New Weighted Inequalities for Functions Whose Higher-Order Partial Derivatives Are Co-Ordinated Convex

Year 2024, Volume: 7 Issue: 2, 77 - 86, 30.06.2024

Samet Erden , Mehmet Zeki Sarıkaya

https://doi.org/10.33401/fujma.1383885

Cited By: 1

Abstract

The purpose of this study is to establish recent inequalities based on double integrals of mappings whose higher-order partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Also, some special cases of results improved in this study are examined.

Keywords

Hermite-Hadamard inequalities, co-ordinated convex mapping, partial derivative functions, Integral inequalities

References

• [1] S.S. Dragomir, On 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{http://dx.doi.org/10.11650/twjm/1500574995}{[\mbox{CrossRef}]} $
• [2] F. Chen, On Hermite-Hadamard type inequalities for s-convex functions on the coordinates via Riemann-Liouville fractional integrals, J. Appl. Math., 2014 (2014), Article ID 248710:1-8. $ \href{https://doi.org/10.1155/2014/248710}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84901778648&origin=resultslist&sort=plf-f&src=s&sid=bcbe6436c0ddc6c0ea845c8c54a2e78b&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+Hermite-Hadamard+type+inequalities+for+s-convex+functions+on+the+coordinates+via+Riemann-Liouville+fractional+integrals%22%29&sl=81&sessionSearchId=bcbe6436c0ddc6c0ea845c8c54a2e78b&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000336289900001}{[\mbox{Web of Science}]} $
• [3] 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&sid=f4c3eebe11d050ef6a99c3daef5b4602&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+Hermite-Hadamard-type+inequalities+for+co-ordinated+convex+function+via+fractional+integrals%22%29&sl=81&sessionSearchId=f4c3eebe11d050ef6a99c3daef5b4602&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000328110600005}{[\mbox{Web of Science}]} $
• [4] M.A. Latif and S.S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions, J. Inequal. Appl., 2012 (2012), 28. $ \href{https://doi.org/10.1186/1029-242X-2012-28}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84870407675&origin=resultslist&sort=plf-f&src=s&sid=f4c3eebe11d050ef6a99c3daef5b4602&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+some+new+inequalities+for+differentiable%22%29&sl=81&sessionSearchId=f4c3eebe11d050ef6a99c3daef5b4602&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000303883200001}{[\mbox{Web of Science}]} $
• [5] S. Erden and M.Z. Sarıkaya, On the Hermite-Hadamard-type and Ostrowski type inequalities for the co-ordinated convex functions, Palestine J. Math., 6(1) (2017), 257-270. $\href{https://pjm.ppu.edu/paper/321}{[\mbox{Web}]} $
• [6] S. Erden and M.Z. Sarıkaya, Some inequalities for double integrals and applications for cubature formula, Acta Univ. Sapientiae, Math., 11(2) (2019), 271-295. $\href{https://doi.org/10.2478/ausm-2019-0021}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85082131341&origin=resultslist&sort=plf-f&src=s&sid=bcbe6436c0ddc6c0ea845c8c54a2e78b&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+inequalities+for+double+integrals+and+applications+for+cubature+formula%22%29&sl=81&sessionSearchId=bcbe6436c0ddc6c0ea845c8c54a2e78b&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000518406900003}{[\mbox{Web of Science}]} $
• [7] M.Z. Sarıkaya, E. Set, M.E. Özdemir and S.S. Dragomir, New some Hadamard’s type inequalities for co-ordineted convex functions, Tamsui Oxf. J. Math. Sci., 28(2) (2010), 137-152. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84875239482&origin=resultslist&sort=plf-f&src=s&sid=f4c3eebe11d050ef6a99c3daef5b4602&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22New+some+Hadamard%27s+type+inequalities+for+co-ordinated+convex+functions%22%29&sl=81&sessionSearchId=f4c3eebe11d050ef6a99c3daef5b4602&relpos=0}{[\mbox{Scopus}]} $
• [8] D.Y. Hwang, K.L. Seng and G.S. Yang, Some Hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane, Taiwanese J. Math.,11(1) (2007), 63-73. $ \href{https://doi.org/10.11650/twjm/1500404635}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-34250215947&origin=resultslist&sort=plf-f&src=s&sid=bcbe6436c0ddc6c0ea845c8c54a2e78b&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+Hadamard%27s+inequalities+for+co-ordinated+convex+functions+in+a+rectangle+from+the+plane%22%29&sl=81&sessionSearchId=bcbe6436c0ddc6c0ea845c8c54a2e78b&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000245526600006}{[\mbox{Web of Science}]} $
• [9] M.A. Latif and M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinetes, Int. Math. Forum, 4(47) (2009), 2327-2338. $ \href{https://www.m-hikari.com/imf-password2009/45-48-2009/latifIMF45-48-2009.pdf}{[\mbox{Web}]} $
• [10] M.A. Latif, S. Hussain and S.S. Dragomir, New Ostrowski type inequalities for co-ordinated convex functions, Transylvanian J. Math. Mech., 4(2) (2012), 125-136. $ \href{http://tjmm.edyropress.ro/journal/12040204.pdf}{[\mbox{Web}]} $
