拟齐次核的Hilbert型级数不等式的最佳搭配参数条件及应用

数学物理学报 ›› 2022, Vol. 42 ›› Issue (1): 26-34.

拟齐次核的Hilbert型级数不等式的最佳搭配参数条件及应用

洪勇¹(),陈强²()

¹ 广州华商学院应用数学系广州 511300
² 广东第二师范学院计算机学院广州 510303

收稿日期:2020-10-28 出版日期:2022-02-01 发布日期:2022-02-23
作者简介:洪勇, E-mail: hongyonggdcc@yeah.net|陈强, E-mail: cq_c120m3@163.com
基金资助:
国家自然科学基金(61772140)

The Best Matching Parameters Conditions of Hilbert-Type Series Inequality with Quasi-Homogeneous Kernel and Applications

Yong Hong¹(),Qiang Chen²()

¹ Department of Applied Mathematics, Guangzhou Huashang College, Guangzhou 511300
² School of Computer, Guangdong University of Education, Guangzhou 510303

Received:2020-10-28 Online:2022-02-01 Published:2022-02-23
Supported by:
the NSFC(61772140)

摘要/Abstract

摘要：

选择搭配参数 $a，b$ ，利用权函数方法可得Hilbert型级数不等式 $\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}K(m, n)a_{m}b_{n}\leq M(a, b)\|\tilde{a}\|_{p, \alpha(a, b)}\|\tilde{b}\|_{q, \beta(a, b)}.$ 该文讨论 $a，b$ 应如何选取才能使具有拟齐次核的不等式中 $M（a，b）$ 为最佳常数因子的问题，得到了 $a，b$ 为最佳搭配参数的充分必要条件及最佳常数因子的表达式.最后讨论其在求算子范数中的应用.

关键词: Hilbert型级数不等式, 拟齐次核, 最佳常数因子, 最佳搭配参数, 算子范数

Abstract:

Choosing $a, b$ as the matching parameters, we can sue the weight function method to obtain Hilbert-type series inequality $\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}K(m, n)a_{m}b_{n}\leq M(a, b)\|\tilde{a}\|_{p, \alpha(a, b)}\|\tilde{b}\|_{q, \beta(a, b)}.$ in the paper, the problem of how to choose $a, b$ in order to make $M(a, b)$ the best constant factor in inequality with quasi-homogeneous kernels are discussed, necessary and sufficient conditions are obtained for $a, b$ to the best matching parameters, the formula for the best constant factor is obtained. Finally, their applications to solving operator morn are discussed.

Key words: Hilbert-type series inequality, Quasi-homogeneous kernel, The best constant factor, The best matching parameter, Operator norm

中图分类号:

O178

洪勇,陈强. 拟齐次核的Hilbert型级数不等式的最佳搭配参数条件及应用[J]. 数学物理学报, 2022, 42(1): 26-34.

Yong Hong,Qiang Chen. The Best Matching Parameters Conditions of Hilbert-Type Series Inequality with Quasi-Homogeneous Kernel and Applications[J]. Acta mathematica scientia,Series A, 2022, 42(1): 26-34.

参考文献 13

1	Brnetic I , Pecaric J . Generalization of inequality of Hardy-Hilbert's type. Math Inequal Appl, 2004, 7 (2): 217- 225
2	洪勇. 关于零阶齐次核的Hardy-Hilbert型不等式. 浙江大学学(理学版), 2013, 40 (1): 15- 18
	Hong Y . On Hardy-Hilbert type integral inequality with homogeneous kernel of 0-degree. Journal of Zhejiang University (Seience Edition), 2013, 40 (1): 15- 18
3	洪勇, 孔荫莹. 含变量可转移函数核的Hilbert型级数不等式. 数学物理学报, 2014, 34A (3): 708- 715
	Hong Y , Kong Y Y . A Hilbert type series inequality with transferable variable kernel. Acta Mathematica Scientia, 2014, 34A (3): 708- 715
4	Kuang J C . On new extensions of Hilbert's integral inequality. Journal of Mathematical Analysis and Applications, 1999, 235: 608- 614 doi: 10.1006/jmaa.1999.6373
5	刘琼, 刘英迪. 一个混合核Hilbert型积分不等式及其算子范数表达式. 数学物理学报, 2020, 40A (2): 369- 378 doi: 10.3969/j.issn.1003-3998.2020.02.009
	Liu Q , Liu Y D . A Hilbert-type integral inequality with the mixed kernel and operator expression with norm. Acta Mathematica Scientia, 2020, 40A (2): 369- 378 doi: 10.3969/j.issn.1003-3998.2020.02.009
6	Yang B C . On new generalizations of Hilbert's inequality. Journal of Mathematical Analysis and Applications, 2000, 248 (1): 29- 40 doi: 10.1006/jmaa.2000.6860
7	杨必成. 一个新的Hilbert型不等式及其推广. 吉林大学学报(理学版), 2005, 43 (5): 580- 584 doi: 10.3321/j.issn:1671-5489.2005.05.005
	Yang B C . A new Hilbert-type integral inequality and its generalzation. Journal of Jini University(Science Edition), 2005, 43 (5): 580- 584 doi: 10.3321/j.issn:1671-5489.2005.05.005
8	杨必成. 关于一个推广的Hardy-Hilbert积分不等式. 数学年刊, 2002, 23A (2): 247- 254 doi: 10.3321/j.issn:1000-8134.2002.02.014
	Yang B C . On an extension of Hardy-Hilbert's inequalty. Chinese Annale of Mathematics, 2002, 23A (2): 247- 254 doi: 10.3321/j.issn:1000-8134.2002.02.014
9	高明哲. Hardy-Riesz拓广的Hilbert不等式的一个改进. 数学研究与评论, 1994, 14 (2): 255- 259
	Gao M Z . An improvement of Hardy-Riesz's extension of the Hilbert inequality. Journal of Mathematical Riesearch and Exposition, 1994, 14 (2): 255- 259
10	Rassia M T , Yang B C . On a Hilbert-type inequality in the plane related to the extended Riemann zeta function. Complex Analysis and Operator Theory, 2019, 13: 1765- 1782 doi: 10.1007/s11785-018-0830-5
11	Krnić M , Vuković P . On multidimensional version of the Hilbert-type inequality. Analysis Mathematical, 2012, 38: 291- 303 doi: 10.1007/s10476-012-0402-2
12	洪勇. 准齐次核的Hardy-Hilbert型级数不等式. 数学年刊, 2012, 33A (6): 679- 686 doi: 10.3969/j.issn.1000-8314.2012.06.003
	Hong Y . Hardy-Hilbert type series inequality with a quasi-homegeneour kernel. Chinese Annals of Mathematics, 2012, 33A (6): 679- 686 doi: 10.3969/j.issn.1000-8314.2012.06.003
13	洪勇, 温雅敏. 齐次核的Hilbert型级数不等式取最佳常数因子的充要条件. 数学年刊, 2016, 37A (3): 329- 336
	Hong Y , Wen Y M . A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor. Chinese Annals of Mathematics, 2016, 37A (3): 329- 336

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