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On New Inequalities Involving AB-fractional Integrals for Some Convexity Classes

Year 2021, Volume: 2 Issue: 2, 127 - 145, 31.07.2021

Abstract

During the past two decades or so, fractional integral operators have been one of the most important tools in the development of inequality theory. By this means, a lot of generalized integral inequalities involving various fractional integral operators have been presented in the literature. Very recently, Atangana-Baleanu fractional integral operators has been introduced by Atangana and Baleanu and with the help of these operators some new integral inequalities are obtained. The main aim of the paper is to establish some new integral inequalities for quasi-convex function and P -function via Atangana-Baleanu integral operators by using identity which was produced by Set et al. in [16].

References

  • [1] Abdeljawad T., Baleanu D., Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, Journal of Nonlinear Sciences and Applications, 10, 1098-1107, 2017.
  • [2] Abdeljawad T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279, 57-66, 2015.
  • [3] Akdemir A.O., Özdemir M.E., Ardic M.A., Yalçın A., Some new generalizations for GA-convex functions, Filomat, 31(4), 1009-1016, 2017.
  • [4] Akdemir A.O., Set E., Özdemir M.E., Yalçın A., New generalizations for functions whose second derivatives are GG-convex, Uzbek Mathematical Journal, 4, 22-34, 2018.
  • [5] Akdemir A.O., Ekinci A., Set E., Conformable fractional integrals and related new integral inequalities, Journal of Nonlinear and Convex Analysis, 18(4), 661-674, 2017.
  • [6] Anderson G.D., Vamanamurthy M.K., Vuorinen M., Generalized convexity and inequalities, Journal of Mathematical Analysis and Applications, 335, 1294-1308, 2007.
  • [7] Atangana A., Baleanu D., New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20(2), 763-769, 2016.
  • [8] Awan M.U., Noor M.A., Mihai M.V., Noor K.I., Conformable fractional Hermite-Hadamard inequalities via preinvex functions, Tbilisi Mathematical Journal, 10(4), 129-141, 2017.
  • [9] Dahmani Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Annals of Functional Analysis, 1(1), 51-58, 2010.
  • [10] Ion D.A., Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of the University of Craiova, Mathematics and Computer Science Series, 34, 82-87, 2007.
  • [11] Latif M.A., New Hermite-Hadamard type integral inequalities for GA-convex functions with applications, Analysis, 34(4), 379-389, 2014.
  • [12] Khalil R., Horani M. Al, Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, 2014.
  • [13] Niculescu C.P., Convexity according to the geometric mean, Mathematical Inequalities and Applications, 3(2), 155-167, 2000.
  • [14] Pearce C.E.M., P -functions, Quasi-convex functions and Hadamard-type inequalities, Journal of Mathematical Analysis and Applications, 240, 92-104, 1999.
  • [15] Samko S.G., Kilbas A.A., Marichev, O.I., Fractional Integral and Derivatives, Theory and Applications, Gordon and Breach, 1993.
  • [16] Set E., Butt S.I., Akdemir A.O., Karaoğlan A., Abdeljawad T., New integral inequalities for differentiable convex functions via Atangana- Baleanu fractional integral operators, Chaos, Solitions and Fractals, 143, 110554, 2021.
  • [17] Set E., Akdemir A.O., Özdemir M.E., Simpson type integral inequalities for convex functions via Riemann-Liouville integrals, Filomat, 31(14), 4415-4420, 2017.
  • [18] Tariboon J., Ntouyas S.K., Sudsutad W., Some new Riemann-Liouville fractional integral inequalities, International Journal of Mathematics and Mathematical Sciences, 2014, Article ID 869434, 6, 2014.
  • [19] Zhang X-M., Chu Y-M., Zhang X-H., The Hermite-Hadamard type inequality of GA-convex functions and its application, Journal of Inequalities and Applications, 2010, 507560, 2010. doi:10.1155/2010/507560.
Year 2021, Volume: 2 Issue: 2, 127 - 145, 31.07.2021

Abstract

References

  • [1] Abdeljawad T., Baleanu D., Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, Journal of Nonlinear Sciences and Applications, 10, 1098-1107, 2017.
  • [2] Abdeljawad T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279, 57-66, 2015.
  • [3] Akdemir A.O., Özdemir M.E., Ardic M.A., Yalçın A., Some new generalizations for GA-convex functions, Filomat, 31(4), 1009-1016, 2017.
  • [4] Akdemir A.O., Set E., Özdemir M.E., Yalçın A., New generalizations for functions whose second derivatives are GG-convex, Uzbek Mathematical Journal, 4, 22-34, 2018.
  • [5] Akdemir A.O., Ekinci A., Set E., Conformable fractional integrals and related new integral inequalities, Journal of Nonlinear and Convex Analysis, 18(4), 661-674, 2017.
  • [6] Anderson G.D., Vamanamurthy M.K., Vuorinen M., Generalized convexity and inequalities, Journal of Mathematical Analysis and Applications, 335, 1294-1308, 2007.
  • [7] Atangana A., Baleanu D., New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20(2), 763-769, 2016.
  • [8] Awan M.U., Noor M.A., Mihai M.V., Noor K.I., Conformable fractional Hermite-Hadamard inequalities via preinvex functions, Tbilisi Mathematical Journal, 10(4), 129-141, 2017.
  • [9] Dahmani Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Annals of Functional Analysis, 1(1), 51-58, 2010.
  • [10] Ion D.A., Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of the University of Craiova, Mathematics and Computer Science Series, 34, 82-87, 2007.
  • [11] Latif M.A., New Hermite-Hadamard type integral inequalities for GA-convex functions with applications, Analysis, 34(4), 379-389, 2014.
  • [12] Khalil R., Horani M. Al, Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, 2014.
  • [13] Niculescu C.P., Convexity according to the geometric mean, Mathematical Inequalities and Applications, 3(2), 155-167, 2000.
  • [14] Pearce C.E.M., P -functions, Quasi-convex functions and Hadamard-type inequalities, Journal of Mathematical Analysis and Applications, 240, 92-104, 1999.
  • [15] Samko S.G., Kilbas A.A., Marichev, O.I., Fractional Integral and Derivatives, Theory and Applications, Gordon and Breach, 1993.
  • [16] Set E., Butt S.I., Akdemir A.O., Karaoğlan A., Abdeljawad T., New integral inequalities for differentiable convex functions via Atangana- Baleanu fractional integral operators, Chaos, Solitions and Fractals, 143, 110554, 2021.
  • [17] Set E., Akdemir A.O., Özdemir M.E., Simpson type integral inequalities for convex functions via Riemann-Liouville integrals, Filomat, 31(14), 4415-4420, 2017.
  • [18] Tariboon J., Ntouyas S.K., Sudsutad W., Some new Riemann-Liouville fractional integral inequalities, International Journal of Mathematics and Mathematical Sciences, 2014, Article ID 869434, 6, 2014.
  • [19] Zhang X-M., Chu Y-M., Zhang X-H., The Hermite-Hadamard type inequality of GA-convex functions and its application, Journal of Inequalities and Applications, 2010, 507560, 2010. doi:10.1155/2010/507560.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Ali Karaoğlan This is me 0000-0002-8726-4716

Barış Çelik 0000-0001-5372-7543

Erhan Set 0000-0003-1364-5396

Ahmet Ocak Akdemir 0000-0003-2466-0508

Publication Date July 31, 2021
Published in Issue Year 2021 Volume: 2 Issue: 2

Cite

19113 FCMS is licensed under the Creative Commons Attribution 4.0 International Public License.