Trapezoid-type Inequalities Based on Generalized Conformable Integrals via Co-ordinated $h$-Convex Mappings

Year 2024, Volume: 7 Issue: 4, 236 - 252, 31.12.2024

Mehmet Eyüp Kiriş , Murat Yücel Ay Gözde Bayrak

https://doi.org/10.33401/fujma.1578534

Abstract

In this study, some new trapezoid type inequalities are generalized for $h-$ convex functions in coordinates by means of generalized conformable fractional integrals. For functions with $h-$convex absolute values of their partial derivatives, some new trapezoid type inequalities are obtained using the well-known Holder and Power Mean inequalities. In addition, some findings of this study include some results based on Riemann Liouville fractional integrals.

Keywords

Trapezoid-type inequalities, Convex functions, $h$-convex function, Co-ordinated $h$-convex functions, Fractional integrals, Riemann-Liouville fractional integrals, Conformable fractional integrals

References

• [1] S.S. Dragomir and C.E.M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, (2000). $ \href{https://rgmia.org/papers/monographs/Master.pdf}{\mbox{[Web]}} $
• [2] S.S. Dragomir, On The Hadamard’s Inequality For Convex Functions on the Co-ordinates in a Rectangle from the Plane, Taiwanese J. Math., 5(4) (2001), 775-788. $ \href{https://doi.org/10.11650/twjm/1500574995}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-1542394739&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28ON+THE+HADAMARD%E2%80%99S+INEQULALITY+FOR+CONVEX+FUNCTIONS+ON+THE+CO-ORDINATES+IN+A+RECTANGLE+FROM+THE+PLANE%29&sessionSearchId=ff199f78906cad6afa4af87f1f80443f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000173556800007}{\mbox{[Web of Science]}} $
• [3] S. Varosanec,On h-convexity, J. Math. Anal. Appl., 326 (2077), 303-311. $ \href{https://doi.org/10.1016/j.jmaa.2006.02.086}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-33750610962&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+h-convexity%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000242730700023}{\mbox{[Web of Science]}} $
• [4] M.A. Latif and M. Alomari, On Hadamard-type inequalities for h-convex functions on the co-ordinates, Int. J. Math. Anal., 3(33-36) (2009), 1645-1656. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-79251558517&origin=resultslist&sort=plf-t&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28On+Hadmard-Type+Inequalities+for+h-Convex+Functions+on+the+Co-ordinates%29&relpos=0}{\mbox{[Scopus]}} $
• [5] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, (1992). $ \href{https://doi.org/10.1016/s0076-5392%2808%29x6162-4}{\mbox{[CrossRef]}}$
• [6] M. Iqbal, S. Qaisar and M. Muddassar, A short note on integral inequality of type Hermite–Hadamard through convexity, J. Comput. Anal. Appl., 21(5) (2016), 946-953. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85014511542&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+short+note+on+integral+inequality+of+type+Hermite--Hadamard+through+convexity%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000368960300012}{\mbox{[Web of Science]}} $
• [7] S.S. Dragomir, R. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91-95. $\href{https://doi.org/10.1016/S0893-9659(98)00086-X}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0002448458&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28Two+inequalities+for+differentiable+mappings+and+applications+to+special+means+of+real+numbers+and+to+trapezoidal+formula%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=1}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000075622100017}{\mbox{[Web of Science]}} $
• [8] M.Z. Sarıkaya, E. Set, H. Yaldız and N. Başak, Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57(9–10) (2013), 2403–2407. $ \href{https://doi.org/10.1016/j.mcm.2011.12.048}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84875664244&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28Hermite%E2%80%93Hadamard%E2%80%99s+inequalities+for+fractional+integrals+and+related+fractional+inequalities%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=125}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000317262100039}{\mbox{[Web of Science]}} $
• [9] M.Z. Sarıkaya, E. Set, M.E. O¨ zdemir, S.S. Dragomir,New some Hadamard’s type inequalities for Co-ordinated convex functions, Tamsui Oxf. J. Math. Sci., 28(2) (2012), 137-152. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84875239482&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22New+some+Hadamard%27s+type+inequalities+for+co-ordinated+convex+functions%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=0}{\mbox{[Scopus]}} $
