﻿ 一个非单调非齐次核的Hilbert型积分不等式
 文章快速检索 高级检索
 浙江大学学报(理学版)  2017, Vol. 44 Issue (2): 150-153  DOI:10.3785/j.issn.1008-9497.2017.02.005 0

### 引用本文 [复制中英文]

[复制中文]
ZHONG Jianhua, CHEN Qiang, ZENG Zhihong. A Hilbert-type integral inequality with a non-monotone and non-homogeneous kernel[J]. Journal of Zhejiang University(Science Edition), 2017, 44(2): 150-153. DOI: 10.3785/j.issn.1008-9497.2017.02.005.
[复制英文]

### 文章历史

1. 广东第二师范学院 数学系, 广东 广州 510303;
2. 广东第二师范学院 计算机科学系, 广东 广州 510303;
3. 广东第二师范学院 学报编辑部, 广东 广州 510303

A Hilbert-type integral inequality with a non-monotone and non-homogeneous kernel
ZHONG Jianhua1 , CHEN Qiang2 , ZENG Zhihong3
1. Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China;
2. Department of Computer Science, Guangdong University of Education, Guangzhou 510303, China;
3. Editorial Department of Journal, Guangdong University of Education, Guangzhou 510303, China
Abstract: By introducing a parameter σ, a Hilbert-type integral inequality with a non-monotone and non-homogeneous kernel and a best constant factor was established by the way of weight functions. The equivalent forms and some particular cases are also considered.
Key words: Hilbert-type integral inequality    weight coefficient    parameter    equivalent form    non-homogeneous kernel

1925年, HARDY[1]利用一对共轭指数 (p, q), $\frac{1}{p} + \frac{1}{q} = 1$, p>1, 当$0 < \smallint _{_0}^{^\infty }{f^p}\left( x \right){\rm{d}}x < \infty$$0 < \smallint _{_0}^{^\infty }{g^q}\left( x \right){\rm{d}}x < \infty$时, 得到不等式:

 $\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{f\left( x \right)g\left( y \right)}}{{x + y}}{\rm{d}}x{\rm{d}}y < \frac{\pi }{{{\rm{sin}}(\pi /p)}} \times } } \\ \;\;\;\;\;\;\;{\left( {\int_0^\infty {{f^p}\left( x \right){\rm{d}}x} } \right)^{1/p}}{\left( {\int_0^\infty {{g^q}\left( x \right){\rm{d}}x} } \right)^{1/q}}. \end{array}$ (1)

 $\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{f\left( x \right)g\left( y \right)}}{{{x^\lambda } + {y^\lambda }}}{\rm{d}}x{\rm{d}}y < \frac{\pi }{{\lambda {\rm{sin}}(\pi /r)}} \times } } \\ \;\;\;\;\;\;\;{\left[ {\int_0^\infty {{x^{p\left( {1 - \lambda /r} \right) - 1}}{f^p}\left( x \right){\rm{d}}x} } \right]^{1/p}}{\left[ {\int_0^\infty {{x^{q\left( {1 - \lambda /s} \right) - 1}}{g^q}\left( x \right){\rm{d}}x} } \right]^{1/q}}, \end{array}$ (2)

 $\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{f\left( x \right)g\left( y \right)}}{{1 + {x^\lambda }{y^\lambda }}}{\rm{d}}x{\rm{d}}y < \frac{\pi }{\lambda } \times } } \\ \;\;\;\;\;\;\;{\left[ {\int_0^\infty {{x^{p\left( {1 - \lambda /2} \right) - 1}}{f^p}\left( x \right){\rm{d}}x} } \right]^{1/p}}{\left[ {\int_0^\infty {{x^{q\left( {1 - \lambda /2} \right) - 1}}{g^q}\left( x \right){\rm{d}}x} } \right]^{1/q}}. \end{array}$ (3)

