Research Article
BibTex RIS Cite

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

Year 2024, Volume: 53 Issue: 3, 667 - 689, 27.06.2024

Zhong-xuan Mao Wen-bin Zhang Jing-feng Tian

https://doi.org/10.15672/hujms.1191725

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] 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, Volume: 53 Issue: 3, 667 - 689, 27.06.2024

Zhong-xuan Mao Wen-bin Zhang Jing-feng Tian

https://doi.org/10.15672/hujms.1191725

Abstract

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.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Zhong-xuan Mao This is me 0000-0001-5089-301X

Wen-bin Zhang This is me 0000-0002-9964-357X

Jing-feng Tian 0000-0003-1711-1113

Early Pub Date September 14, 2023
Publication Date June 27, 2024
Published in Issue Year 2024 Volume: 53 Issue: 3

Cite

APA Mao, Z.-x., Zhang, W.-b., & Tian, J.-f. (2024). On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral. Hacettepe Journal of Mathematics and Statistics, 53(3), 667-689. https://doi.org/10.15672/hujms.1191725
AMA Mao Zx, Zhang Wb, Tian Jf. On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral. Hacettepe Journal of Mathematics and Statistics. June 2024;53(3):667-689. doi:10.15672/hujms.1191725
Chicago Mao, Zhong-xuan, Wen-bin Zhang, and Jing-feng Tian. “On Hardy and Hermite-Hadamard Inequalities for $N$-Tuple Diamond-Alpha Integral”. Hacettepe Journal of Mathematics and Statistics 53, no. 3 (June 2024): 667-89. https://doi.org/10.15672/hujms.1191725.
EndNote Mao Z-x, Zhang W-b, Tian J-f (June 1, 2024) On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral. Hacettepe Journal of Mathematics and Statistics 53 3 667–689.
IEEE Z.-x. Mao, W.-b. Zhang, and J.-f. Tian, “On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 3, pp. 667–689, 2024, doi: 10.15672/hujms.1191725.
ISNAD Mao, Zhong-xuan et al. “On Hardy and Hermite-Hadamard Inequalities for $N$-Tuple Diamond-Alpha Integral”. Hacettepe Journal of Mathematics and Statistics 53/3 (June 2024), 667-689. https://doi.org/10.15672/hujms.1191725.
JAMA Mao Z-x, Zhang W-b, Tian J-f. On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral. Hacettepe Journal of Mathematics and Statistics. 2024;53:667–689.
MLA Mao, Zhong-xuan et al. “On Hardy and Hermite-Hadamard Inequalities for $N$-Tuple Diamond-Alpha Integral”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 3, 2024, pp. 667-89, doi:10.15672/hujms.1191725.
Vancouver Mao Z-x, Zhang W-b, Tian J-f. On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral. Hacettepe Journal of Mathematics and Statistics. 2024;53(3):667-89.

https://dergipark.org.tr/en/pub/hujms