On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral

Abstract

In this paper, we aim to construct $n$ dimensional Jensen, Hardy and Hermite-Hadamard type inequalities for multiple diamond-alpha integral on time scales. The cases of Hardy type inequality with a weighted function and Hermite-Hadamard type inequality with three variables are also considered minutely.

Keywords

• Hardy type inequality
• Hermite-Hadamard type inequality
• diamond-alpha integral
• diamond-alpha derivative
• time scales

References

1. [1] M. Adil Khan, T. Ali, S.S. Dragomir and M. Z. Sarikaya, Hermite-Hadamard type inequalities for conformable fractional integrals, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112 (4), 1033–1048, 2018.
2. [2] M. Adil Khan, Y.-M. Chu, T.U. Khan and J. Khan, Some new inequalities of Hermite- Hadamard type for s-convex functions with applications, Open Math. 15 (1), 1414– 1430, 2017.
3. [3] M. Adil Khan, N. Mohammad, E.R. Nwaeze and Y.-M. Chu, Quantum Hermite- Hadamard inequality by means of a Green function, Adv. Difference Equ. 2020, 99, 2020.
4. [4] R.P. Agarwal, E. Çetin and A. Özbekler, Lyapunov type inequalities for second-order forced dynamic equations with mixed nonlinearities on time scales, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 111 (1), 231–246, 2017.
5. [5] R.P. Agarwal, A. Denk Oğuz and A. Özbekler, Lyapunov-type inequalities for Lidstone boundary value problems on time scales, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2), 98, 2020.
6. [6] S.-P. Bai, F. Qi and S.-H.Wang, Some new integral inequalities of Hermite-Hadamard type for $(\alpha,m;P)$-convex functions on co-ordinates, J. Appl. Anal. Comput. 6 (1), 171–178, 2016.
7. [7] M. Bilal, K.A. Khan, H. Ahmad, et al. Some dynamic inequalities via Diamond integrals for function of several variables, Fractal Fract. 5 (4), 207, 2021.
8. [8] M. Bohner and S.G. Georgiev, Multivariable dynamic calculus on time scales, Springer, 2016.

