On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral
Year 2024,
, 667 - 689, 27.06.2024
Zhong-xuan Mao
Wen-bin Zhang
Jing-feng Tian
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.
References
- [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] 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] 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] 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] 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] 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] 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] M. Bohner and S.G. Georgiev, Multivariable dynamic calculus on time scales,
Springer, 2016.
- [9] M. Bohner and A. Peterson, Dynamic Equations on time scales, Springer, 2001.
- [10] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Springer,
2003.
- [11] Y.-M. Chu, Q. Xu and X.-M. Zhang, A note on Hardy’s inequality, J. Inequal. Appl.,
2014, 217, 2014.
- [12] C. Dinu, Hermite-Hadamard inequality on time scales, J. Inequal. Appl. 2008, 287947,
2008.
- [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] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, PhD
thesis, Universität Würzburg, 1988.
- [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] 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] 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] 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] Z.-X. Mao, Y.-R. Zhu, B.-H. Guo, et al. Qi type Diamond-Alpha integral inequalities,
Math. 9 (4), 449, 2021.
- [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] 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] 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] 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] U.M. Ozkan and H. Yildirim, Hardy-Knopp-type inequalities on time scales, Dynam.
Systems Appl. 17 (3-4), 477–486, 2008.
- [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] 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] 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] 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] 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] 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] 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] J.-F. Tian, Triple Diamond-Alpha integral and Hölder-type inequalities, J. Inequal.
Appl. 2018, 111, 2018.
- [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] B.-C. Yang, On a reverse of a Hardy-Hilbert type inequality, JIPAM. J. Inequal. Pure
Appl. Math. 7 (3), 115, 2006.
- [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] 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] 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] 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] 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.
Year 2024,
, 667 - 689, 27.06.2024
Zhong-xuan Mao
Wen-bin Zhang
Jing-feng Tian
References
- [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] 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] 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] 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] 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] 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] 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] M. Bohner and S.G. Georgiev, Multivariable dynamic calculus on time scales,
Springer, 2016.
- [9] M. Bohner and A. Peterson, Dynamic Equations on time scales, Springer, 2001.
- [10] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Springer,
2003.
- [11] Y.-M. Chu, Q. Xu and X.-M. Zhang, A note on Hardy’s inequality, J. Inequal. Appl.,
2014, 217, 2014.
- [12] C. Dinu, Hermite-Hadamard inequality on time scales, J. Inequal. Appl. 2008, 287947,
2008.
- [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] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, PhD
thesis, Universität Würzburg, 1988.
- [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] 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] 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] 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] Z.-X. Mao, Y.-R. Zhu, B.-H. Guo, et al. Qi type Diamond-Alpha integral inequalities,
Math. 9 (4), 449, 2021.
- [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] 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] 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] 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] U.M. Ozkan and H. Yildirim, Hardy-Knopp-type inequalities on time scales, Dynam.
Systems Appl. 17 (3-4), 477–486, 2008.
- [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] 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] 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] 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] 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] 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] 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] J.-F. Tian, Triple Diamond-Alpha integral and Hölder-type inequalities, J. Inequal.
Appl. 2018, 111, 2018.
- [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] B.-C. Yang, On a reverse of a Hardy-Hilbert type inequality, JIPAM. J. Inequal. Pure
Appl. Math. 7 (3), 115, 2006.
- [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] 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] 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] 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] 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.