ON SOME BULLEN-TYPE INEQUALITIES VIA CONFORMABLE FRACTIONAL INTEGRALS

Year 2019, Volume: 3 Issue: 4, 285 - 298, 29.10.2019

Musa Çakmak

https://doi.org/10.26900/jsp.3.030

Abstract

In this study, the Author has established a new lemma for α-differentiable function and some inequalities of Bullen-type inequalities for conformable fractional integrals. Some applications are also given. Examples are given to show the results.

Keywords

Bullen type inequality , Conformable fractional integrals , Hölder's inequality , Power-mean inequality

References

• [1] Bessenyei, M., Pales, Z., Characterization of convexity via Hadamard's inequality, Math. Inequal. Appl., 9(2006), 53-62.
• [2] Dragomir, SS., Agarwal, RP., Cerone, P., On Simpson's inequality and applications, J. of Ineq. and Appl., 5(2000), 533-579.
• [3] Dragomir, SS., Pearce, CEM., Selected topics on Hermite-Hadamard inequalities and applications, RGMIA monographs, Victoria University, 2000. [Online: http://www.staff.vu.edu.au/RGMIA/monographs/hermitehadamard.html].
• [4] Hadamard, J., Etude sur les proprietes des fonctions entieres en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl., 58(1893), 171-215.
• [5] Mitrinovic, DS, Lackovic, IB., Hermite and convexity, Aequationes Math., 28(1985), 229-232.
• [6] Mitrinovic, DS., Pecaric, JE., Fink, AM., Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.
• [7] Niculescu, CP., Persson, L., Convex Functions and their Applications, A Contemporary Approach, in: CMS Books in Mathematics, 23, Springer-Verlag, New York, 2006.
• [8] Sarikaya, M.,Z., Set, E., Ozdemir, ME., On new inequalities of Simpson's type for convex functions, RGMIA Res. Rep. Coll., 13(2)(2010), Article2.
• [9] Wu, S., Debnath, L., Inequalities for convex sequences and their applica- tions, Comp&Math Appl., 54(2007), 525-534.
• [10] Xi, B., Qi,F., Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means, Journal of Function Spaces and Appl., Vol 2012, Article ID 980438, 14 p., doi:10.1155/2012/980438.
• [11] C_ akmak, M.,Re_nements of Hadamard's Type Inequalities for s; m; (a;m)-Convex Functions, (2018), submitted.
• [12] Abdeljawad, T., On conformable fractional calculus. Journal of Computational and Applied Mathematics, (2015), 279, 57{66.
• [13] Anderson D.R., Taylors formula and integral inequalities for conformable fractional derivatives. In: Pardalos, P., Rassias, T. (eds) Contributions in Mathematics and Engineering. Springer, Cham, (2016), 25- 43 https://doi.org/10.1007/978-3-319-31317- 7-2.
• [14] Khalil, R., Al horani, M., Yousef, A. and Sababheh, M., A new definition of fractional derivative. Journal of Computational Applied Mathematics, (2014), 264, 65-70.
• [15] Iyiola, O.S. and Nwaeze, E.R., Some new results on the new conformable fractional calculus with application using D'Alambert approach. Progr. Fract. Di_er. Appl., (2016), 2(2), 115-122.
• [16] Abu Hammad, M. and Khalil, R., Conformable fractional heat differential equations. International Journal of Differential Equations and Applications, (2014), 13( 3), 177-183.
• [17] Abu Hammad, M. and Khalil, R., Abel's formula and wronskian for conformable fractional differential equations. InternationalJournal of Differential Equations and Applications, (2014), 13(3), 177-183.
• [18] Akkurt, A., Yıldırım, M.E. and Yıldırım, H., On some integral inequalities for conformable fractional integrals. Asian Journal of Mathematics and Computer Research, (2017), 15(3), 205-212.
• [19] Akkurt, A., Yıldırım, M.E. and Yıldırım, H., A new generalized fractional derivative and integral. Konuralp Journal of Mathematics, (2017), 5(2), 248{259.
• [20] Budak, H., Usta, F., Sarikaya, M.Z. and Ozdemir, M.E. On generalization of midpoint type inequalities with generalized fractional integral operators. Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas, (2018), https://doi.org/10.1007/s13398-018-0514-z.
• [21] Usta, F., Budak, H., Sarikaya, M.Z. and Set, E., On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators. Filomat, (2018), 32(6).