1]c_i^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right\}\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}.\hfill \end{array}$$

(10)

In particular:

(1)

For $s_0=s,0<c_1\le \cdots \le c_s\le 1,$ we have

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& \sum _{i=0}^{s-1}\frac{1}{{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}^2}\hfill \\ & & \times \left\{[1-({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_{i+1}]c_{i+1}^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right.\hfill \\ & & -\left.[1-({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_i]c_i^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right\}\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \\ & & +\frac{2-[1+({\lambda }_2+\alpha )ln{c}_s]c_s^{-({\lambda }_2+\alpha )}}{{({\lambda }_2+\alpha )}^2}\prod _{k=1}^{s}c{}_k^{\frac{\alpha }{s}};\hfill \end{array}$$

(2)

For $s_0=0,1<c_1\le \cdots \le c_s,$ we have

$$\begin{array}{ccc}\hfill k({\lambda }_1& )=& \frac{[({\lambda }_1+\alpha )ln{c}_1-1]c_1^{({\lambda }_1+\alpha )}+2}{{({\lambda }_1+\alpha )}^2}\frac{1}{{\prod }_{k=1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}}\hfill \\ & & +\sum _{i=1}^s\frac{1}{{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}^2}\left\{[({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_{i+1}-1]c_{i+1}^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right.\hfill \\ & & -\left.[({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)ln{c}_i-1]c_i^{({\lambda }_1+\alpha -\frac{\lambda +2\alpha }{s}i)}\right\}\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda +\alpha }{s}}};\hfill \end{array}$$

(3)

For $s=1$ (or $c_s=\cdots =c_1),$

$$h(u)=\frac{|lnu|{(min\{u,c_1\})}^{\alpha }}{{(max\{u,c_1\})}^{\lambda +\alpha }},$$

in view of (1) and (2), we deduce that

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& {\int }_0^{\infty }|lnu|\frac{{(min\{u,c_1\})}^{\alpha }{u}^{{\lambda }_1-1}}{{(max\{u,c_1\})}^{\lambda +\alpha }}du\hfill \\ & =& \left\{\begin{array}{c}\left[\frac{1-({\lambda }_1+\alpha )ln{c}_1}{{({\lambda }_1+\alpha )}^2}+\frac{2c_1^{({\lambda }_2+\alpha )}-1-({\lambda }_2+\alpha )ln{c}_1}{{({\lambda }_2+\alpha )}^2}\right]\frac{1}{c{}_1^{{\lambda }_2}},c_1\le 1\\ \left[\frac{({\lambda }_1+\alpha )ln{c}_1-1+2^{-({\lambda }_1+\alpha )}}{{({\lambda }_1+\alpha )}^2}+\frac{1+({\lambda }_2+\alpha )ln{c}_1}{{({\lambda }_2+\alpha )}^2}\right]\frac{1}{c{}_1^{{\lambda }_2}},c_1>1\end{array}\right.;\hfill \end{array}$$

(4)

For $\alpha =0,$

$$h(u)=\frac{|lnu|}{{\prod }_{k=1}^s{(max\{u,c_k\})}^{\frac{\lambda }{s}}},{\lambda }_1,{\lambda }_2>0,$$

we get that

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& \sum _{i=0}^{s_0-1}\frac{1}{{({\lambda }_1-\frac{\lambda i}{s})}^2}\left\{[1-({\lambda }_1-\frac{\lambda i}{s})ln{c}_{i+1}]c_{i+1}^{({\lambda }_1-\frac{\lambda i}{s})}\right.\hfill \\ & & -\left.[1-({\lambda }_1-\frac{\lambda i}{s})ln{c}_i]c_i^{({\lambda }_1-\frac{\lambda i}{s})}\right\}\frac{1}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda }{s}}}\hfill \\ & & +\frac{1-[1-({\lambda }_1-\frac{\lambda }{s}{s}_0)ln{c}_{s_0}]c_{s_0}^{({\lambda }_1-\frac{\lambda }{s}{s}_0)}}{{({\lambda }_1-\frac{\lambda }{s}{s}_0)}^2}\frac{1}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\lambda }{s}}}\hfill \\ & & +\frac{[({\lambda }_1-\frac{\lambda }{s}{s}_0)ln{c}_{s_0+1}-1]c_{s_0+1}^{({\lambda }_1-\frac{\lambda }{s}{s}_0)}+1}{{({\lambda }_1-\frac{\lambda }{s}{s}_0)}^2}\frac{1}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\lambda }{s}}}\hfill \\ & & +\sum _{i=s_0+1}^s\frac{1}{{({\lambda }_1-\frac{\lambda i}{s})}^2}\left\{[({\lambda }_1-\frac{\lambda i}{s})ln{c}_{i+1}-1]c_{i+1}^{({\lambda }_1-\frac{\lambda i}{s})}\right.\hfill \\ & & -\left.[({\lambda }_1-\frac{\lambda i}{s})ln{c}_i-1]c_i^{({\lambda }_1-\frac{\lambda i}{s})}\right\}\frac{1}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\lambda }{s}}};\hfill \end{array}$$