 $\begin{array}{l} \int_0^\infty {\int_0^\infty {\frac{{f\left( x \right)g\left( y \right)}}{{{{\left| {1 - xy} \right|}^\lambda }}}{\rm{d}}x{\rm{d}}y < 2B\left( {1 - \lambda ,\frac{\lambda }{2}} \right) \times } } \\ {\left[ {\int_0^\infty {{x^{p\left( {1 - \lambda /2} \right) - 1}}{f^p}\left( x \right){\rm{d}}x} } \right]^{1/p}}{\left[ {\int_0^\infty {{x^{q\left( {1 - \lambda /2} \right) - 1}}{g^q}\left( x \right){\rm{d}}x} } \right]^{1/q}}, \end{array}$ (4)

2012年, 文献[6]研究了非齐次核的Hilbert型不等式的一般理论, 得到了一个重要的推广:

 $0 < \int_0^\infty {{x^{p\left( {1 - \lambda /2} \right) - 1}}{f^p}\left( x \right){\rm{d}}x} < \infty$

 $0 < \int_0^\infty {{x^{q\left( {1 - \lambda /2} \right) - 1}}{g^q}\left( x \right){\rm{d}}x} < \infty$

 $\begin{array}{l} \int_0^\infty {\int_0^\infty {h\left( {xy} \right)f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y} } < \\ {k_\lambda }{\left[ {\int_0^\infty {{x^{p\left( {1 - \lambda /2} \right) - 1}}{f^p}\left( x \right){\rm{d}}x} } \right]^{1/p}}{\left[ {\int_0^\infty {{x^{q\left( {1 - \lambda /2} \right) - 1}}{g^q}\left( x \right){\rm{d}}x} } \right]^{1/q}}, \end{array}$ (5)

 $h\left( t \right) = {{\rm{e}}^{ - t}}\tanh t = {{\rm{e}}^{ - t}}\frac{{{{\rm{e}}^t} - {{\rm{e}}^{ - t}}}}{{{{\rm{e}}^t} + {{\rm{e}}^{ - t}}}}\left( {0 < t < \infty } \right),$ (6)

 $h\left( {xy} \right) = {{\rm{e}}^{ - xy}}\tanh \left( {xy} \right) = {{\rm{e}}^{ - xy}}\frac{{{{\rm{e}}^{xy}} - {{\rm{e}}^{ - xy}}}}{{{{\rm{e}}^{xy}} + {{\rm{e}}^{ - xy}}}}.$

1 引理

 ${\omega _\sigma }\left( y \right): = \int_0^\infty {{y^\sigma }{x^{\sigma - 1}}h\left( {xy} \right){\rm{d}}x} ,\;\;\;\;y > 0,$ (7)

ωσ(y) 是与y无关的正数, 且

 ${\omega _\sigma }\left( y \right) = k\left( \sigma \right): = \Gamma \left( \sigma \right)\eta \left( \sigma \right).$ (8)

 $\eta \left( \sigma \right): = 2\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {2k + 1} \right)}^\sigma }}} - 1 > 0.}$

 $0 < {\omega _\sigma }\left( y \right) = \int_0^\infty {{u^{\sigma - 1}}h\left( u \right){\rm{d}}u} .$ (9)

 $\begin{array}{*{20}{c}} {h\left( u \right) = {{\rm{e}}^{ - u}}\frac{{1 - {{\left( {{{\rm{e}}^{ - u}}} \right)}^2}}}{{1 + {{\left( {{{\rm{e}}^{ - u}}} \right)}^2}}} = {{\rm{e}}^{ - u}}\left[ {1 - \frac{{2{{\left( {{{\rm{e}}^{ - u}}} \right)}^2}}}{{1 + {{\left( {{{\rm{e}}^{ - u}}} \right)}^2}}}} \right] = }\\ {{{\rm{e}}^{ - u}} - 2\sum\limits_{k = 0}^\infty {{{\left( { - 1} \right)}^k}{{\rm{e}}^{ - \left( {2k + 3} \right)u}}} ,} \end{array}$

