New integral type inequalities via Raina-convex functions and its applications

Abstract

In this work, we discuss and introduce the novel literature about Raina-convex function and its algebraic properties. In addition, We elaborate and investigate Hermite-Hadamard and Fejer-type inequalities for newly discussed definition. Finally, using newly introduced definition, we find and prove amazing new integral type inequalities and applications for mean to positive real numbers. The amazing techniques and wonderful ideas of this paper may inspire and motivate for further activities and research in this direction furthermore.

Keywords

• Hermite-Hadamard inequality
• Hölder's inequality
• power mean inequality
• Raina-convex function

References

1. Butt, S. I., Nadeem, M., Qaisar, S., Akdemir, A. O., Abdeljawad, T., Hermite-Jensen-Mercer type inequalities for conformable integrals and related results, Adv. Differ. Equ., 1 (2020), 1–24. https://doi.org/10.1186/s13662-020-02968-4
2. Chu, H. H., Kalsoom, H., Rashid, S., Idrees, M., Safdar, F., Chu, Y-M., Baleanu, D., Quantum analogs of Ostrowski type inequalities for Rainas function correlated with coordinated generalized η–convex functions, Symmetry, 12(2) (2020), 1–26. https://doi.org/10.3390/sym12020308
3. Dragomir, S. S., Fitzpatrik, S., The Hadamard’s inequality for s–convex functions in the second sense, Demonstratio Math., 32(4) (1999), 687-696. https://doi.org/10.1515/dema-1999-0403
4. Eftekhari, N., Some remarks on (s,m)–convexity in the second sense, J. Math. Inequal., 8 (2014), 489-495. dx.doi.org/10.7153/jmi-08-36
5. Fejer, L., Über die Fourierreihen, II. Math. Naturwiss. Anz Ungar. Akad. Wiss., 24 (1906).
6. 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. http://eudml.org/doc/234668
7. Hernandez, H., Jorge, E., Vivas–Cortez, M., Hermite–Hadamard inequalities type for Raina’s Fractional integral operator using η–convex functions, Revista de Mathematica Teoriay Aplicaciones., 26(1) (2019), 1–20. http://dx.doi.org/10.15517/rmta.v26i1.35515
8. Khan, S., Khan, M. A., Butt, S. I., Chu, Y-M., A new bound for the Jensen gap pertaining twice differentiable functions with applications, Adv. Differ. Equ., 1 (2020), 1–11. https://doi.org/10.1186/s13662-020-02794-8

9. Mehmood, N., Butt, S. I., Pecaric, D., Pecaric, J., Generalizations of cyclic refinements of Jensena’s inequality by Lidstonea’s polynomial with applications in Information Theory, J. Math. Inequal., 14(1) (2020), 249–271. dx.doi.org/10.7153/jmi-2020-14-17
10. Niculescu, C. P., Persson, L. E., Convex Functions and Their Applications, Springer, New York, 2006. https://doi.org/10.1007/0-387-31077-0
11. Özdemir, M. E., Yildiz, C., Akdemir, A. O., Set, E., On some inequalities for s–convex functions and applications, J. Ineq. Appl., 333 (2013), 2–11. https://doi.org/10.1186/1029-242X-2013-333
12. Raina, R. K., On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2) (2005), 191–203.
13. Sarikaya, M. Z., Saglam, A., Yildirim, H., On some Hadamard type inequalities for h–convex functions, J. Math. Anal., 2(3) (2008), 335–341. https://doi.org/10.1186/s13660-019-2151-2
14. Set, E., Noor, M. A., Awan, M. U., Gözpinar, A., Generalized Hermite–Hadamard type inequalities involving fractional integral operator, J. Inequal. Appl., 169 (2017), 1–10. https://doi.org/10.1186/s13660-017-1444-6
15. Set, E., Some new generalizations of Ostrowski type inequalities for s-convex functions via fractional integral operators, Filomat., 32(16) (2018), 5595–5609. https://doi.org/10.2298/FIL1816595S
16. Toader, G., Some generalizations of the convexity, Proceedings of The Colloquium on Approximation and Optimization, Univ. Cluj–Napoca, Cluj–Napoca, (1985), 329–338.
17. Xi, B. Y., Q, F., Some integral inequalities of Hermite–Hadamard type for convex functions with applications to means, J. Funct. Spaces. Appl., 2012 Article ID 980438, (2012), 1–14. https://doi.org/10.1155/2012/980438
18. Butt, S. I., Tariq, M., Aslam, A., Ahmad, H., Nofal, T. A., Hermite-Hadamard type inequalities via generalized harmonic exponential convexity and applications, Journal of Function Spaces, 2021 Article ID 5533491 (2021), 12 pages. https://doi.org/10.1155/2021/5533491
19. Butt, S. I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A., Gao, W., n-polynomial exponential type p-convex function with some related inequalities and their applications, Heliyon, 6(11) (2020), e05420 ISSN 2405-8440. https://doi.org/10.1016/j.heliyon.2020.e05420
20. Butt, S. I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A., Gao, W., Hermite-Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications, Adv. Differ. Equ., 508 (2020). https://doi.org/10.1186/s13662-020-02967-5
21. Gao, W., Kashuri, A., Butt, S. I., Tariq, M., Aslam, A., Nadeem, M., New inequalities via n-polynomial harmonically exponential type convex functions, AIMS Mathematics, 5(6) (2020), 6856-6873. doi: 10.3934/math.2020440
22. Butt, S. I., Kashuri, A., Umar, M., Aslam, A., Gao, W., Hermite-Jensen-Mercer type inequalities via ψ-Riemann-Liouville k-fractional integrals, AIMS Mathematics, 5(5) (2020), 5193-5220. doi:10.3934/math.2020334