(5)

For $\lambda =0,$

$$h(u)=|lnu|\prod _{k=1}^s{\left(\frac{min\{u,c_k\}}{max\{u,c_k\}}\right)}^{\frac{\alpha }{s}},|{\lambda }_1|<\alpha (\alpha >0),$$

we have

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& \sum _{i=0}^{s_0-1}\frac{1}{{({\lambda }_1+\alpha -\frac{2\alpha }{s}i)}^2}\left\{[1-({\lambda }_1+\alpha -\frac{2\alpha }{s}i)ln{c}_{i+1}]c_{i+1}^{({\lambda }_1+\alpha -\frac{2\alpha }{s}i)}\right.\hfill \\ & & -\left.[1-({\lambda }_1+\alpha -\frac{2\alpha }{s}i)ln{c}_i]c_i^{({\lambda }_1+\alpha -\frac{2\alpha }{s}i)}\right\}\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\alpha }{s}}}\hfill \\ & & +\frac{1-[1-({\lambda }_1+\alpha -\frac{2\alpha }{s}{s}_0)ln{c}_{s_0}]c_{s_0}^{({\lambda }_1+\alpha -\frac{2\alpha }{s}{s}_0)}}{{({\lambda }_1+\alpha -\frac{2\alpha }{s}{s}_0)}^2}\frac{{\prod }_{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\alpha }{s}}}\hfill \\ & & +\frac{[({\lambda }_1+\alpha -\frac{2\alpha }{s}{s}_0)ln{c}_{s_0+1}-1]c_{s_0+1}^{({\lambda }_1+\alpha -\frac{2\alpha }{s}{s}_0)}+1}{{({\lambda }_1+\alpha -\frac{2\alpha }{s}{s}_0)}^2}\frac{{\prod }_{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=s_0+1}^{s}c{}_k^{\frac{\alpha }{s}}}\hfill \\ & & +\sum _{i=s_0+1}^s\frac{1}{{({\lambda }_1+\alpha -\frac{2\alpha }{s}i)}^2}\left\{[({\lambda }_1+\alpha -\frac{2\alpha }{s}i)ln{c}_{i+1}-1]c_{i+1}^{({\lambda }_1+\alpha -\frac{2\alpha }{s}i)}\right.\hfill \\ & & -\left.[({\lambda }_1+\alpha -\frac{2\alpha }{s}i)ln{c}_i-1]c_i^{({\lambda }_1+\alpha -\frac{2\alpha }{s}i)}\right\}\frac{{\prod }_{k=1}^{i}c{}_k^{\frac{\alpha }{s}}}{{\prod }_{k=i+1}^{s}c{}_k^{\frac{\alpha }{s}}};\hfill \end{array}$$

(6)

For $\lambda =-\alpha (\alpha >0),$

$$h(u)=|lnu|\prod _{k=1}^s{(min\{u,c_k\})}^{\frac{\alpha }{s}},$$

we derive that

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& \sum _{i=0}^{s_0-1}\frac{1}{{({\lambda }_1+\alpha -\frac{\alpha i}{s})}^2}\left\{[1-({\lambda }_1+\alpha -\frac{\alpha i}{s})ln{c}_{i+1}]c_{i+1}^{({\lambda }_1+\alpha -\frac{\alpha i}{s})}\right.\hfill \\ & & -\left.[1-({\lambda }_1+\alpha -\frac{\alpha i}{s})ln{c}_i]c_i^{({\lambda }_1+\alpha -\frac{\alpha i}{s})}\right\}\prod _{k=1}^{i}c{}_k^{\frac{\alpha }{s}}\hfill \\ & & +\frac{1-[1-({\lambda }_1+\alpha -\frac{\alpha }{s}{s}_0)ln{c}_{s_0}]c_{s_0}^{({\lambda }_1+\alpha -\frac{\alpha }{s}{s}_0)}}{{({\lambda }_1+\alpha -\frac{\alpha }{s}{s}_0)}^2}\prod _{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}\hfill \\ & & +\frac{[({\lambda }_1+\alpha -\frac{\alpha }{s}{s}_0)ln{c}_{s_0+1}-1]c_{s_0+1}^{({\lambda }_1+\alpha -\frac{\alpha }{s}{s}_0)}+1}{{({\lambda }_1+\alpha -\frac{\alpha }{s}{s}_0)}^2}\prod _{k=1}^{s_0}c{}_k^{\frac{\alpha }{s}}\hfill \\ & & +\sum _{i=s_0+1}^s\frac{1}{{({\lambda }_1+\alpha -\frac{\alpha i}{s})}^2}\left\{[({\lambda }_1+\alpha -\frac{\alpha i}{s})ln{c}_{i+1}-1]c_{i+1}^{({\lambda }_1+\alpha -\frac{\alpha i}{s})}\right.\hfill \\ & & -\left.[({\lambda }_1+\alpha -\frac{\alpha i}{s})ln{c}_i-1]c_i^{({\lambda }_1+\alpha -\frac{\alpha i}{s})}\right\}\prod _{k=1}^{i}c{}_k^{\frac{\alpha }{s}}.\hfill \end{array}$$