• [10] M.Z. Sarıkaya, A. Akuurt, H. Budak, M.E. Yıldırım and H. Yıldırım, Hermite-Hadamard’s inequalities for conformable fractional integrals, Int. J. Optim. Control: Theor. Appl., 9(1), 49-59(2019). $ \href{https://doi.org/10.11121/ijocta.01.2019.00559}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85066103606&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Hermite-Hadamard%27s+inequalities+for+conformable+fractional+integrals%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=1}{\mbox{[Scopus]}} $
• [11] G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, (1993). $ \href{https://www.semanticscholar.org/paper/Fractional-Integrals-and-Derivatives%3A-Theory-and-Samko-Kilbas/897fb8e45375c8fee443db9f69016c48f4129176}{\mbox{[Web]}} $
• [12] I. Podlubny, Fractional Differantial Equations, Academic Press, eBook ISBN: 9780080531984, San Diego CA, (1999).
• [13] S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley, (1993). $ \href{http://lib.ysu.am/disciplines_bk/b6c0b30496074d6bc08b794381aca81a.pdf}{\mbox{[Web]}} $
• [14] A. Akkurt, M.E. Yıldırım and H. Yıldırım, A new Generalized fractional derivative and integral, Konuralp J. Math., 5(2) (2017), 248-259. $ \href{https://dergipark.org.tr/en/pub/konuralpjournalmath/issue/28490/315229}{\mbox{[Web]}} $
• [15] A. Akkurt, M.Z. Sarıkaya, H. Budak and H. Yıldırım, On the Hadamard’s type inequalities for Co-ordinated convex functions via fractional integrals, J. King Saud Univ. Sci., 29(3) (2017), 380-387. $\href{https://doi.org/10.1016/j.jksus.2016.06.003}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85003794207&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+Hadamard%27s+type+inequalities+for+Co-ordinated+convex+functions+via+fractional+integrals%22%29&sessionSearchId=f23218ac34d3c1d4fd0d144855621200&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000456305200012}{\mbox{[Web of Science]}} $
• [16] M.K. Bakula, An improvement of the Hermite-Hadamard inequality for functions convex on the coordinates, Aust. J. Math. Anal. Appl., 11(1) (2014), 1-7. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84932173392&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22An+improvement+of+the+Hermite-Hadamard+inequality+for+functions+convex+on+the+coordinates%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=0}{\mbox{[Scopus]}} $
• [17] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. $ \href{https://doi.org/10.1016/j.cam.2014.01.002}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84893186929&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28A+new+definition+of+fractional+derivative%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=497}{\mbox{[Scopus]}} $
• [18] D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54(3) (2017), 903-917. $ \href{http://dx.doi.org/10.1007/s10092-017-0213-8}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85009170962&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28General+conformable+fractional+derivative+and+its+physical+interpretation%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000410163200012}{\mbox{[Web of Science]}} $
• [19] T.U. Khan and M.A.Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378-389. $ \href{https://doi.org/10.1016/j.cam.2018.07.018}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85050987862&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Generalized+conformable+fractional+operators%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=4}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000449246500026}{\mbox{[Web of Science]}} $
• [20] H. Budak, New version of Simpson type inequality for Y-Hilfer fractional integrals, Adv. Anal. Appl. Math., 1(1) (2024), 1-11. $\href{https://doi.org/10.62298/advmath.3}{\mbox{[CrossRef]}} $
• [21] M. Maaz, M. Muawwaz, U. Ali, M.D. Faiz, E.A. Rahman and A.Qayyum, New extension in Ostrowski’s type inequalities by using 13-step linear kernel, Adv. Anal. Appl. Math., 1(1) (2024), 55-67. $ \href{https://doi.org/10.62298/advmath.4}{\mbox{[CrossRef]}} $
• [22] M.Z. Sarıkaya, H. Budak and F. Usta, On generalized the conformable fractional calculus, TWMS J. App. Eng. Math., 9(4) (2019), 792-799. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85096570607&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+generalized+the+conformable+fractional+calculus%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000488642900011}{\mbox{[Web of Science]}} $