 $\begin{array}{l} {\omega _\sigma }\left( y \right) = \int_0^\infty {{u^{\sigma - 1}}\left[ {{{\rm{e}}^{ - u}} - 2\sum\limits_{k = 0}^\infty {{{\left( { - 1} \right)}^k}{{\rm{e}}^{ - \left( {2k + 3} \right)u}}} } \right]} {\rm{d}}u = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\Gamma \left( \sigma \right) - \sum\limits_{k = 0}^\infty {{{\left( { - 1} \right)}^k}\int_0^\infty {{{\rm{e}}^{ - \left( {2k + 3} \right)u}}{u^{\sigma - 1}}{\rm{d}}u} } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\Gamma \left( \sigma \right)2 - \sum\limits_{k = 0}^\infty {{{\left( { - 1} \right)}^k}\frac{1}{{{{\left( {2k + 3} \right)}^\sigma }}}\int_0^\infty {{{\rm{e}}^{ - t}}{u^{\sigma - 1}}{\rm{d}}t} } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\Gamma \left( \sigma \right)\left[ {1 - 2\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {2k + 3} \right)}^\sigma }}}} } \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\Gamma \left( \sigma \right)\left[ {2\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {2k + 1} \right)}^\sigma }}} - 1} } \right] > 0, \end{array}$

 $\eta \left( \sigma \right) = 2\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {2k + 1} \right)}^\sigma }}} - 1} > 0,$

 $\begin{array}{*{20}{c}} {J: = \int_0^\infty {{y^{p\sigma - 1}}} {{\left[ {\int_0^\infty {h\left( {xy} \right)f\left( x \right){\rm{d}}x} } \right]}^p}{\rm{d}}y \le }\\ {{k^p}\left( \sigma \right)\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} ,} \end{array}$ (10)

k(σ) 的定义见式 (8).

 $\begin{array}{l} \int_0^\infty {h\left( {xy} \right)f\left( x \right){\rm{d}}x} = \\ \int_0^\infty {h\left( {xy} \right)} \left[ {\frac{{{x^{\left( {1 - \sigma } \right)/q}}}}{{{y^{\left( {1 - \sigma } \right)/p}}}}f\left( x \right)} \right]\left[ {\frac{{{y^{\left( {1 - \sigma } \right)/p}}}}{{{x^{\left( {1 - \sigma } \right)/q}}}}} \right]{\rm{d}}x \le \\ {\left[ {\int_0^\infty {h\left( {xy} \right)} \frac{{{x^{\left( {1 - \sigma } \right)/q}}}}{{{y^{1 - \sigma }}}}{f^p}\left( x \right){\rm{d}}x} \right]^{1/p}}{\left[ {\int_0^\infty {h\left( {xy} \right)} \frac{{{y^{\left( {1 - \sigma } \right)q/p}}}}{{{x^{1 - \sigma }}}}{\rm{d}}x} \right]^{1/q}} = \\ {\left\{ {{\omega _\sigma }\left( y \right)} \right\}^{1/q}}{y^{1/p - \sigma }}{\left[ {\int_0^\infty {h\left( {xy} \right)} \frac{{{x^{\left( {1 - \sigma } \right)\left( {p - 1} \right)}}}}{{{y^{1 - \sigma }}}}{f^p}\left( x \right){\rm{d}}x} \right]^{1/p}}. \end{array}$ (11)

 $\begin{array}{l} J: = \int_0^\infty {{y^{p\sigma - 1}}} {\left[ {\int_0^\infty {h\left( {xy} \right)f\left( x \right){\rm{d}}x} } \right]^p}{\rm{d}}y \le \\ {\left[ {{\omega _\sigma }\left( y \right)} \right]^{p - 1}}\int_0^\infty {\left[ {\int_0^\infty {h\left( {xy} \right)} \frac{{{x^{\left( {1 - \sigma } \right)\left( {p - 1} \right)}}}}{{{y^{1 - \sigma }}}}{f^p}\left( x \right){\rm{d}}x} \right]} {\rm{d}}y = \\ {k^{p - 1}}\left( \sigma \right)\int_0^\infty {\left[ {\int_0^\infty {h\left( {xy} \right)} {x^\sigma }{y^{\sigma - 1}}{\rm{d}}y} \right]} {x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x = \\ {k^{p - 1}}\left( \sigma \right)\int_0^\infty {{\omega _\sigma }\left( x \right){x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} = {k^p}\left( \sigma \right)\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} , \end{array}$

