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  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
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引用本文 [复制中英文]

钟建华, 陈强, 曾志红. 一个非单调非齐次核的Hilbert型积分不等式[J]. 浙江大学学报(理学版), 2017, 44(2): 150-153. DOI: 10.3785/j.issn.1008-9497.2017.02.005.
[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]

基金项目

国家自然科学基金资助项目(61370186,61640222);广东省省级科技计划项目(2013A011403002,2014B010116001);广东第二师范学院教授科研专项经费研究项目(2015ARF25)

作者简介

钟建华 (1962-), ORCID:http://orcid.org/0000-0002-6094-7920, 男, 副教授, 主要从事几何与Hilbert型不等式研究

通信作者

陈强, ORCID:http://orcid.org/0000-0001-8010-6398, E-mail:cq_c@gdei.edu.cn

文章历史

收稿日期:2016-04-01
一个非单调非齐次核的Hilbert型积分不等式
钟建华1 , 陈强2 , 曾志红3     
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
摘要: 通过引入参数σ和应用权函数的方法,建立了一个具有最佳常数因子的非单调且非齐次核的Hilbert型积分不等式及其等价形式,并考虑了特殊结果.
关键词: Hilbert型积分不等式    权系数    参数    等价式    非齐次核    
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)

式 (1) 是经典的-1齐次核Hardy-Hilbert积分不等式, 常数因子$ \frac{{\rm{\pi }}}{{{\rm{sin}}({\rm{\pi }}/p)}}$为最佳值, 它在分析学中有很多重要应用[2-3].后来, 一系列具有参量化的结果推广了此不等式.

文献[4]引入了2对共轭指数 (p, q) 与 (r, s), 当λ>0, f, g≥0时, 有如下-λ齐次核的Hilbert型积分不等式:

$ \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)

这里,常数因子$ \frac{{\rm{\pi }}}{{\lambda {\rm{sin}}({\rm{\pi }}/r)}} $为最佳值.由式 (2) 的核可衍生如下具有最佳常数因子的非齐次核的Hilbert型积分不等式[5]:

$ \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)

关于非齐次核的Hilbert型不等式的研究不断推陈出新, 文献[5]得到一个非齐次核的Hilbert型不等式:

$ \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)

其中,常数因子$ 2B\left( {1-\lambda, \frac{\lambda }{2}} \right)\left( {0 < \lambda < 1} \right) $为最佳值.

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:

给出一对共轭指数 (p, q), 且p>1, 参数λR, h(u) 是 (0, ∞) 上的非负可测函数, $ {k_\lambda } = \int_{_0}^\infty h \left( u \right){u^{\lambda /2-1}}{\rm{d}}u \in {{\bf{R}}_ + } $.若f, g≥0, 则当

$ 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 $

时, 有

$ \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)

其中, 常数kλ是最佳值.特别地,设$ h\left( u \right) = \frac{1}{{|1-u{|^\lambda }}}\left( {0 < \lambda < 1} \right) $, 可得${k_\lambda } = \int_{_0}^\infty h \frac{1}{{|1-u{|^\lambda }}}{u^{\lambda /2-1}}{\rm{d}}u = 2B\left( {1-\lambda, \frac{\lambda }{2}} \right) $, 式 (5) 则变成式 (4).

本文应用一个非单调双曲正割函数[7]:

$ 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)

其所确定的非齐次核:

$ 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}}}}. $

受式 (3)~(5) 的研究思路启发, 在此引入参数σ>0, 应用权系数及实分析方法, 得到了一个具有最佳常数因子的Hilbert型不等式和等价式, 并考虑了一些特殊结果.

