Inequalities for Strongly $r$-Convex Functions on Time Scales
Download PDF
Authors: S. G. GEORGIEV, V. DARVISH, M. RAZEGHI AND B. KAYMAKçALAN
DOI: 10.46793/KgJMat2506.933G
Abstract:
In this paper, first we establish the Hermite-Hadamard type inequality based on diamond-α integral for a subset of strongly r-convex functions. Then we prove several new inequalities for n-times continuously differentiable strongly r-convex functions on time scales by virtue of some techniques and introducing new quantities.
Keywords:
Time scales, strongly r-convex functions, Hermite-Hadamard inequality, delta differentiable function, diamond integral.
References:
[1] N. Atasever and B. Kaymakçalan, G. Lešaja and K. Taş, Generalized diamond-α dynamic opial inequalities, Adv. Difference Equ. 2012(109) (2012), 1–9. https://doi.org/10.1186/1687-1847-2012-109
[2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston and Berlin, 2001. https://doi.org/10.1007/978-1-4612-0201-1
[3] F. Chen and X. Liu, Refinements on the Hermite-Hadamard inequalities for r-convex functions, J. Appl. Math. 2013 (2013), Article ID 978493, 5 pages. https://doi.org/10.1155/2013/978493
[4] S. Georgiev, Integral Inequalities on Time Scales, De Gruyter, 2020. https://doi.org/10.1515/9783110705553
[5] G. S. Guseinov and B. Kaymakçalan, 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
[6] A. F. Güvenilir, B. Kaymakçalan and N. N. Pelen, 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
[7] İ. İscan, H. Kadakal and M. Kadakal, Some new integral inequalities for n-times differentiable log-convex functions, New Trends in Mathematical Sciences 5(2) (2017), 10–15. http://dx.doi.org/10.20852/ntmsci.2017.150
[8] H. Kadakal, New inequalities for strongly r-convex functions, J. Funct. Spaces 2019 (2019), Article ID 1219237, 10 pages. https://doi.org/10.1155/2019/1219237
[9] M. Kadakal, H. Kadakal and İ. İscan, Some new integral inequalities for n-times differentiable s-convex functions in the first sense, Turkish Journal of Analysis and Number Theory 5(2) (2017), 63–6. https://doi.org/10.12691/tjant-5-2-4
[10] M. Kadakal, H. Kadakal and İ. İscan, Some new integral inequalities for n-times differentiable strongly convex functions, Karaelmas Science and Engineering Journal 8(1) (2018), 147–150. https://doi.org/10.7212/2Fzkufbd.v8i1.919
[11] A. Malinowska and D. Torres, On the diamond-α Riemann integral and mean value theorems on time scales, Dynamic Systems and Applications 18 (2009), 469–482.
[12] J. W. Rogers Jr. and Q. Sheng, Notes on the diamond-α dynamic derivative on time scales, J. Math. Anal. Appl. 326(1) (2007), 228–241. https://doi.org/10.1016/j.jmaa.2006.03.004
[13] Q. Sheng, M. Fadag, J. Henderson and J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl. 7(3) (2006), 395–413. https://doi.org/10.1016/j.nonrwa.2005.03.008