2 主要结果

 $0 < \int_0^\infty {{x^{q\left( {1 - \sigma } \right) - 1}}{g^q}\left( x \right){\rm{d}}x} < \infty$

 $\begin{array}{l} I: = \int_0^\infty {\int_0^\infty {h\left( {xy} \right)f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y} } < \\ \;\;\;\;\;\;k\left( \sigma \right){\left\{ {\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} } \right\}^{1/p}} \times \\ \;\;\;\;\;\;{\left\{ {\int_0^\infty {{x^{q\left( {1 - \sigma } \right) - 1}}{f^q}\left( x \right){\rm{d}}x} } \right\}^{1/q}}, \end{array}$ (12)
 $\begin{array}{l} J = \int_0^\infty {{y^{p\sigma - 1}}} {\left[ {\int_0^\infty {h\left( {xy} \right)f\left( x \right){\rm{d}}x} } \right]^p}{\rm{d}}y < \\ \;\;\;\;\;\;{k^p}\left( \sigma \right)\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} , \end{array}$ (13)

 $\frac{{A{x^{\left( {1 - \sigma } \right)p/q}}}}{{{y^{1 - \sigma }}}}{f^p}\left( x \right) = \frac{{B{y^{\left( {1 - \sigma } \right)q/p}}}}{{{x^{1 - \sigma }}}}{\rm{a}}{\rm{.e}}{\rm{.}}\;于\left( {0,\infty } \right).$

 ${x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right) = {y^{q\left( {1 - \sigma } \right)}}\frac{B}{{Ax}}{\rm{a}}{\rm{.e}}{\rm{.}}\;于\left( {0,\infty } \right).$

 $\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} = \frac{B}{A}{y^{q\left( {1 - \sigma } \right)}}\mathop {\lim }\limits_{\varepsilon \to {0^ + }} \ln x\left| {_\varepsilon ^{ + \infty }} \right. = + \infty ,$

 $\begin{array}{l} I: = \int_0^\infty {\left( {{y^{1/p - \sigma }}g\left( y \right)} \right)\left[ {{y^{\sigma - 1/p}}\int_0^\infty {h\left( {xy} \right)f\left( x \right){\rm{d}}x} } \right]} {\rm{d}}y \le \\ \;\;\;\;\;\;\;{J^{1/p}}{\left( {\int_0^\infty {{y^{q\left( {1 - \sigma } \right) - 1}}{g^q}\left( y \right){\rm{d}}y} } \right)^{1/q}}, \end{array}$ (14)

 $\begin{array}{l} 0 < J = \int_0^\infty {{y^{q\left( {1 - \sigma } \right) - 1}}{g^q}\left( y \right){\rm{d}}y} = I < \\ \;\;\;\;\;k\left( \sigma \right){\left\{ {\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} } \right\}^{1/p}} \times \\ \;\;\;\;\;{\left\{ {\int_0^\infty {{y^{q\left( {1 - \sigma } \right) - 1}}{g^q}\left( y \right){\rm{d}}y} } \right\}^{1/q}}, \end{array}$ (15)

 $\begin{array}{l} {J^{1/p}} = {\left\{ {\int_0^\infty {{y^{q\left( {1 - \sigma } \right) - 1}}{g^q}\left( y \right){\rm{d}}y} } \right\}^{1/p}} < \\ \;\;\;\;\;\;\;\;\;\;k\left( \sigma \right){\left\{ {\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{f^p}\left( x \right){\rm{d}}x} } \right\}^{1/p}}. \end{array}$ (16)

 $\tilde f\left( x \right): = \left\{ \begin{array}{l} {x^{\left( {\sigma + \varepsilon } \right)/\left( {p - 1} \right)}},\;\;\;\;\;x \in \left( {0,1} \right],\\ 0,\;\;\;\;\;\;x \in \left( {1,\infty } \right); \end{array} \right.$
 $\tilde g\left( y \right): = \left\{ \begin{array}{l} 0,\;\;\;\;\;\;y \in \left( {0,1} \right),\\ {y^{\left( {\sigma - \varepsilon } \right)/\left( {q - 1} \right)}},\;\;\;\;\;\;y \in \left[ {1,\infty } \right). \end{array} \right.$

