ON HERMITE-HADAMARD TYPE INEQUALITIES VIA KATUGAMPOLA FRACTIONAL INTEGRALS

Abstract

In this paper, we give new denitons related to Katugampola fractional integral for two variables functions. We are interested in giving the Hermite{Hadamard inequality for a rectangle in plane via convex functions on co-ordinates involving Katugampola fractional integral.

Keywords

• Convex function,co-ordinated convex function,Hermite-Hadamard inequalities,Katugampola fractional integral.

References

1. Bakula M. K. and Pecaric J., (2006), On the Jensen’s inequality for convex functions on the co- ordinates in a rectangle from the plane, Taiwanese Journal of Mathematics, 10(5), 1271-1292.
2. Chen F., (2014),A note on the Hermite-Hadamard inequality for convex functions on the co-ordinates,J.of Math. Inequalities, 8(4), 915-923.
3. Chen, H. and Katugampola U.N., (2017), Hermite-Hadamard and Hermite-Hadamard-Fej˘er type in- equalities for generalized fractional integrals, J. Math. Anal. Appl., 446 , 1274-1291.
4. Dragomir S. S., (2001), On Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese Journal of Mathematics, 4, 775-788.
5. Dragomir S.S. and Pearce C.E.M., (2000), Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, Victoria University.
6. Ekinci A., Akdemir A. O. and ¨Ozdemir M. E., (2017), On Hadamard-type inequalities for co-ordinated r-convex functions, AIP Conference Proceedings 1833.
7. Gorenflo R. and Mainardi F., (1997), Fractional calculus: integral and differential equations of frac- tional order, Springer Verlag, Wien, 223-276.
8. Hadamard J., (1893), Etude sur les proprietes des fonctions entieres et en particulier d’une fonction considree par, Riemann, J. Math. Pures. et Appl. 58, 171–215.

9. Hwang D. Y., Tseng K. L. and Yang G. S., (2007), Some Hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane, Taiwanese Journal of Mathematics, 11, 63-73.
10. Katugampola U.N., (2011), New approach to a generalized fractional integrals, Appl. Math. Comput., (4) , 860-865.
11. Katugampola U.N., (2014), New approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (4) , 1-15.
12. Kilbas A. A., Srivastava H. M. and Trujillo J. J., (2006), Theory and applications of fractional differ- ential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam.
13. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, USA, 1993, p.2.
14. ¨Ozdemir M. E., Set E. and Sarıkaya M. Z., (2011), Some new Hadamard’s type inequalities for coordinated m-convex and (α, m)-convex functions, Hacettepe J. of Math. and Statistics, 40, 219-229.
15. Sarıkaya M. Z., Set E., Yaldiz H. and Basak N., ( 2013), Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. ,Math Comput Model., 57(9–10):2403–2407.
16. Sarıkaya M. Z., Set E., Ozdemir M.E. and Dragomir S. S. , (2011), New some Hadamard’s type inequalities for co-ordinated convex functions, Tamsui Oxford J of Information and Math. Sciences , (2), 137-152.
17. Sarıkaya M. Z., (2014), On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integral Transforms and Special Functions, Vol. 25, No. 2, 134-147.
18. Set E., Dahmani Z. and Mumcu ´I., (2018), New extensions of Chebyshev type inequalities using generalized Katugampola integrals via P˘olya-Szeg¨o inequality, Vol. 8, No. 2, 137-144.
19. Sarıkaya M. Z. and Yaldız H., (2013), On the Hadamard’s type inequalities for L-Lipschitzian mapping,Konuralp Journal of Mathematics, Volume 1, No. 2, pp. 33-40.
20. Yaldız H. , Sarıkaya M. Z. and Dahmani Z., (2017), On the Hermite-Hadamard-Fejer-type inequalities for co-ordinated convex functions via fractional integrals, An International Journal of Optimization and Control: Theories & Applications, Vol.7, No.2, pp.205-215.