For $n\in \mathbf{N}$, we consider the following two expressions:

$$I_1:={\int }_1^{\infty }\left\{{\int }_0^1|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]x^{{\lambda }_1+\frac{1}{pn}-1}dx\right\}{y}^{{\sigma }_1-\frac{1}{qn}-1}dy,$$

(11)

$$I_2:={\int }_0^1\left\{{\int }_1^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]x^{{\lambda }_1-\frac{1}{pn}-1}dx\right\}{y}^{{\sigma }_1+\frac{1}{qn}-1}dy.$$

(12)

Setting $u=xy$ in (11) and (12), by Fubini’s theorem (cf. [27]), we obtain

$$\begin{array}{ccc}\hfill I_1& =& {\int }_1^{\infty }\left\{{\int }_0^y|lnu|\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]{\left(\frac{u}{y}\right)}^{{\lambda }_1+\frac{1}{pn}-1}\frac{du}{y}\right\}{y}^{{\sigma }_1-\frac{1}{qn}-1}dy\hfill \\ & =& {\int }_1^{\infty }{y}^{({\sigma }_1-{\lambda }_1)-\frac{1}{n}-1}\left\{{\int }_0^y|lnu|\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\right\}dy\hfill \\ & =& {\int }_1^{\infty }{y}^{({\sigma }_1-{\lambda }_1)-\frac{1}{n}-1}dy{\int }_0^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\hfill \\ & +& {\int }_1^{\infty }{y}^{({\sigma }_1-{\lambda }_1)-\frac{1}{n}-1}{\int }_1^{y}lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}dudy\hfill \\ & =& {\int }_1^{\infty }{y}^{({\sigma }_1-{\lambda }_1)-\frac{1}{n}-1}dy{\int }_0^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\hfill \\ & +& {\int }_1^{\infty }[{\int }_u^{\infty }{y}^{({\sigma }_1-{\lambda }_1)-\frac{1}{n}-1}dy]lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill I_2& =& {\int }_0^1\left\{{\int }_y^{\infty }|lnu|\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]{\left(\frac{u}{y}\right)}^{{\lambda }_1-\frac{1}{pn}-1}\frac{du}{y}\right\}{y}^{{\sigma }_1+\frac{1}{qn}-1}dy\hfill \\ & =& {\int }_0^1{y}^{({\sigma }_1-{\lambda }_1)+\frac{1}{n}-1}\left\{{\int }_y^{\infty }|lnu|\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{pn}-1}du\right\}dy\hfill \\ & =& {\int }_0^1{y}^{({\sigma }_1-{\lambda }_1)+\frac{1}{n}-1}dy{\int }_y^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{pn}-1}du\hfill \\ & & +{\int }_0^1{y}^{({\sigma }_1-{\lambda }_1)+\frac{1}{n}-1}{\int }_1^{\infty }lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{pn}-1}dudy\hfill \\ & =& {\int }_0^1\left[{\int }_0^u{y}^{({\sigma }_1-{\lambda }_1)+\frac{1}{n}-1}dy\right](-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{pn}-1}du\hfill \\ & & +{\int }_0^1{y}^{({\sigma }_1-{\lambda }_1)+\frac{1}{n}-1}dy{\int }_1^{\infty }lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{pn}-1}du.\hfill \end{array}$$

(14)

Lemma 1.

Suppose that $p>1,\frac{1}{p}+\frac{1}{q}=1,$${\sigma }_1\in \mathbf{R}.$ If there exists a constant M, such that for any non-negative measurable functions $f(x)$ and $g(y)$ in $(0,\infty ),$ the following inequality

$$\begin{array}{ccc}\hfill I& :=& {\int }_0^{\infty }{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]f(x)g(y)dxdy\hfill \\ & \le & M{\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-{\sigma }_1)-1}{g}^q(y)dy\right]}^{\frac{1}{q}}\hfill \end{array}$$

(15)

holds, then we have ${\sigma }_1={\lambda }_1.$ When ${\sigma }_1={\lambda }_1,$ we have $M\ge k({\lambda }_1).$

Proof. 