 $\begin{array}{l} \tilde L: = {\left\{ {\int_0^\infty {{x^{p\left( {1 - \sigma } \right) - 1}}{{\tilde f}^p}\left( x \right){\rm{d}}x} } \right\}^{1/p}} \times \\ \;\;\;\;\;\;\;{\left\{ {\int_0^\infty {{y^{q\left( {1 - \sigma } \right) - 1}}{{\tilde g}^q}\left( y \right){\rm{d}}y} } \right\}^{1/q}} = 1/\varepsilon . \end{array}$

 $\begin{array}{l} \tilde I: = \int_0^\infty {\int_0^\infty {h\left( {xy} \right)\tilde f\left( x \right)\tilde g\left( y \right){\rm{d}}x{\rm{d}}y} } = \\ \;\;\;\int_1^\infty {{y^{\sigma - \varepsilon /q - 1}}\left[ {\int_0^1 {h\left( {xy} \right){x^{\sigma + \varepsilon /p - 1}}{\rm{d}}x} } \right]} {\rm{d}}y = \\ \;\;\;\int_1^\infty {{y^{ - \varepsilon - 1}}\left[ {\int_0^y {{u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} } \right]} {\rm{d}}y = \\ \;\;\;\int_1^\infty {{y^{ - \varepsilon - 1}}\left[ {\int_0^1 {{u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} } \right]} {\rm{d}}y + \\ \;\;\;\int_1^\infty {{y^{ - \varepsilon - 1}}\left[ {\int_1^y {{u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} } \right]} {\rm{d}}y + \\ \;\;\;\frac{1}{\varepsilon }\int_0^1 {{u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} + \int_1^\infty {\left( {\int_u^\infty {{y^{ - \varepsilon - 1}}{\rm{d}}y} } \right){u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} = \\ \;\;\;\frac{1}{\varepsilon }\left[ {\int_0^1 {{u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} + \int_1^\infty {{u^{\sigma - \varepsilon /q - 1}}h\left( u \right){\rm{d}}u} } \right]. \end{array}$

 $\begin{array}{l} \int_0^1 {{u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} + \\ \;\;\;\;\;\int_1^\infty {{u^{\sigma - \varepsilon /q - 1}}h\left( u \right){\rm{d}}u} = \varepsilon \tilde I < \varepsilon k\tilde L = k. \end{array}$ (17)

 $\begin{array}{l} k\left( \sigma \right) = \int_0^1 {\mathop {\lim }\limits_{\varepsilon \to {0^ + }} {u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} + \\ \;\;\;\;\;\;\;\;\;\;\;\int_1^\infty {\mathop {\lim }\limits_{\varepsilon \to {0^ + }} {u^{\sigma - \varepsilon /q - 1}}h\left( u \right){\rm{d}}u} \le \\ \;\;\;\;\;\;\;\;\;\;\;\mathop {\lim }\limits_{\varepsilon \to {0^ + }} \left( {\int_0^1 {{u^{\sigma + \varepsilon /p - 1}}h\left( u \right){\rm{d}}u} + \int_1^\infty {{u^{\sigma - \varepsilon /q - 1}}h\left( u \right){\rm{d}}u} } \right) \le k. \end{array}$

 $\begin{array}{l} \int_0^\infty {\int_0^\infty {h\left( {xy} \right)f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y} } < \\ \;\;\;\;\;\;\;k\left( {\frac{1}{2}} \right){\left\{ {\int_0^\infty {{x^{\frac{p}{2} - 1}}{f^p}\left( x \right){\rm{d}}x} } \right\}^{1/p}}{\left\{ {\int_0^\infty {{x^{\frac{q}{2} - 1}}{g^q}\left( x \right){\rm{d}}x} } \right\}^{1/q}}, \end{array}$ (18)
 $\begin{array}{l} {\int_0^\infty {{y^{p/2 - 1}}\left[ {\int_0^\infty {h\left( {xy} \right)f\left( x \right){\rm{d}}x} } \right]} ^p}{\rm{d}}y < \\ \;\;\;\;\;\;\;{k^p}\left( {\frac{1}{2}} \right)\int_0^\infty {{x^{\frac{p}{2} - 1}}{f^p}\left( x \right){\rm{d}}x} , \end{array}$ (19)

