Araştırma Makalesi
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Yıl 2025, Cilt: 8 Sayı: 1, 21 - 29, 25.03.2025

Christophe Chesneau

https://doi.org/10.32323/ujma.1590154

Öz

Kaynakça

  • [1] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
  • [2] E. F. Beckenbach, E. Bellman, Inequalities, Springer, Berlin, 1961.
  • [3] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970.
  • [4] D. Bainov, P. Simeonov, Integral Inequalities and Applications, Mathematics and Its Applications, vol. 57. Kluwer Academic, Dordrecht, 1992.
  • [5] B.C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates, 2009.
  • [6] A. Kufner, L. E. Persson, N. Samko, Weighted Inequalities of Hardy Type, Second Edition, World Scientific, 2017.
  • [7] B. Opic, A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, Essex, 1990.
  • [8] G. H. Hardy, Note on a Theorem of Hilbert, Math. Z., 6, (1920), 314-317.
  • [9] N. Levinson, Generalizations of an inequality of Hardy, Duke Math. J., 31 (1964), 389-394.
  • [10] H. Brezis, M. Marcus, Hardy’s inequalities revisited, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217-237.
  • [11] H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: The sub-critical case, Math. Nachr., 208 (1999), 167-178.
  • [12] S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.
  • [13] F. Gazzola, H. C. Grunau, E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Am. Math. Soc., 356 (2004), 2149-2168.
  • [14] H. Bahouri, J. Y. Chemin, I. Gallagher, Refined Hardy inequalities, Ann. Sc. Norm. Super. Pisa, 5 (2006), 375-391.
  • [15] R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.
  • [16] W. T. Sulaiman, Some Hardy type integral inequalities, Appl. Math. Lett., 25 (2012), 520-525.
  • [17] S. Machihara, T. Ozawa, H. Wadade, Hardy type inequalities on balls, Tohoku Math. J., 65 (2013), 321-330.
  • [18] B. Sroysang, More on some Hardy type integral inequalities, J. Math. Inequalities, 8 (2014), 497-501.
  • [19] B. Dyda, A. V. V¨ah¨akangas, Characterizations for fractional Hardy inequality, Adv. Calc. Var., 8 (2015), 173-182.
  • [20] S. Wu, B. Sroysang, S. Li, A further generalization of certain integral inequalities similar to Hardy’s inequality, J. Nonlinear Sci. Appl., 9 (2016), 1093-1102.
  • [21] K. Mehrez, Some generalizations and refined Hardy type integral inequalities, Afr. Mat., 28 (2016), 451-457.
  • [22] B. Devyver, Y. Pinchover, G. Psaradakis, Optimal Hardy inequalities in cones, Proc. R. Soc. Edinb. A, 147 (2017), 89-124.
  • [23] P. Mironescu, The role of the Hardy type inequalities in the theory of function spaces, Rev. Roum. Math. Pures Appl., 63 (2018), 447-525.
  • [24] B. Benaissa, M. Sarikaya, A. Senouci, On some new Hardy-type inequalities, Math. Methods Appl. Sci., 43 (2020), 8488-8495.
  • [25] S. Yin, Y. Ren, C. Liu, A sharp Lp-Hardy type inequality on the n-sphere, ScienceAsia, 46 (2020), 746-752.
  • [26] S. Thongjob, K. Nonlaopon, J. Tariboon, S. Ntouyas, Generalizations of some integral inequalities related to Hardy type integral inequalities via (p;q)-calculus, J. Inequalities Appl., 2021 (2021), 1-17.
  • [27] H. M. Rezk, M. E. Bakr, O. Balogun, A. Saied, Exploring generalized Hardy-type inequalities via nabla calculus on time scales, Symmetry, 15 (2023), 1-17.
  • [28] M. Aldovardi, J. Bellazzini, A note on the fractional Hardy inequality, Bolletino Unione Mat. Ital., 16 (2023), 667-676.
  • [29] H. M. Rezk, O. Balogun, M. Bakr, Unified generalizations of Hardy-type inequalities through the nabla framework on time scales, Axioms, 13 (2024), 1-16.
  • [30] E. M. Stein, R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, 2011.

Contributions to the Fractional Hardy Integral Inequality

Yıl 2025, Cilt: 8 Sayı: 1, 21 - 29, 25.03.2025

Christophe Chesneau

https://doi.org/10.32323/ujma.1590154

Öz

This article makes three contributions to the fractional Hardy integral inequality. First, we refine an existing result in the literature by improving the main constant and relaxing some assumptions on the parameters. We then propose a fractional-type Hardy integral inequality for an under-studied case, with a significant adaptation of the existing general proof. Finally, a version of this result is established when the integral domain is finite. The proofs are given in detail, with the exact expression of the constants involved at each step. We also mention that almost no intermediate results are used.

