Certain Weighted Fractional Integral Inequalities for Convex Functions

Abstract

In this study, by using the monotonicity properties of functions, several inequalities for convex functions are obtained with the help of a weighted fractional integral operator which provides a function f to be integrated in fractional order with respect to another function. It is also seen that the results obtained were generalizations of the previous results presented.

Keywords

• Convex functions
• Weighted fractional operators
• Fractional integral inequality.

References

1. Liu W.J., Ngo Q.A., Huy V.N., Several interesting integral inequalities, Journal of Mathematical Inequalities, 3, 201-212, 2009.
2. Mitrinović D.S., Pečarić J.E., Fink A.M., Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.
3. Jarad F., Abdeljawad T., Shah K., On the weighted fractional operators of a function with respect to another function, Fractals, 28(08), 20040011, 2020.
4. Osler T.J., The fractional derivative of a composite function, SIAM Journal on Mathematical Analysis, 1, 288-293, 1970.
5. Almeira R., A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, 44, 460-481, 2017.
6. Kilbas A.A., Hadamard-type fractional calculus, Journal of the Korean Mathematical Society, 38, 1191-1204, 2001.
7. Ngo Q.A., Thang D.D., Dat T.T., Tuan D.A., Notes on an integral inequality, Journal of Inequalities in Pure and Applied Mathematics, 7(4), 120, 2006.
8. Liu W.J., Cheng G.S., Li C.C., Further development of an open problem concerning an integral inequality, Journal of Inequalities in Pure and Applied Mathematics, 9(1), 14, 2008.

9. Dahmani Z., A note on some new fractional results involving convex functions, Acta Mathematica Universitatis Comenianae, LXXXI, 241-246, 2012.
10. Rahman G., Nisar K.S., Abdeljawad T., Ullah S., Certain fractional proportional integral inequalities via convex functions, Mathematics, 8, 222, 2020.
11. Rahman G., Nisar K.S., Abdeljawad T., Tempered fractional integral inequalities for convex functions, Mathematics, 8(4), 500, 2020.
12. Rahman G., Abdeljawad T., Jarad F., Khan A., Nisar K.S., Certain inequalities via generalized proportional Hadamard fractional integral operators, Advances in Differential Equations, 2019:454, 2019.