 $k\left( {\frac{1}{2}} \right) = \Gamma \left( {\frac{1}{2}} \right)\eta \left( {\frac{1}{2}} \right) = \sqrt \pi \left( {2\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{{{\left( {2k - 1} \right)}^{1/2}}}} - 1} } \right).$

σ=1时, 有

 $k\left( 1 \right) = \Gamma \left( 1 \right)\eta \left( 1 \right) = \eta \left( 1 \right) = 2\sum\limits_{k = 0}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{2k + 1}} - 1 = \frac{\pi }{2} - 1} .$

 $\begin{array}{l} \int_0^\infty {\int_0^\infty {h\left( {xy} \right)f\left( x \right)g\left( y \right){\rm{d}}x{\rm{d}}y} } < \\ \;\;\;\;\;\;\;\;\left( {\frac{\pi }{2} - 1} \right){\left\{ {\int_0^\infty {{x^{ - 1}}{f^p}\left( x \right){\rm{d}}x} } \right\}^{1/p}}{\left\{ {\int_0^\infty {{x^{ - 1}}{g^q}\left( x \right){\rm{d}}x} } \right\}^{1/q}}, \end{array}$ (20)
 $\begin{array}{l} {\int_0^\infty {{y^{p - 1}}\left[ {\int_0^\infty {h\left( {xy} \right)f\left( x \right){\rm{d}}x} } \right]} ^p}{\rm{d}}y < \\ \;\;\;\;\;\;\;\;{\left( {\frac{\pi }{2} - 1} \right)^p}\int_0^\infty {{x^{ - 1}}{f^p}\left( x \right){\rm{d}}x} . \end{array}$ (21)

 [1] HARDY G H. Note on a theorem of Hilbert concerning series of positive term[J]. Proceeding of the London Math Society, 1925, 23(2): 45–46. [2] HARDY G H, LITTLEWOOD J E, POLYE G. Inequalities[M]. Cambridge: Cambridge Univ Press, 1952. [3] MINTRINOVIC D S, PECARIC J E, FINK A M. Inequalities Involving Functions and Their Integrals and Derivatives[M]. Boston: Kluwer Academic Publishers, 1991. [4] YANG B C. On an extension of Hilbert's inequality with some parameters[J]. The Australian Journal of Mathematical Analysis and Applications, 2004, 1(1): 1–8. [5] 杨必成. 算子范数与Hilbert型不等式[M]. 北京: 科学出版社, 2009: 300-307. YANG B C. The Norm of Operator and Hilbert-type Inequalities[M]. Beijing: The Science Press, 2009: 300-307. [6] 杨必成. 关于一个非齐次核的Hilbert型积分算子[J]. 应用泛函分析学报, 2012, 14(1): 84–88. YANG B C. On a Hilbert-type integral operator with the none-homogeneous kernels[J]. Acta Analysis Functionalis Applicata, 2012, 14(1): 84–88. DOI:10.3724/SP.J.1160.2012.00084 [7] 钟玉泉. 复变函数论[M]. 北京: 高等教育出版社, 2003. ZHONG Y Q. Theory of Functions of Complex Variable[M]. Beijing: Higher Education Press, 2003. [8] 匡继昌. 常用不等式[M]. 济南: 山东科学技术出版社, 2004: 4-5. KUANG J C. Applied Inequalities[M]. Jinan: Shandong Science and Technology Press, 2004: 4-5. [9] 匡继昌. 实分析引论[M]. 长沙: 湖南教育出版社, 1996: 45-46. KUANG J C. Real Analysis[M]. Changsha: Hunan Educational Press, 1996: 45-46.