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Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2026 / Volume: 71 Issue: 2

The complementary nabla Bennett-Leindler type inequalities

Zeynep Kayar(VAN YUZUNCU YIL UNIVERSITY)0000-0002-8309-7930
Billur Kaymakçalan(CANKAYA UNIVERSITY)0000-0002-0763-9128
Pages
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Publication Date
2022-06-30
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Some of our results are novel even in the special cases.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calculus by changing the exponent from $0<\zeta< 1$ to $\zeta>1.$ Different from the literature, the directions of the new inequalities, where $\zeta>1,$ are the same as that of the previous nabla Bennett-Leindler type inequalities obtained for $0<\zeta< 1$.
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Key Findings & Data:By these settings, we not only complement existing nabla Bennett-Leindler type inequalities but also generalize them by involving more exponents.
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Conclusion & Impact:Some of our results are novel even in the special cases.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Time scale calculus · Hardy's inequality · Bennett's inequality

Abstract

We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calculus by changing the exponent from $0<\zeta< 1$ to $\zeta>1.$ Different from the literature, the directions of the new inequalities, where $\zeta>1,$ are the same as that of the previous nabla Bennett-Leindler type inequalities obtained for $0<\zeta< 1$. By these settings, we not only complement existing nabla Bennett-Leindler type inequalities but also generalize them by involving more exponents. The dual results for the delta approach and the special cases for the discrete and continuous ones are obtained as well. Some of our results are novel even in the special cases.

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Cite This Article

Kayar, Z., Kaymakçalan, B. (2022). The complementary nabla Bennett-Leindler type inequalities. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 71 Issue: 2. https://doi.org/10.31801/cfsuasmas.930138

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References (55)