• [23] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279, 2015, 57-66. $\href{https://doi.org/10.1016/j.cam.2014.10.016}{\mbox{[CrossRef]}} \href{https://www.sciencedirect.com/science/article/pii/S0377042714004622}{\mbox{[Web]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000348259100005}{\mbox{[Web of Science]}} $
• [24] A.A. Abdelhakim, The flaw in the conformable calculus: it is conformable because it is not fractional, Fract. Calc. Appl. Anal., 22(2) (2019), 242-254. $ \href{https://doi.org/10.1515/fca-2019-0016}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85067095094&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+flaw+in+the+conformable+calculus%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000468966600002}{\mbox{[Web of Science]}} $
• [25] A. Hyder, A.H. Soliman, A new generalized q-conformable calculus and its applications in mathematical physics, Phys. Scr., 96(1) (2020), 015208. $ \href{https://doi.org/10.1088/1402-4896/abc6d9}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85096538146&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28A+new+generalized+%CE%B8-conformable+calculus+and+its+applications+in+mathematical+physics%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000588257500001}{\mbox{[Web of Science]}} $
• [26] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, (2006). $ \href{https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/204/suppl/C}{\mbox{[Web]}} $
• [27] M.Z. Sarıkaya, On the Hermite-Hadamard-type inequalities for Co-ordinated convex function via fractional integrals, Integral Transforms Spec. Funct., 25(2) (2014), 134-147. $ \href{https://doi.org/10.1080/10652469.2013.824436}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84890430251&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28On+the+Hermite%E2%80%93Hadamard-type+inequalities+for+co-ordinated+convex+function+via+fractional+integrals%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=5}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000328110600005}{\mbox{[Web of Science]}} $
• [28] F. Jarad, E. Uğurlu, T. Abdeljawad, and D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 247. $ \href{https://doi.org/10.1186/s13662-017-1306-z}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85028006940&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28On+a+new+class+of+fractional+operators%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=744}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000408335400002}{\mbox{[Web of Science]}}$
• [29] M. Bozkurt, A. Akkurt and H. Yıldırım, Conformable derivatives and integrals for the functions of two variables, Konuralp J. Math., 9(1) (2021), 49-59. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85186523687&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Conformable+derivatives+and+integrals+for+the+functions+of+two+variables%22%29&sessionSearchId=a3046c90cf8101a01c3cbc88438ba9c4&relpos=0}{\mbox{[Scopus]}} $
• [30] M.E. Kiriş and G. Bayrak, New version of Hermite-Hadamard inequality for Co-ordinated convex function via generalized conformable integrals, Filomat, 38(16) (2024), 5575–5589. $ \href{https://doi.org/10.2298/FIL2416575K}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85194053809&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28New+version+of+Hermite-Hadamard+inequality+for+co-ordinated+convex+function+via+generalized+conformable+integrals%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001337381700001}{\mbox{[Web of Science]}} $
• [31] A.A. Hyder, A.A. Almoneef, H. Budak, and M.A. Barakat, On new fractional version of generalized Hermite-Hadamard inequalities, Mathematics, 10(18) (2022), 3337. $ \href{https://doi.org/10.3390/math10183337}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85138598109&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28On+New+Fractional+Version+of+Generalized+Hermite-Hadamard+Inequalities%29&sessionSearchId=21e029aecd428d56366970d3fdd8de5e&relpos=19}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000857766500001}{\mbox{[Web of Science]}}$
• [32] M.E. Kiriş and G. Bayrak, Coordinated convex mapping approach to trapezoid type inequalities with generalized conformable integrals, Filomat 38(33) (2024), (Accepted).