A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters

Christophe Chesneau

doi:10.36753/mathenot.1702817

EN

A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters

Abstract

In this article, we extend the theory of the Hardy-Hilbert integral inequality by developing a novel three-dimensional variant. This variant depends on three norm parameters and five adjustable parameters, including two power parameters that influence the structure of the denominator in a novel way. We determine the exact conditions under which the inequality holds for these parameters, together with the expression of the constant factor in terms of beta functions. Two illustrative examples are presented, and two additional results are derived, including a lower bound. Thus, this work adds to a classical topic in analysis by exploring an original extension in both dimensionality and parametric complexity.

Keywords

Beta function, Convex inequality, Generalized H\"older integral inequality, Hardy-Hilbert integral inequality

References

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Details

Primary Language

English

Subjects

Mathematical Methods and Special Functions

Journal Section

Research Article

Authors

Christophe Chesneau ^*
0000-0002-1522-9292
France

Early Pub Date

August 22, 2025

Publication Date

September 6, 2025

Submission Date

May 20, 2025

Acceptance Date

August 17, 2025

Published in Issue

Year 2025 Volume: 13 Number: 3

DOI

https://doi.org/10.36753/mathenot.1702817

IZ

https://izlik.org/JA72LC89UR

APA

Chesneau, C. (2025). A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters. Mathematical Sciences and Applications E-Notes, 13(3), 165-178. https://doi.org/10.36753/mathenot.1702817

AMA

1.Chesneau C. A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters. Math. Sci. Appl. E-Notes. 2025;13(3):165-178. doi:10.36753/mathenot.1702817

Chicago

Chesneau, Christophe. 2025. “A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions With Five Parameters”. Mathematical Sciences and Applications E-Notes 13 (3): 165-78. https://doi.org/10.36753/mathenot.1702817.

EndNote

Chesneau C (September 1, 2025) A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters. Mathematical Sciences and Applications E-Notes 13 3 165–178.

IEEE

[1]C. Chesneau, “A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters”, Math. Sci. Appl. E-Notes, vol. 13, no. 3, pp. 165–178, Sept. 2025, doi: 10.36753/mathenot.1702817.

ISNAD

Chesneau, Christophe. “A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions With Five Parameters”. Mathematical Sciences and Applications E-Notes 13/3 (September 1, 2025): 165-178. https://doi.org/10.36753/mathenot.1702817.

JAMA

1.Chesneau C. A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters. Math. Sci. Appl. E-Notes. 2025;13:165–178.

MLA

Chesneau, Christophe. “A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions With Five Parameters”. Mathematical Sciences and Applications E-Notes, vol. 13, no. 3, Sept. 2025, pp. 165-78, doi:10.36753/mathenot.1702817.

Vancouver

1.Christophe Chesneau. A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters. Math. Sci. Appl. E-Notes. 2025 Sep. 1;13(3):165-78. doi:10.36753/mathenot.1702817