Asia Mathematika, Vol 9, issue 3, pages: 12--19.

Two contributions to the absolute-value-Hardy-Hilbert-type integral inequalities

Christophe Chesneau
Department of Mathematics, LMNO, University of Caen-Normandie, Caen, France.

Received: 04 Sep 2025 | Accepted: 02 Dec 2025 | Final Version: 31 Dec 2025

Abstract

This article presents two new Hardy-Hilbert-type integral inequalities involving the absolute value function. The first result can be viewed as an analog of a well-known inequality of this type, while the second is distinguished by its originality, incorporating the sine function into the integrand. Complete and detailed proofs are provided for both results.

Keywords:
Hardy-Hilbert integral inequality, absolute value function, trigonometric functions, change of variables technique.

1. Introduction

The Hardy-Hilbert integral inequality is one of the most celebrated results in mathematical analysis. Its precise formulation is given below. Let \(p>1, q=p/(p-1)\) such that \(1/p+1/q=1\), and \(f,g:(0,+\infty)\to (0,+\infty)\) be two (measurable) functions such that \(\int_{0}^{+\infty}f^p(x)dx < +\infty, \int_{0}^{+\infty}g^q(x)dx < +\infty.\) Then we have \begin{equation} \int_{0}^{+\infty}\int_{0}^{+\infty} \frac{1}{x+y}f(x)g(y)dxdy\le \frac{\pi}{\sin(\pi/p)}\left[\int_{0}^{+\infty}f^p(x)dx\right]^{1/p}\left[\int_{0}^{+\infty}g^q(x)dx\right]^{1/q}... \end{equation}

References

  1. Adiyasuren V, Batbold T, Krnić M. Hilbert-type inequalities involving differential operators, the best constants and applications. Math Inequal Appl 2015; 18: 111–124.
  2. Adiyasuren V, Batbold T, Krnić M. Multiple Hilbert-type inequalities involving some differential operators. Banach J Math Anal 2016; 10: 320–337.
  3. Azar LE. The connection between Hilbert and Hardy inequalities. J Inequal Appl 2013; 2013: 1–10.
  4. Batbold T, Sawano Y. Sharp bounds for m-linear Hilbert-type operators on the weighted Morrey spaces. Math Inequal Appl 2017; 20: 263–283.
  5. Bényi A, Oh CT. Best constant for certain multilinear integral operator. J Inequal Appl 2006; 2006: 1–12.
  6. Chen Q, Yang BC. A survey on the study of Hilbert-type inequalities. J Inequal Appl 2015; 2015: 1–29.
  7. Chesneau C. Some four-parameter trigonometric generalizations of the Hilbert integral inequality. Asia Math 2024; 8: 45–59.
  8. Chesneau C. A new two-parameter optimal integral inequality of the Hardy-Hilbert type. Asia Math 2025; 9: 21–35.
  9. Chesneau C. Statement and proof of a new general Hardy-Hilbert-type integral inequality theorem. Asia Math 2025; 9: 51–62.
  10. Du H, Miao Y. Several new Hardy-Hilbert's inequalities. Filomat 2011; 25: 153–162.
  11. Hardy GH. Note on a theorem of Hilbert concerning series of positive terms. Proc London Math Soc 1925; 23(2): 45–46.
  12. Hardy GH, Littlewood JE, Polya G. Inequalities. Cambridge, UK: Cambridge University Press, 1934.
  13. Hong Y. On multiple Hardy-Hilbert integral inequalities with some parameters. J Inequal Appl 2006; 2006: 1–11.
  14. Huang Z, Yang BC. A multidimensional Hilbert-type integral inequality. J Inequal Appl 2015; 2015: 1–13.
  15. Jian SW, Yang FZ. All-sided generalization about Hardy-Hilbert integral inequalities. Acta Math Sinica (China) 2001; 44: 619–626.
  16. Li Y, Qian Y, He B. On further analogs of Hilbert's inequality. Int J Math Math Sci 2007; 2007: 1–6.
  17. Sulaiman WT. On Hardy-Hilbert's integral inequality. J Inequal Pure Appl Math 2004; 5: 1–9.
  18. Sulaiman WT. On three inequalities similar to Hardy-Hilbert's integral inequality. Acta Math Univ Comenianae 2007; 2: 273–278.
  19. Sulaiman WT. Hardy-Hilbert's integral inequalities via homogeneous functions and some other generalizations. Acta Comment Univ Tartu Math 2007; 11: 23–32.
  20. Sulaiman WT. A different type of Hardy-Hilbert's integral inequality. AIP Conf Proc 2008; 1046: 135–141.
  21. Sulaiman WT. General new forms of Hardy-Hilbert's integral inequality via new ideas. Int J Contemp Math Sci 2008; 3: 1059–1067.
  22. Sulaiman WT. New types of Hardy-Hilbert's integral inequality. Gen Math Notes 2011; 2: 111–118.
  23. Sun B. A multiple Hilbert-type integral inequality with the best constant factor. J Inequal Appl 2007; 2007: 1–14.
  24. Ullrich DC. A simple elementary proof of Hilbert's inequality. Am Math Monthly 2013; 120: 161–164.
  25. Xu JS. Hardy-Hilbert's inequalities with two parameters. Adv Math 2007; 36: 63–76.
  26. Yang BC. On Hilbert's integral inequality. J Math Anal Appl 1998; 220: 778–785.
  27. Yang BC. A multiple Hardy-Hilbert integral inequality. Chin Ann Math Ser A 2003; 24: 743–750.
  28. Yang BC. On the norm of an integral operator and applications. J Math Anal Appl 2006; 321: 182–192.
  29. Yang BC. On the norm of a Hilbert's type linear operator and applications. J Math Anal Appl 2007; 325: 529–541.
  30. Yang BC. The Norm of Operator and Hilbert-Type Inequalities. Beijing, China: Science Press, 2009.
  31. Yang BC. Hilbert-Type Integral Inequalities. Sharjah, UAE: Bentham Science Publishers, 2009.
  32. Yang BC, Krnić M. On the norm of a multi-dimensional Hilbert-type operator. Sarajevo J Math 2011; 7: 223–243.
  33. Zhong WY, Yang BC. On a multiple Hilbert-type integral inequality with the symmetric kernel. J Inequal Appl 2007; 2007: 1–17.