If ${\sigma }_1<{\lambda }_1,$ then for

$$n>\frac{1}{{\lambda }_1-{\sigma }_1}(n\in \mathbf{N}),$$

we set the following two functions

$$f_n(x):=\left\{\begin{array}{c}0,0<x<1\\ x^{{\lambda }_1-\frac{1}{pn}-1},x\ge 1\end{array}\right.,g_n(y):=\left\{\begin{array}{c}{y}^{{\sigma }_1+\frac{1}{qn}-1},0<y\le 1\\ 0,y>1\end{array}\right..$$

Hence, we derive that

$$\begin{array}{ccc}\hfill J_2& :=& {\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}_n^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-{\sigma }_1)-1}{g}_n^q(y)dy\right]}^{\frac{1}{q}}\hfill \\ & =& {\left({\int }_1^{\infty }{x}^{-\frac{1}{n}-1}dx\right)}^{\frac{1}{p}}{\left({\int }_0^1{y}^{\frac{1}{n}-1}dy\right)}^{\frac{1}{q}}=n.\hfill \end{array}$$

By (14) and (15), we have

$$\begin{array}{ccc}& & {\int }_0^1\left[{\int }_0^u{y}^{({\sigma }_1-\sigma )+\frac{1}{n}-1}dy\right](-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{pn}-1}du\hfill \\ & \le & I_2={\int }_0^{\infty }{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]f_n(x)g_n(y)dxdy\hfill \\ & \le & M{J}_2=Mn.\hfill \end{array}$$

(16)

Since

$$({\sigma }_1-{\lambda }_1)+\frac{1}{n}<0,$$

it follows that for any $u\in (0,1),$

$${\int }_0^u{y}^{({\sigma }_1-{\lambda }_1)+\frac{1}{n}-1}dy=\infty .$$

By (16), in view of

$$(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{pn}-1}>0,u\in (0,1),$$

we obtain that $\infty \le Mn<\infty ,$ which is a contradiction.

If ${\sigma }_1>{\lambda }_1,$ then for

$$n>\frac{1}{{\sigma }_1-{\lambda }_1}(n\in \mathbf{N}),$$

we set

$${\tilde{f}}_n(x):=\left\{\begin{array}{c}{x}^{{\lambda }_1+\frac{1}{pn}-1},0<x\le 1\\ 0,x>1\end{array}\right.,{\tilde{g}}_n(y):=\left\{\begin{array}{c}0,0<y<1\\ y^{{\sigma }_1-\frac{1}{qn}-1},y\ge 1\end{array}\right.,$$

and find that

$$\begin{array}{ccc}\hfill {\tilde{J}}_2& :=& {\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{\tilde{f}}_n^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-{\sigma }_1)-1}{\tilde{g}}_n^q(y)dy\right]}^{\frac{1}{q}}\hfill \\ & =& {\left({\int }_0^1{x}^{\frac{1}{n}-1}dx\right)}^{\frac{1}{p}}{\left({\int }_1^{\infty }{y}^{-\frac{1}{n}-1}dy\right)}^{\frac{1}{q}}=n.\hfill \end{array}$$

By (13) and (15), we have

$$\begin{array}{ccc}& & {\int }_1^{\infty }{y}^{({\sigma }_1-{\lambda }_1)-\frac{1}{n}-1}dy{\int }_0^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\hfill \\ & \le & I_1={\int }_0^{\infty }{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]{\tilde{f}}_n(x){\tilde{g}}_n(y)dxdy\hfill \\ & \le & M{\tilde{J}}_2=Mn.\hfill \end{array}$$

(17)

Since $({\sigma }_1-{\lambda }_1)-\frac{1}{n}>0,$ it follows that

$${\int }_1^{\infty }{y}^{({\sigma }_1-{\lambda }_1)-\frac{1}{n}-1}dy=\infty .$$

By (17), in view of

$${\int }_0^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du>0,$$

we have $\infty \le Mn<\infty ,$ which is a contradiction.

Hence, we conclude that ${\sigma }_1={\lambda }_1.$

For ${\sigma }_1={\lambda }_1,$ we reduce (13) and then use (17) as follows:

$$\begin{array}{ccc}\hfill \frac{1}{n}{I}_1& =& \frac{1}{n}\left\{{\int }_1^{\infty }{y}^{-\frac{1}{n}-1}dy{\int }_0^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\right.\hfill \\ & & +\left.{\int }_1^{\infty }\left({\int }_u^{\infty }{y}^{-\frac{1}{n}-1}dy\right)lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\right\}\hfill \\ & =& {\int }_0^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\hfill \\ & & +{\int }_1^{\infty }lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{qn}-1}du\le \frac{1}{n}M{\tilde{J}}_2=M.\hfill \end{array}$$

(18)

Since

$$(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}$$

($lnu\left[{\prod }_{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{qn}-1}$) is nonnegative and increasing in $(0,1)$ ($(1,\infty )$), by Levi’s theorem (cf. [27]), we derive that

$$\begin{array}{ccc}\hfill k({\lambda }_1)& =& {\int }_0^1\underset{n\to \infty }{lim}(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\hfill \\ & & +{\int }_1^{\infty }\underset{n\to \infty }{lim}lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{qn}-1}du\hfill \\ & =& \underset{n\to \infty }{lim}\left\{{\int }_0^1(-lnu)\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1+\frac{1}{pn}-1}du\right.\hfill \\ & & \left.+{\int }_1^{\infty }lnu\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-\frac{1}{qn}-1}du\right\}\le M<\infty .\hfill \end{array}$$

(19)

This completes the proof of the lemma. □

3. Main Results and Operator Expressions

Theorem 1.