  1. Agarwal, R., Bohner, M., Peterson, A., Inequalities on time scales: A survey, Math. Inequal. Appl. 4(4) (2001), 535–557. https://doi.org/dx.doi.org/10.7153/mia-04-48
  2. Agarwal, R. P., Mahmoud, R. R., Saker, S., Tun¸c, C., New generalizations of N´emeth-Mohapatra type inequalities on time scales, Acta Math. Hungar. 152(2) (2017), 383-403. https://doi.org/10.1007/s10474-017-0718-2
  3. Agarwal, R., O’Regan, D. and Saker, S., Dynamic Inequalities on Time Scales, Springer, Cham, 2014. https://doi.org/10.1007/978-3-319-11002-8
  4. Agarwal, R., O’Regan, D., Saker, S., Hardy Type Inequalities on Time Scales, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-44299-0
  5. Anderson, D. R., Time-scale integral inequalities, J. Inequal. Pure Appl. Math., 6(3) Article 66 (2005), 1-15.
  6. Atici, F. M., Guseinov, G. S., On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141(1-2) (2002), 75-99. https://doi.org/10.1016/S0377-0427(01)00437-X
  7. Balinsky, A. A., Evans, W. D., Lewis, R. T., The Analysis and Geometry of Hardy’s Inequality, Springer International Publishing, Switzerland, 2015. https://doi.org/10.1007/978-3-319-22870-9
  8. Beesack, P. R., Hardy’s inequality and its extensions, Pacific J. Math., 11(1) (1961), 39-61. http://projecteuclid.org/euclid.pjm/1103037533
  9. Bennett, G., Some elementary inequalities, Quart. J. Math. Oxford Ser. (2), 38(152) (1987), 401-425. https://doi.org/10.1093/qmath/38.4.401
  10. Bennett, G., Some elementary inequalities II., Quart. J. Math., 39(4) (1988), 385-400. https://doi.org/10.1093/qmath/39.4.385
  11. Bohner, M., Mahmoud, R., Saker, S. H, Discrete, continuous, delta, nabla, and diamond-alpha Opial inequalities, Math. Inequal. Appl., 18(3) (2015), 923-940. https://doi.org/10.7153/mia-18-69
  12. Bohner, M., Peterson, A., Dynamic Equations on Time Scales, An Introduction With Applications, Birkhauser Boston, Inc., Boston, MA, 2001. https://doi.org/10.1007/978-1-4612-0201-1
  13. Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser Boston, Inc., Boston, MA, 2003. https://doi.org/10.1007/978-0-8176-8230-9
  14. Chu, Y.-M., Xu, Q., Zhang, X.-M., A note on Hardy’s inequality, J. Inequal. Appl., 2014(271) (2014), 1-10. https://doi.org/10.1186/1029-242X-2014-271
  15. Copson, E. T., Note on series of positive terms, J. London Math. Soc., 3(1) (1928), 49-51. https://doi.org/10.1112/jlms/s1-3.1.49
  16. Copson, E. T., Some integral inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 75(2) (1976), 157-164. https://doi.org/10.1017/S0308210500017868
  17. El-Deeb, A. A., Elsennary, H. A., Khan, Z. A., Some reverse inequalities of Hardy type on time scales, Adv. Difference Equ., 2020(402) (2020), 1-18. https://doi.org/10.1186/s13662-020-02857-w
  18. El-Deeb, A. A., Elsennary, H. A., Dumitru, B., Some new Hardy-type inequalities on time scales, Adv. Difference Equ., 2020(441) (2020), 1-22. https://doi.org/10.1186/s13662-020-02883-8
  19. Gao, P., Zhao, H. Y., On Copson’s inequalities for 0 < p < 1, J. Inequal. Appl., 2020(72) (2020), 1-13. https://doi.org/10.1186/s13660-020-02339-3
  20. Guseinov, G. S., Kaymak¸calan, B., Basics of Riemann delta and nabla integration on time scales, J. Difference Equ. Appl., 8(11) (2002), 1001-1017. https://doi.org/10.1080/10236190290015272
  21. Gürses, M., Guseinov, G. S., Silindir, B., Integrable equations on time scales, J. Math. Phys., 46(11) 113510 (2005), 1-22. https://doi.org/10.1063/1.2116380
  22. Güvenilir, A. F., Kaymakçalan, B., Pelen, N. N., Constantin’s inequality for nabla and diamond-alpha derivative, J. Inequal. Appl., 2015(167) (2015), 1-17. https://doi.org/10.1186/s13660-015-0681-9
  23. Hardy, G. H., Littlewood, J. E., Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, Journal f¨ur die reine und angewandte Mathematik, 157 (1927), 141-158. https://doi.org/10.1515/crll.1927.157.141
  24. Hardy, G. H., Note on a theorem of Hilbert, Math. Z., 6(3-4) (1920), 314-317. https://doi.org/10.1007/BF01199965
  25. Hardy, G. H., Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math., 54(3) (1925), 150-156.
  26. Hardy, G. H., Littlewood and P´olya, G., Inequalities, Cambridge University Press, London, 1934.
  27. Iddrisu, M. M., Okpoti, A. C., Gbolagade, A. K., Some proofs of the classical integral Hardy inequality, Korean J. Math., 22(3) 2014, 407-417. https://doi.org/10.11568/kjm.2014.22.3.407
  28. Kayar, Z., Kaymakçalan, B., Pelen, N. N., Bennett-Leindler type inequalities for time scale nabla calculus, Mediterr. J. Math., 18(14) (2021). https://doi.org/10.1007/s00009-020-01674-5