• [11] M.Z. Sarıkaya, Some inequalities for differentiable co-ordinated convex mappings, Asian-Eur J. Math., 8(3)1550058(2015), 1-21. $ \href{https://doi.org/10.1142/S1793557115500588}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84951746799&origin=resultslist&sort=plf-f&src=s&sid=f4c3eebe11d050ef6a99c3daef5b4602&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+inequalities+for+differentiable+coordinated+convex+mappings%22%29&sl=81&sessionSearchId=f4c3eebe11d050ef6a99c3daef5b4602&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000214277200021}{[\mbox{Web of Science}]} $
• [12] M.A. Latif and M. Alomari, On the Hadamard-type inequalities for h-convex functions on the co-ordinetes, Int. J. Math. Anal., 3(33) (2009), 1645-1656. $ \href{https://www.m-hikari.com/ijma/ijma-password-2009/ijma-password33-36-2009/latifIJMA33-36-2009.pdf}{[\mbox{Web}]} $
• [13] M.E. Özdemir, E. Set and M.Z. Sarıkaya, Some new hadamard type inequalities for co-ordinated m-Convex and (a;m)-Convex Functions, Hacettepe J. Math. Stat., 40(2) (2011), 219-229. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-80052739706&origin=resultslist&sort=plf-f&src=s&sid=f4c3eebe11d050ef6a99c3daef5b4602&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22SOME+NEW+HADAMARD+TYPE+INEQUALITIES+FOR+CO-ORDINATED%22%29&sl=81&sessionSearchId=f4c3eebe11d050ef6a99c3daef5b4602&relpos=1}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000291713800008}{[\mbox{Web of Science}]} $
• [14] J. Park, Some Hadamard’s type inequalities for co-ordinated (s;m)-convex mappings in the second sense, Far East J. Appl. Math., 51(2) (211), 205–216. $ \href{https://www.researchgate.net/publication/267163168_Some_Hadamard's_type_inequalities_for_co-ordinated_sm-convex_mappings_in_the_second_sense}{[\mbox{Web}]} $
• [15] G. Anastassiou, Ostrowski type inequalities, Proc. Am. Math. Soc., 123(12) (1995), 375-378. $ \href{https://doi.org/10.2307/2161906}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-21844481304&origin=resultslist&sort=plf-f&src=s&sid=276bfa7203558b76452503c68fcf40c6&sot=b&sdt=cl&cluster=scoexactsrctitle%2C%22Proceedings+Of+The+American+Mathematical+Society%22%2Ct&s=TITLE-ABS-KEY%28%22Ostrowski+type+inequalities%22%29&sl=42&sessionSearchId=276bfa7203558b76452503c68fcf40c6&relpos=1}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1995TM66500027}{[\mbox{Web of Science}]} $
• [16] P. Cerone, S.S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32(4) (1999), 698–712. $ \href{http://dx.doi.org/10.1515/dema-1999-0404}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-18544380048&origin=resultslist&sort=plf-f&src=s&sid=bcbe6436c0ddc6c0ea845c8c54a2e78b&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+Ostrowski+type+inequalities+for+n-time+differentiable+mappings+and+applications%22%29&sl=81&sessionSearchId=bcbe6436c0ddc6c0ea845c8c54a2e78b&relpos=0}{[\mbox{Scopus}]} $
• [17] M.A. Fink, Bounds on the deviation of a function from its averages, Czec. Math. J., 42 (117) (1992), 289-310. $ \href{http://dml.cz/dmlcz/128336}{[\mbox{Web}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1992KC73300010}{[\mbox{Web of Science}]} $
• [18] S. Erden, M.Z. Sarıkaya and H. Budak, New weighted inequalities for higher order derivatives and applications, Filomat, 32(12) (2018), 4419-4433. $ \href{https://dx.doi.org/10.2298/FIL1812419E}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85061454688&origin=resultslist&sort=plf-f&src=s&sid=bcbe6436c0ddc6c0ea845c8c54a2e78b&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22New+weighted+inequalities+for+higher+order+derivatives+and+applications%22%29&sl=81&sessionSearchId=bcbe6436c0ddc6c0ea845c8c54a2e78b&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000461182200027}{[\mbox{Web of Science}]} $
• [19] Z. Changjian and W.S. Cheung,On Ostrowski-type inequalities heigher-order partial derivatives, J. Inequal. Appl., 2010 (2010), Article ID: 960672:1-8. $ \href{https://doi.org/10.1155/2010/960672}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-80052691180&origin=resultslist&sort=plf-f&src=s&sid=bcbe6436c0ddc6c0ea845c8c54a2e78b&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+Ostrowski-Type+Inequalities+for+Higher-Order+Partial+Derivatives%22%29&sl=81&sessionSearchId=bcbe6436c0ddc6c0ea845c8c54a2e78b&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000276301800001}{[\mbox{Web of Science}]} $
• [20] G. Hanna, S.S. Dragomir and P. Cerone, A General Ostrowski type inequality for double integrals, Tamkang J. Math., 33(4) (2002), 319-333. $ \href{https://doi.org/10.5556/j.tkjm.33.2002.280}{[\mbox{CrossRef}]} $
• [21] N. Ujevic, Ostrowski-Grüss type inequalities in two dimensional, J. of Ineq. in Pure and Appl. Math., 4(5), Article 101 (2003), 1-12. $ \href{https://www.emis.de/journals/JIPAM/images/003_03/003_03_www.pdf}{[\mbox{Web}]} $
• [22] S. Erden and M.Z. Sarıkaya, On the Hermite- Hadamard’s and Ostrowski’s inequalities for the co-ordinated convex functions, New Trend Math. Sci., 5(3) (2017):33-45. $ \href{http://dx.doi.org/10.20852/ntmsci.2017.182}{[\mbox{CrossRef}]} $