Suppose that $p>1,\frac{1}{p}+\frac{1}{q}=1,{\sigma }_1\in \mathbf{R}.$ The following statements are equivalent:

(i) 

There exists a constant M such that for any $f(x)\ge 0$, with

$$0<{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx<\infty ,$$

the following inequality holds true:

$$\begin{array}{ccc}\hfill J& :=& {\left\{{\int }_0^{\infty }{y}^{p{\sigma }_1-1}{\left[{\int }_0^{\infty }|lnxy|\left(\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right)f(x)dx\right]}^{p}dy\right\}}^{\frac{1}{p}}\hfill \\ & <& M{\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}};\hfill \end{array}$$

(20)

(ii) 

There exists a constant $M,$ such that for any $f(x),g(y)\ge 0,$ with

$$0<{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx<\infty$$

and

$$0<{\int }_0^{\infty }{y}^{q(1-{\sigma }_1)-1}{g}^q(y)dy<\infty ,$$

the following inequality holds true:

$$\begin{array}{ccc}\hfill I& =& {\int }_0^{\infty }{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]f(x)g(y)dxdy\hfill \\ & <& M{\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-{\sigma }_1)-1}{g}^q(y)dy\right]}^{\frac{1}{q}};\hfill \end{array}$$

(21)

(iii) 

${\sigma }_1={\lambda }_1.$

If Condition (iii) is satisfied, then $M\ge k({\lambda }_1)$ and the constant factor $M=k({\lambda }_1)$ in (20) and (21) is the best possible.

Proof. 

“$(i)\Rightarrow (ii)$”. By Hölder’s inequality (cf. [28]), we have

$$\begin{array}{ccc}\hfill I& =& {\int }_0^{\infty }\left[y^{{\sigma }_1-\frac{1}{p}}{\int }_0^{\infty }|lnxy|\left(\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right)f(x)dx\right]\left(y^{\frac{1}{p}-{\sigma }_1}g(y)\right)dy\hfill \\ & \le & J{\left[{\int }_0^{\infty }{y}^{q(1-{\sigma }_1)-1}{g}^q(y)dy\right]}^{\frac{1}{q}}.\hfill \end{array}$$

(22)

Then by (20), we deduce (21).

“$(ii)\Rightarrow (iii)$”. By Lemma 1, we have ${\sigma }_1={\lambda }_1.$

“$(iii)\Rightarrow (i)$”. Setting $u=xy,$ we obtain the following weight function:

For $y>0,$

$$\begin{array}{ccc}\hfill \omega ({\lambda }_1,y)& :=& y^{{\lambda }_1}{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]x^{{\lambda }_1-1}dx\hfill \\ & =& {\int }_0^{\infty }|lnu|\left[\prod _{k=1}^s\frac{{(min\{u,c_k\})}^{\frac{\alpha }{s}}}{{(max\{u,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]u^{{\lambda }_1-1}du=k({\lambda }_1).\hfill \end{array}$$

(23)

By Hölder’s inequality with weight and (23), we obtain that

$$\begin{array}{ccc}& & {\left\{{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]f(x)dx\right\}}^p\hfill \\ & =& {\left\{{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]\frac{y^{({\lambda }_1-1)/p}f(x)}{x^{({\lambda }_1-1)/q}}\frac{x^{({\lambda }_1-1)/q}}{y^{({\lambda }_1-1)/p}}dx\right\}}^p\hfill \\ & \le & {\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]\frac{y^{{\lambda }_1-1}}{x^{({\lambda }_1-1)p/q}}{f}^p(x)dx\hfill \\ & & \times {\left\{{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]\frac{x^{{\lambda }_1-1}}{y^{({\lambda }_1-1)q/p}}dx\right\}}^{p/q}\hfill \\ & =& {\left[\frac{\omega ({\lambda }_1,y)}{y^{q({\lambda }_1-1)+1}}\right]}^{p-1}{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]\frac{y^{{\lambda }_1-1}{f}^p(x)}{x^{({\lambda }_1-1)p/q}}dx\hfill \\ & =& \frac{{(k({\lambda }_1))}^{p-1}}{y^{p{\lambda }_1-1}}{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]\frac{y^{{\lambda }_1-1}{f}^p(x)}{x^{({\lambda }_1-1)p/q}}dx.\hfill \end{array}$$

(24)

If (24) assumes the form of equality for some $y\in (0,\infty )$, then (cf. [28]) there exist constants A and B, such that they are not all zero and

$$A\frac{y^{{\lambda }_1-1}}{x^{({\lambda }_1-1)p/q}}{f}^p(x)=B\frac{x^{{\lambda }_1-1}}{y^{({\lambda }_1-1)q/p}}\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbf{R}}_{+}.$$

We suppose that $A\ne 0$ (otherwise, $B=A=0$). Then, it follows that

$$x^{p(1-{\lambda }_1)-1}{f}^p(x)=y^{q(1-{\lambda }_1)}{\frac{B}{Ax}}^{}\mathrm{a}.\mathrm{e}.{}^{}\mathrm{i}{\mathrm{n}}^{}{\mathbf{R}}_{+},$$

which contradicts the fact that

$$0<{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx<\infty .$$

Hence, (24) assumes the form of strict inequality.