9. [9] M. Bohner and A. Peterson, Dynamic Equations on time scales, Springer, 2001.
10. [10] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Springer, 2003.
11. [11] Y.-M. Chu, Q. Xu and X.-M. Zhang, A note on Hardy’s inequality, J. Inequal. Appl., 2014, 217, 2014.
12. [12] C. Dinu, Hermite-Hadamard inequality on time scales, J. Inequal. Appl. 2008, 287947, 2008.
13. [13] A.A. El-Deeb, H.A. El-Sennary and P. Agarwal, Some Opial-type inequalities with higher order delta derivatives on time scales, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (1), 29, 2020.
14. [14] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, PhD thesis, Universität Würzburg, 1988.
15. [15] X.-M. Hu, J.-F. Tian, Y.-M. Chu, et al. On Cauchy-Schwarz inequality for N-tuple diamond-alpha integral, J. Inequal. Appl. 2020, 8, 2020.
16. [16] A. Iqbal, M. Adil Khan, S. Ullah, et al. Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Space. 2020, 9845407, 2020.
17. [17] Z.-X. Mao and J.-F. Tian, Delta complete monotonicity and completely monotonic degree on time scales, Bull. Malays. Math. Sci. Soc. 46 (4), 142, 2023.
18. [18] Z.-X. Mao, J.-F. Tian and Y.-R. Zhu, Psi, polygamma functions and Q-complete monotonicity on time scales, J. Appl. Anal. Comput. 13 (3), 1137–1154, 2023.
19. [19] Z.-X. Mao, Y.-R. Zhu, B.-H. Guo, et al. Qi type Diamond-Alpha integral inequalities, Math. 9 (4), 449, 2021.
20. [20] Z.-X. Mao, Y.-R. Zhu,J.-P. Hou, et al. Multiple Diamond-Alpha integral in general form and their properties, applications, Math. 9 (10), 1123, 2021.
21. [21] Z.-X. Mao, Y.-R. Zhu and J.-F. Tian, Higher dimensions Opial diamond-alpha inequalities on time scales, J. Math. Inequal. 15 (3), 1055–1074, 2021.
22. [22] E.R. Nwaeze, Time scale version of the Hermite-Hadamard inequality for functions convex on the coordinates, Adv. Dyn. Syst. Appl. 12 (2), 159–171, 2017.
23. [23] U.M. Özkan and B.Kaymakçalan, Basics of diamond-$\alpha$ partial dynamic calculus on time scales, Math. Comput. Modelling, 50 (9-10), 1253–1261, 2009.
24. [24] U.M. Ozkan and H. Yildirim, Hardy-Knopp-type inequalities on time scales, Dynam. Systems Appl. 17 (3-4), 477–486, 2008.
25. [25] F. Qi, M.A. Latif, W.-H. Li, et al. Some integral inequalities of Hermite-Hadamard type for functions whose n-times derivatives are $(\alpha,m)$-convex, Turk. J. Anal. Number Theory, 2 (4), 140–146, 2014.
26. [26] F. Qi, T.-Y. Zhang and B.-Y. Xi, Hermite-Hadamard-type integral inequalities for functions whose first derivatives are convex, Ukrainian Math. J. 67 (4), 625-640, 2015.
27. [27] F. Qi, P.O. Mohammed, J.C. Yao, et al. Generalized fractional integral inequalities of Hermite-Hadamard type for $(\alpha,m)$-convex functions, J. Inequal. Appl. 2019, 135, 2019.
28. [28] F. Qi and B.-Y. Xi, Some Hermite-Hadamard type inequalities for geometrically quasiconvex functions, Proc. Indian Acad. Sci. Math. Sci. 124 (3), 333–342, 2014.
29. [29] G.-Z. Qin and C. Wang, Lebesgue-Stieltjes combined $\Diamond_\alpha$-measure and integral on time scales, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115 (2), 50, 2021.
30. [30] Q. Sheng, M. Fadag, J. Henderson, et al. An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl. 7 (3), 395–413, 2006.
31. [31] M.R. Sidi Ammi and D.F.M. Torres, Hölder’s and Hardy’s two dimensional Diamondalpha inequalities on time scales, An. Univ. Craiova Ser. Mat. Inform. 37 (1), 1–11, 2010.
32. [32] J.-F. Tian, Triple Diamond-Alpha integral and Hölder-type inequalities, J. Inequal. Appl. 2018, 111, 2018.
33. [33] J.-F. Tian, Y.-R. Zhu and W.-S. Cheung, N-tuple Diamond-Alpha integral and inequalities on time scales, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113 (3), 2189–2200, 2019.
34. [34] B.-C. Yang, On a reverse of a Hardy-Hilbert type inequality, JIPAM. J. Inequal. Pure Appl. Math. 7 (3), 115, 2006.
35. [35] B.-C. Yang, On a general Hardy-Hilbert’s integral inequality with a best value, Chinese Ann. Math. 21A, 401–408, 2000. (In Chinese)
36. [36] B.-C. Yang, B. Ilko, K. Mario, et al. Generalization of Hilbert and Hardy-Hilbert integral inequalities, Math. Inequal. Appl. 8 (2), 259–272, 2005.
37. [37] B.-C. Yang and Z.-H. Zeng, Note on new generalizations of Hardy’s integral inequality, J. Math. Anal. Appl. 217 (6), 321–327, 1998.
38. [38] Y.-R. Zhu, Z.-X. Mao, S.-P. Liu, et al. Oscillation criteria of second-order dynamic equations on time scales, Math. 9 (16), 1867, 2021.
39. [39] Y.-R. Zhu, Z.-X. Mao, J.-F. Tian, et al. Oscillation and nonoscillatory criteria of higher order dynamic equations on time scales, Math. 10 (5), 717, 2022.