  29. Kayar, Z., Kaymakçalan, B., Hardy-Copson type inequalities for nabla time scale calculus, Turk. J. Math., 45(2) (2021), 1040-1064. https://doi.org/10.3906/mat-2011-38
  30. Kayar, Z., Kaymakçalan, B., Some extended nabla and delta Hardy-Copson type inequalities with applications in oscillation theory, Bull. Iran. Math. Soc., Accepted. https://doi.org/10.1007/s41980-021-00651-2.
  31. Kayar, Z., Kaymakçalan, B., Complements of nabla and delta Hardy-Copson type inequalities and their applications, Submitted.
  32. Kufner, A., Maligranda, L., Persson, L. E., The Hardy Inequality. About Its History and Some Related Results, Vydavatelsk´y Servis, Pilsen, 2007.
  33. Kufner, A., Persson, L. E., Samko, N., Weighted Inequalities of Hardy Type, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. https://doi.org/10.1142/10052
  34. Lefevre, P., A short direct proof of the discrete Hardy inequality, Arch. Math. (Basel)., 114(2) (2020), 195-198. https://doi.org/10.1007/s00013-019-01395-6
  35. Leindler, L., Some inequalities pertaining to Bennett’s results, Acta Sci. Math. (Szeged)., 58(1-4) (1993), 261-279.
  36. Leindler, L., Further sharpening of inequalities of Hardy and Littlewood, Acta Sci. Math., 54(3–4) (1990), 285–289.
  37. Liao, Z.-W., Discrete Hardy-type inequalities, Adv. Nonlinear Stud., 15(4) (2015), 805-834. https://doi.org/10.1515/ans-2015-0404
  38. Masmoudi, N., About the Hardy Inequality, in: An Invitation to Mathematics. From Competitions to Research, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-19533-4_11
  39. Nikolidakis, E. N., A sharp integral Hardy type inequality and applications to Muckenhoupt weights on R, Ann. Acad. Sci. Fenn. Math., 39(2) (2014), 887-896. https://doi.org/10.5186/aasfm.2014.3947
  40. Özkan, U. M., Sarikaya, M. Z., Yildirim, H., Extensions of certain integral inequalities on time scales, Appl. Math. Lett., 21(10) (2008), 993-1000. https://doi.org/10.1016/j.aml.2007.06.008
  41. Pachpatte, B. G., On Some Generalizations of Hardy’s Integral Inequality, J. Math. Anal. Appl., 234(1) (1999), 15-30. https://doi.org/10.1006/jmaa.1999.6294
  42. Pecaric, J., Hanjs, Z., On some generalizations of inequalities given by B. G. Pachpatte, An. Şttiint¸. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 45(1) (1999), 103-114.
  43. Pelen, N. N., Hardy-Sobolev-Mazya inequality for nabla time scale calculus, Eskisehir Technical University Journal of Science and Technology B - Theoretical Sciences, 7(2) (2019), 133-145. https://doi.org/10.20290/estubtdb.609525
  44. Rehak, P., Hardy inequality on time scales and its application to half-linear dynamic equations, J. Inequal. Appl., 2005(5) (2005), 495–507. https://doi.org/10.1155/JIA.2005.495
  45. Renaud, P., A reversed Hardy inequality, Bull. Austral. Math. Soc., 34 (1986), 225-232. https://doi.org/10.1017/S0004972700010091
  46. Saker, S. H., Dynamic inequalities on time scales: A survey, J. Fractional Calc. & Appl., 3(S)(2) (2012), 1-36.
  47. Saker, S. H., Hardy–Leindler Type Inequalities on Time Scales, Appl. Math. Inf. Sci., 8(6) (2014), 2975-2981. https://doi.org/10.12785/amis/080635
  48. Saker, S. H., Mahmoud, R. R., A connection between weighted Hardy’s inequality and half-linear dynamic equations, Adv. Difference Equ., 2014(129) (2019), 1-15. https://doi.org/10.1186/s13662-019-2072-x
  49. Saker, S. H., Mahmoud, R. R., Peterson, A., Some Bennett-Copson type inequalities on time scales, J. Math. Inequal., 10(2) (2016), 471-489. https://doi.org/10.7153/jmi-10-37
  50. Saker, S. H., Mahmoud, R. R., Osman, M. M., Agarwal, R. P., Some new generalized forms of Hardy’s type inequality on time scales, Math. Inequal. Appl., 20(2) (2017), 459-481. https://doi.org/10.7153/mia-20-31
  51. Saker, S. H., O’Regan, D., Agarwal, R. P., Dynamic inequalities of Hardy and Copson type on time scales, Analysis, 34(4) (2014), 391-402. https://doi.org/10.1515/anly-2012-1234
  52. Saker, S. H., O’Regan, D., Agarwal, R. P., Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales, Math. Nachr., 287(5-6) (2014), 686-698. https://doi.org/10.1002/mana.201300010
  53. Saker, S. H., Osman, M. M., O’Regan, D., Agarwal, R. P., Inequalities of Hardy type and generalizations on time scales, Analysis, 38(1) (2018), 47–62. https://doi.org/10.1515/anly-2017-0006
  54. Saker, S. H., Mahmoud, R. R., Peterson, A., A unified approach to Copson and Beesack type inequalities on time scales, Math. Inequal. Appl., 21(4) (2018), 985-1002. https://doi.org/10.7153/mia-2018-21-67
  55. Saker, S. H., O’Regan, D., Agarwal, R. P., Converses of Copson’s inequalities on time scales, Math. Inequal. Appl., 18(1) (2015), 241-254. https://doi.org/10.7153/mia-18-18