Therefore, for ${\sigma }_1={\lambda }_1,$ by Fubini’s theorem, we derive that

$$\begin{array}{ccc}\hfill J& <& {(k({\lambda }_1))}^{\frac{1}{q}}\left\{{\int }_0^{\infty }{\int }_0^{\infty }|lnxy|\left[\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right]\right.{\left.\frac{y^{{\lambda }_1-1}}{x^{({\lambda }_1-1)p/q}}{f}^p(x)dxdy\right\}}^{\frac{1}{p}}\hfill \\ & =& {(k({\lambda }_1))}^{\frac{1}{q}}{\left\{{\int }_0^{\infty }\left[{\int }_0^{\infty }|lnxy|\left(\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right)\frac{y^{{\lambda }_1-1}dy}{x^{({\lambda }_1-1)p/q}}\right]f^p(x)dx\right\}}^{\frac{1}{p}}\hfill \\ & =& {(k({\lambda }_1))}^{\frac{1}{q}}{\left[{\int }_0^{\infty }\omega ({\lambda }_1,x)x^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}}\hfill \\ & =& k({\lambda }_1){\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}}.\hfill \end{array}$$

Setting $M\ge k({\lambda }_1),$ (20) follows.

Thus, the conditions (i), (ii) and (iii) are equivalent.

When Condition (iii) is satisfied, if there exists a constant $M\le k({\lambda }_1),$ such that (21) holds true, then by Lemma 1 we have that $M\ge k({\lambda }_1).$ Then the constant factor $M=k({\lambda }_1)$ in (21) is the best possible. The constant factor $M=k({\lambda }_1)$ in (20) is still the best possible. Otherwise, by (22) (for ${\sigma }_1={\lambda }_1$), we would conclude that the constant factor $M=k({\lambda }_1)$ in (21) is not the best possible.

This completes the proof of the theorem. □

Setting $y=\frac{1}{Y}$, $G(Y)=Y^{\lambda -2}g(\frac{1}{Y}),{\sigma }_2=\lambda -{\sigma }_1$ in Theorem 1, then replacing Y ($G(Y)$) by y ($g(y)$), we derive the following corollary.

Corollary 1.

Suppose that $p>1,\frac{1}{p}+\frac{1}{q}=1,{\sigma }_2\in \mathbf{R}.$ The following conditions are equivalent:

(i) 

There exists a constant $M,$ such that for any $f(x)\ge 0$ satisfying

$$0<{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx<\infty ,$$

we have the following Hilbert-type inequality with the homogeneous kernel:

$$\begin{array}{ccc}& & {\left\{{\int }_0^{\infty }{y}^{p{\sigma }_2-1}{\left[{\int }_0^{\infty }|ln\frac{x}{y}|\left(\prod _{k=1}^s\frac{{(min\{x,c_{k}y\})}^{\frac{\alpha }{s}}}{{(max\{x,c_{k}y\})}^{\frac{\lambda +\alpha }{s}}}\right)f(x)dx\right]}^{p}dy\right\}}^{\frac{1}{p}}\hfill \\ & <& M{\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}};\hfill \end{array}$$

(25)

(ii) 

There exists a constant $M,$ such that for any $f(x),g(y)\ge 0,$ satisfying

$$0<{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx<\infty ,$$

and

$$0<{\int }_0^{\infty }{y}^{q(1-{\sigma }_2)-1}{g}^q(y)dy<\infty ,$$

we have the following inequality:

$$\begin{array}{ccc}& & {\int }_0^{\infty }{\int }_0^{\infty }|ln\frac{x}{y}|\left[\prod _{k=1}^s\frac{{(min\{x,c_{k}y\})}^{\frac{\alpha }{s}}}{{(max\{x,c_{k}y\})}^{\frac{\lambda +\alpha }{s}}}\right]f(x)g(y)dxdy\hfill \\ & <& M{\left[{\int }_0^{\infty }{x}^{p(1-{\lambda }_1)-1}{f}^p(x)dx\right]}^{\frac{1}{p}}{\left[{\int }_0^{\infty }{y}^{q(1-{\sigma }_2)-1}{g}^q(y)dy\right]}^{\frac{1}{q}};\hfill \end{array}$$

(26)

(iii) 

${\sigma }_2={\lambda }_2.$

If Condition (iii) is satisfied, then we have $M\ge k({\lambda }_2),$ and the constant $M=k({\lambda }_2)$ in (25) and (26) is the best possible.

Remark 1.