1 引理

引理1 若σ>0, 且h(t) 如式 (6) 所定义, 定义如下权系数:

$ {\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)

这里, $ \Gamma \left( \sigma \right) = \smallint _{_0}^{^\infty }{{\rm{e}}^{-u}}{u^{\sigma-1}}{\rm{d}}u $是Γ函数[5], 且

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

证明 对式 (7) 做u=xy变换,则有

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

利用公式$\frac{1}{{1 + {x^2}}} = \sum\limits_{k = 0}^\infty {{{\left( {-1} \right)}^k}} {x^{2k}}\left( {0 < x < 1} \right) $, 则有

$ \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} $

将其代入式 (9), 并令t=(2k+3)u, 得

$ \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} $

在上式中自然有

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

则有ωσ(y)=Γ(σ)η(σ).即式 (8) 成立.证毕.

引理2 若p>1, $\frac{1}{p} + \frac{1}{q} = 1 $, 其他条件如引理1所设, f(x) 在 (0, ∞) 上非负可测, 则有

$ \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).

证明配方并由带权的Hölder不等式[8]及式 (7), 有

$ \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)

由式 (11)、Fubini定理[9]及式 (7) 和 (8), 有

$ \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} $

即式 (10) 成立.证毕.

2 主要结果

定理1 设p>1, $ \frac{1}{p} + \frac{1}{q} = 1 $, σ>0, 且h(t) 如式 (6) 所定义, f(x), g(x)≥0, k(σ) 如式 (8) 所定义, 则当

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

时,有如下等价式:

$ \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)

其中, k(σ) 及kp(σ) 均为最佳值.

证明 当$ 0 < \smallint _{_0}^{^\infty }{x^{p\left( {1-\sigma } \right)-1}}{f^p}\left( x \right){\rm{d}}x < \infty $时, 假设存在y>0使式 (10) 取等号, 则必有不全为零的常数A, B, 使得[7]

$ \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). $

不妨设A≠0(否则, A=B=0), 则有

$ {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). $

由于$ \mathop {{\rm{lim}}}\limits_{\varepsilon \to {0^ + }} {\rm{ln}}\;\varepsilon =-\infty $, 因此

$ \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 , $

这与$ 0 < \smallint _{_0}^{^\infty }{x^{p\left( {1-\sigma } \right)-1}}{f^p}\left( x \right){\rm{d}}x < \infty $矛盾, 故式 (11) 严格取不等号, 因此, 式 (10) 亦严格取不等号, 得到式 (13) 成立.

通过配方,并由Hölder不等式[8], 有

$ \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)

再由式 (13), 得到式 (12) 成立.反之, 设式 (12) 成立, 取$g\left( y \right): = {y^{p\sigma- 1}}{\left[{\smallint _{_0}^{^\infty }h\left( {xy} \right)f\left( x \right){\rm{d}}x} \right]^{p -1}} $, 则有$ J = \smallint _{_0}^{^\infty }{y^{q\left( {1-\sigma } \right)-1}}{g^q}\left( y \right){\rm{d}}y $.由式 (10), J<∞.若J=0, 则式 (13) 自然成立; 设0<J<∞, 则由式 (12), 有

$ \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)

对式 (16) 两边p次方, 可得式 (13), 且式 (13) 与式 (12) 等价.

任给ε>0, 定义$ \tilde f(x) $, $ \tilde g(y) $如下:

$ \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. $

则可算得

$ \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} $

由Fubini定理[9], 并对下式中的内积分做u=xy变换, 可得

$ \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} $

假设有正数kk(σ), 使其取代式 (12) 的常数因子k(σ) 后仍能成立, 则由符合定理条件的$ \tilde f(x) $, $ \tilde g(y) $, 计算得

$ \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)

运用Fatou引理[9]及式 (17), 有

$ \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} $

这与假设矛盾, 故k=k(σ) 必为式 (12) 的最佳值.式 (13) 的常数因子kp(σ) 必为最佳值, 否则, 由式 (14), 必导出式 (12) 的常数因子非最佳值的矛盾结论.证毕.

评注 在定理1的条件下, 当σ=12时, 式 (12) 和 (13) 变为如下具有最佳常数因子的等价式:

$ \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)

其中, 常数因子

$ 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} . $

式 (12) 和 (13) 变为如下具有最佳常数因子的等价不等式:

$ \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)
参考文献
[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.