一个涉及变上限积分函数和高阶导数的 Hilbert 型积分不等式的构建条件及其应用

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1424-1431.

一个涉及变上限积分函数和高阶导数的 Hilbert 型积分不等式的构建条件及其应用

洪勇^1,²(),张丽娟^1,^*

¹广州华商学院人工智能学院广州 511300
²广东财经大学统计与数学学院广州 510320

收稿日期:2025-01-24 修回日期:2025-05-12 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 张丽娟
作者简介:洪勇,E-mail:hongyongdcc@yeah.net
基金资助:
广东省基础与应用基础研究基金(2022A1515012429);广州华商学院特色科研项目(2024HSTS08)

Conditions for the Construction of a Hilbert-Type Integral Inequality Involving Variable Upper Limit Integral Function and Higher Order Derivative and Its Applications

Yong Hong^1,²(),Lijuan Zhang^1,^*

¹Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300
²College of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320

Received:2025-01-24 Revised:2025-05-12 Online:2025-10-26 Published:2025-10-14
Supported by:
Guangdong Basic and Applied Basic Research Fund(2022A1515012429);Guangzhou Huashang College Featured Research Programs(2024HSTS08)

摘要/Abstract

摘要：

利用齐次核 Hilbert 型积分不等式的构造定理, 讨论一个涉及变上限积分函数和高阶导数的 Hilbert 型积分不等式, 得到构建该不等式的充分必要条件及最佳常数因子的表达公式, 推广和改进了现有的结果. 最后, 利用所得 Hilbert 型不等式, 讨论相关算子的有界性与算子范数.

关键词: 变上限积分函数, 高阶导数, Hilbert 型积分不等式, 有界积分算子, 算子范数, Gamma 函数

Abstract:

Using the construction theorem of the homogeneous kernel Hilbert-type integral inequality, a Hilbert-type integral inequality involving a variable upper limit integral function and a higher order derivative is discussed, and sufficient necessary conditions for constructing this inequality and an expression for the optimal constant factor are obtained, which generalizes and improves the existing results. Finally, the resulting Hilbert-type inequality is utilized to discuss the problems of boundedness and operator norm for the relevant integral operator.

Key words: variable upper limit integral function, higher-order derivative, Hilbert-type integral inequality, bounded integral operator, operator norm, Gamma function

中图分类号:

O178

洪勇, 张丽娟. 一个涉及变上限积分函数和高阶导数的 Hilbert 型积分不等式的构建条件及其应用[J]. 数学物理学报, 2025, 45(5): 1424-1431.

Yong Hong, Lijuan Zhang. Conditions for the Construction of a Hilbert-Type Integral Inequality Involving Variable Upper Limit Integral Function and Higher Order Derivative and Its Applications[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1424-1431.

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