On the other hand, setting $y=\frac{1}{Y}$, $G(Y)=Y^{\lambda -2}g(\frac{1}{Y}),{\sigma }_1=\lambda -{\sigma }_2,$ in Corollary 1, then replacing Y ($G(Y))$ by y ($g(y)),$ we deduce Theorem 1. Hence, Theorem 1 and Corollary 1 are equivalent.

For $p>1,\frac{1}{p}+\frac{1}{q}=1,$ we set the following functions:

$$\phi (x):=x^{p(1-{\lambda }_1)-1},\psi (y):=y^{q(1-{\lambda }_1)-1},\varphi (y):=y^{q(1-{\lambda }_2)-1},$$

wherefrom,

$${\psi }^{1-p}(y)=y^{p{\lambda }_1-1},{\varphi }^{1-p}(y)=y^{p{\lambda }_2-1}(x,y\in {\mathbf{R}}_{+}).$$

Define the following real normed linear spaces:

$$\begin{array}{ccc}\hfill L_{p,\phi }({\mathbf{R}}_{+})& =& \left\{{f:||f||}_{p,\phi }:={\left({\int }_0^{\infty }\phi (x){|f(x)|}^{p}dx\right)}^{\frac{1}{p}}<\infty \right\},\hfill \\ \hfill L_{q,\psi }({\mathbf{R}}_{+})& =& \left\{{g:||g||}_{q,\psi }:={\left({\int }_0^{\infty }\psi (y){|g(y)|}^{q}dy\right)}^{\frac{1}{q}}<\infty \right\},\hfill \\ \hfill L_{q,\varphi }({\mathbf{R}}_{+})& =& \left\{{g:||g||}_{q,\varphi }:={\left({\int }_0^{\infty }\varphi (y){|g(y)|}^{q}dy\right)}^{\frac{1}{q}}<\infty \right\},\hfill \\ \hfill L_{p,{\psi }^{1-p}}({\mathbf{R}}_{+})& =& \left\{{h:||h||}_{p,{\psi }^{1-p}}={\left({\int }_0^{\infty }{\psi }^{1-p}(y){|h(y)|}^{p}dy\right)}^{\frac{1}{p}}<\infty \right\},\hfill \\ \hfill L_{q,{\varphi }^{1-p}}({\mathbf{R}}_{+})& =& \left\{{h:||h||}_{p,{\varphi }^{1-p}}={\left({\int }_0^{\infty }{\varphi }^{1-p}(y){|h(y)|}^{p}dy\right)}^{\frac{1}{p}}<\infty \right\}.\hfill \end{array}$$

(a) In view of Theorem 1 (with ${\sigma }_1={\lambda }_1)$, for $f\in L_{p,\phi }({\mathbf{R}}_{+}),$ setting

$$h_1(y):={\int }_0^{\infty }|lnxy|\left(\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right)f(x)d{x}^{}(y\in {\mathbf{R}}_{+}),$$

by (20), we obtain that

$$||h_1{||}_{p,{\psi }^{1-p}}={\left({\int }_0^{\infty }{\psi }^{1-p}(y)h_1^p(y)dy\right)}^{\frac{1}{p}}{<M||f||}_{p,\phi }<\infty .$$

(27)

Definition 1.

Define a Hilbert-type integral operator with the non-homogeneous kernel  $T^{(1)}:L_{p,\phi }({\mathbf{R}}_{+})\to L_{p,{\psi }^{1-p}}({\mathbf{R}}_{+})$ as follows:

For any $f\in L_{p,\phi }({\mathbf{R}}_{+}),$ there exists a unique representation $T^{(1)}f=h_1\in L_{p,{\psi }^{1-p}}({\mathbf{R}}_{+}),$ satisfying $T^{(1)}f(y)=h_1(y),$ for any $y\in {\mathbf{R}}_{+}.$

In view of (27), it follows that

$$||T^{(1)}{f||}_{p,{\psi }^{1-p}}=||h_1{||}_{p,{\psi }^{1-p}}\le {M||f||}_{p,\phi },$$

and then the operator $T^{(1)}$ is bounded satisfying

$$||T^{(1)}||=\underset{f(\ne \theta )\in L_{p,\phi }({\mathbf{R}}_{+})}{sup}\frac{||T^{(1)}f{||}_{p,{\psi }^{1-p}}}{{||f||}_{p,\phi }}\le M.$$

If we define the formal inner product of $T^{(1)}f$ and g as follows:

$$(T^{(1)}f,g):={\int }_0^{\infty }\left[{\int }_0^{\infty }|lnxy|\left(\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}\right)f(x)dx\right]g(y)dy,$$

then we can rewrite Theorem 1 (for ${\sigma }_1={\lambda }_1$) as follows:

Theorem 2.

