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

Abstract

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

CLC Number:

O178

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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References

[1]	Hardy G H. Note on a theorem of Hilbert concerning series of positive terms. Proc Math Soc, 1925, 23: 45-46
[2]	Hardy G H, Littlewood J E, Polya G. Inequalities. Cambridge: Cambridge University Press, 1934
[3]	Liu Q. On a mixed kernel Hilbert-type integral inequality and its operator expressions with norm. Mathematical Methods in the Applied Sciences, 2021, 44(1): 593-604
[4]	Huang X, Yang B, Luo R. A new reverse Hardy-Hilbert inequality with the power function as intermediate variables. Journal of Inequalities and Applications, 2022, 2022(1): Article 49
[5]	Krnic M, Pecaric J. Extension of Hilbert's inequality. J Math Anal Appl, 2006, 324(1): 150-160
[6]	Rassias M T, Yang B. On half-discrete Hilbert's inequality. Applied Mathematics and Computation, 2013, 220: 75-93
[7]	Kuang J C. On new extension of Hilbert's integral inequality. J Math Anal Appl, 1999, 235: 608-614
[8]	Rassias M T, Yang B C. A multidimensional half-discrete Hilbert-type inequality And the Riemann zeta function. Applied Mathematics and Computation, 2013, 225: 263-277
[9]	Azar L E. The connection between Hilbert and Hardy inequalities. Journal of Inequalities and Applications, 2013, 2013: Article 452
[10]	He B. A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor. J Math Anal Appl, 2015, 431: 902-990
[11]	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
[11]	Hong Y and Wen Y. 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
[12]	Liu Q. The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications. Appl Math Comput, 2025, 487: 129076
[13]	Rassias M T, Yang B, Raigorodskii A. Equivalent properties of two kinds of Hardy-type integral inequalities. Symmetry, 2021, 13: Article 1006
[14]	Hong Y. On the structure character of Hilbert's type integral inequality with homogeneous kernel and applications. J Jilin Univ Sci Ed, 2017, 55(2): 189-194
[15]	Rassias M, Yang B C. Equivalent properties of a Hilbert-type integral inequality with the best constant factor related the Hurwitz zeta function. Ann Funct Anal, 2018, 9(2): 282-295
[16]	Cao J F, He B, Hong Y and Yang B C. Equivalent condition and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernel. Journal of Inequalities and Applications, 2018, 2018: Article 206
[17]	Wang A ZH, Yang B C, Chen Q. Equivalent properties of a reverse half-discrete Hilbert's inequality. J Inequal Appl, 2019, 279: 1-12
[18]	洪勇, 陈强. 广义齐次核重积分算子最佳搭配参数的等价条件及应用. 中国科学: 数学, 2023, 53(5): 717-728
[18]	Hong Y, Chen Q. Equivalence conditions for the best matching parameters of multiple integral operator with generalized homogeneous kernel and applications. Scienpia Sinica Mathematica, 2023, 53(5): 717-728
[19]	Hong Y, Huang Q L, Yang B C, Liao J Q. The necessary and sufficient conditions for the existence of a kind Hilbert-type multiple integral inequality with the non-homogeneous kernel and its application. J Inequalities Appl, 2017, 2017: Article 316
[20]	洪勇. 具有齐次核的 Hilbert 型积分不等式的构造特征及应用. 吉林大学学报 (理学版), 2017, 55(3): 189-194
[20]	Hong Y. Structural characteristics and applications of Hilbert's type integral inequalities with homogeneous kernel. Journal of Jilin University (Science Edition), 2017, 55(3): 189-194
[21]	洪勇, 和炳. Hilbert 型不等式的理论与应用. 北京: 科学出版社, 2023
[21]	Hong Y, He B. Theory and Applications of Hilbert-Type Inequalities. Beijing: Science Press, 2023
[22]	Zhong J H, Yang B C. On a multiple Hilbert-type integral inequality involving the upper limit functions. Journal of Inequalities and Applications, 2021, 2021: Article 17
[23]	Yang B, Rassias M T. A new Hardy Hilbert-type integral inequality involving one multiple upper limit function and one derivative function of higher order. Axioms, 2023, 12(5): Article 499
[24]	Mo H, Yang B. On a new Hilbert-type integral inequality involving the upper limit functions. Journal of Inequalities and Applications, 2020, 2020: Article 5

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

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