Anahtar Kelimeler

Fractional Hardy integral inequality Fubini-Tonelli integral theorem Hardy integral inequality Integral inequalities

Kaynakça

  • [1] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
  • [2] E. F. Beckenbach, E. Bellman, Inequalities, Springer, Berlin, 1961.
  • [3] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970.
  • [4] D. Bainov, P. Simeonov, Integral Inequalities and Applications, Mathematics and Its Applications, vol. 57. Kluwer Academic, Dordrecht, 1992.
  • [5] B.C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates, 2009.
  • [6] A. Kufner, L. E. Persson, N. Samko, Weighted Inequalities of Hardy Type, Second Edition, World Scientific, 2017.
  • [7] B. Opic, A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, Essex, 1990.
  • [8] G. H. Hardy, Note on a Theorem of Hilbert, Math. Z., 6, (1920), 314-317.
  • [9] N. Levinson, Generalizations of an inequality of Hardy, Duke Math. J., 31 (1964), 389-394.
  • [10] H. Brezis, M. Marcus, Hardy’s inequalities revisited, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217-237.
  • [11] H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: The sub-critical case, Math. Nachr., 208 (1999), 167-178.
  • [12] S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.
  • [13] F. Gazzola, H. C. Grunau, E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Am. Math. Soc., 356 (2004), 2149-2168.
  • [14] H. Bahouri, J. Y. Chemin, I. Gallagher, Refined Hardy inequalities, Ann. Sc. Norm. Super. Pisa, 5 (2006), 375-391.
  • [15] R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.
  • [16] W. T. Sulaiman, Some Hardy type integral inequalities, Appl. Math. Lett., 25 (2012), 520-525.
  • [17] S. Machihara, T. Ozawa, H. Wadade, Hardy type inequalities on balls, Tohoku Math. J., 65 (2013), 321-330.
  • [18] B. Sroysang, More on some Hardy type integral inequalities, J. Math. Inequalities, 8 (2014), 497-501.
  • [19] B. Dyda, A. V. V¨ah¨akangas, Characterizations for fractional Hardy inequality, Adv. Calc. Var., 8 (2015), 173-182.
  • [20] S. Wu, B. Sroysang, S. Li, A further generalization of certain integral inequalities similar to Hardy’s inequality, J. Nonlinear Sci. Appl., 9 (2016), 1093-1102.
  • [21] K. Mehrez, Some generalizations and refined Hardy type integral inequalities, Afr. Mat., 28 (2016), 451-457.
  • [22] B. Devyver, Y. Pinchover, G. Psaradakis, Optimal Hardy inequalities in cones, Proc. R. Soc. Edinb. A, 147 (2017), 89-124.
  • [23] P. Mironescu, The role of the Hardy type inequalities in the theory of function spaces, Rev. Roum. Math. Pures Appl., 63 (2018), 447-525.
  • [24] B. Benaissa, M. Sarikaya, A. Senouci, On some new Hardy-type inequalities, Math. Methods Appl. Sci., 43 (2020), 8488-8495.
  • [25] S. Yin, Y. Ren, C. Liu, A sharp Lp-Hardy type inequality on the n-sphere, ScienceAsia, 46 (2020), 746-752.
  • [26] S. Thongjob, K. Nonlaopon, J. Tariboon, S. Ntouyas, Generalizations of some integral inequalities related to Hardy type integral inequalities via (p;q)-calculus, J. Inequalities Appl., 2021 (2021), 1-17.
  • [27] H. M. Rezk, M. E. Bakr, O. Balogun, A. Saied, Exploring generalized Hardy-type inequalities via nabla calculus on time scales, Symmetry, 15 (2023), 1-17.
  • [28] M. Aldovardi, J. Bellazzini, A note on the fractional Hardy inequality, Bolletino Unione Mat. Ital., 16 (2023), 667-676.
  • [29] H. M. Rezk, O. Balogun, M. Bakr, Unified generalizations of Hardy-type inequalities through the nabla framework on time scales, Axioms, 13 (2024), 1-16.
  • [30] E. M. Stein, R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, 2011.
Toplam 30 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Temel Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Christophe Chesneau 0000-0002-1522-9292

Erken Görünüm Tarihi 23 Şubat 2025
Yayımlanma Tarihi 25 Mart 2025
Gönderilme Tarihi 23 Kasım 2024
Kabul Tarihi 20 Şubat 2025
Yayımlandığı Sayı Yıl 2025 Cilt: 8 Sayı: 1

Kaynak Göster

APA Chesneau, C. (2025). Contributions to the Fractional Hardy Integral Inequality. Universal Journal of Mathematics and Applications, 8(1), 21-29. https://doi.org/10.32323/ujma.1590154
AMA Chesneau C. Contributions to the Fractional Hardy Integral Inequality. Univ. J. Math. Appl. Mart 2025;8(1):21-29. doi:10.32323/ujma.1590154
Chicago Chesneau, Christophe. “Contributions to the Fractional Hardy Integral Inequality”. Universal Journal of Mathematics and Applications 8, sy. 1 (Mart 2025): 21-29. https://doi.org/10.32323/ujma.1590154.
EndNote Chesneau C (01 Mart 2025) Contributions to the Fractional Hardy Integral Inequality. Universal Journal of Mathematics and Applications 8 1 21–29.
IEEE C. Chesneau, “Contributions to the Fractional Hardy Integral Inequality”, Univ. J. Math. Appl., c. 8, sy. 1, ss. 21–29, 2025, doi: 10.32323/ujma.1590154.
ISNAD Chesneau, Christophe. “Contributions to the Fractional Hardy Integral Inequality”. Universal Journal of Mathematics and Applications 8/1 (Mart 2025), 21-29. https://doi.org/10.32323/ujma.1590154.
JAMA Chesneau C. Contributions to the Fractional Hardy Integral Inequality. Univ. J. Math. Appl. 2025;8:21–29.
MLA Chesneau, Christophe. “Contributions to the Fractional Hardy Integral Inequality”. Universal Journal of Mathematics and Applications, c. 8, sy. 1, 2025, ss. 21-29, doi:10.32323/ujma.1590154.
Vancouver Chesneau C. Contributions to the Fractional Hardy Integral Inequality. Univ. J. Math. Appl. 2025;8(1):21-9.
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