Suppose that $p>1,\frac{1}{p}+\frac{1}{q}=1.$ The following conditions are equivalent:

(i)

There exists a constant $M,$ such that for any $f(x)\ge 0,$$f\in L_{p,\phi }$$({\mathbf{R}}_{+}),$${||f||}_{p,\phi }>0,$ we have the following inequality:

$$||T^{(1)}{f||}_{p,{\psi }^{1-p}}{<M||f||}_{p,\phi };$$

(28)

(ii)

There exists a constant $M,$ such that for any $f(x),g(y)\ge 0,$$f\in L_{p,\phi }({\mathbf{R}}_{+}),$$g\in L_{q,\psi }({\mathbf{R}}_{+}),$${||f||}_{p,\phi }{,||g||}_{q,\psi }>0,$ we have the following inequality:

$$(T^{(1)}f,g){<M||f||}_{p,\phi }{||g||}_{q,\psi }.$$

(29)

We still have $||T^{(1)}||=k({\lambda }_1)\le M.$

(b) In view of Corollary 1 (with ${\sigma }_2={\lambda }_2$), for $f\in L_{p,\phi }({\mathbf{R}}_{+}),$ setting

$$h_2(y):={\int }_0^{\infty }|ln\frac{x}{y}|\left[\prod _{k=1}^s\frac{{(min\{x,c_{k}y\})}^{\frac{\alpha }{s}}}{{(max\{x,c_{k}y\})}^{\frac{\lambda +\alpha }{s}}}\right] f(x)d{x}^{}(y\in {\mathbf{R}}_{+}),$$

by (27), we have

$$||h_2{||}_{p,{\varphi }^{1-p}}={\left({\int }_0^{\infty }{\varphi }^{1-p}(y)h_2^p(y)dy\right)}^{\frac{1}{p}}{<M||f||}_{p,\phi }<\infty .$$

(30)

Definition 2.

Define a Hilbert-type integral operator with the homogeneous kernel $T^{(2)}:L_{p,\phi }({\mathbf{R}}_{+})\to L_{p,{\varphi }^{1-p}}({\mathbf{R}}_{+})$ as follows:

For any $f\in L_{p,\phi }(\mathbf{R}),$ there exists a unique representation $T^{(2)}f=h_2\in L_{p,{\varphi }^{1-p}}({\mathbf{R}}_{+}),$ satisfying $T^{(2)}f(y)=h_2(y),$ for any $y\in {\mathbf{R}}_{+}$.

In view of (30), it follows that

$$||T^{(2)}{f||}_{p,{\varphi }^{1-p}}=||h_2{||}_{p,{\varphi }^{1-p}}\le {M||f||}_{p,\phi },$$

and then the operator $T^{(2)}$ is bounded satisfying

$$||T^{(2)}||=\underset{f(\ne \theta )\in L_{p,\phi }({\mathbf{R}}_{+})}{sup}\frac{||T^{(2)}f{||}_{p,{\varphi }^{1-p}}}{{||f||}_{p,\phi }}\le M.$$

If we define the formal inner product of $T^{(2)}f$ and g as follows:

$$(T^{(2)}f,g):={\int }_0^{\infty }\left\{{\int }_0^{\infty }|ln\frac{x}{y}|\left[\prod _{k=1}^s\frac{{(min\{x,c_{k}y\})}^{\frac{\alpha }{s}}}{{(max\{x,c_{k}y\})}^{\frac{\lambda +\alpha }{s}}}\right]f(x)dx\right\}g(y)dy,$$

then we can rewrite Corollary 1 (for ${\sigma }_2={\lambda }_2$) as follows:

Corollary 2.

Suppose that $p>1,\frac{1}{p}+\frac{1}{q}=1.$ The following conditions are equivalent:

(i) 

There exists a constant $M,$ such that for any $f(x)\ge 0,$$f\in L_{p,\phi }$$({\mathbf{R}}_{+}),$${||f||}_{p,\phi }>0,$ we have the following inequality:

$$||T^{(2)}{f||}_{p,{\varphi }^{1-p}}{<M||f||}_{p,\phi };$$

(31)

(ii) 

There exists a constant $M,$ such that for any $f(x),g(y)\ge 0,$$f\in L_{p,\phi }$$({\mathbf{R}}_{+}),$$g\in L_{q,\varphi }({\mathbf{R}}_{+}),$ ${||f||}_{p,\phi }{, ||g||}_{q,\varphi }>0,$ we have the following inequality:

$$(T^{(2)}f,g){<M||f||}_{p,\phi }{||g||}_{q,\varphi }.$$

(32)

We still have $||T^{(2)}||=k({\lambda }_1)\le M.$

Remark 2.

Theorem 2 and Corollary 2 are equivalent.

4. Conclusions

In this paper, by means of real analysis, an equivalent form related to a Hilbert-type integral inequality with the non-homogeneous kernel

$$|lnxy|\prod _{k=1}^s\frac{{(min\{xy,c_k\})}^{\frac{\alpha }{s}}}{{(max\{xy,c_k\})}^{\frac{\lambda +\alpha }{s}}}$$

and a best possible constant factor is given in Theorem 1. We also consider the case of the homogeneous kernel and the operator expressions in Corollary 1, Corollary 2 and Theorem 2. The lemmas and theorems provide an extensive account of this type of inequalities.