On Hermite-Hadamard type inequalities for interval-valued multiplicative integrals

Abstract

In this work, we define multiplicative integrals for interval-valued functions. We establish some new Hermite-Hadamard type inequalities in the setting of interval-valued multiplicative calculus and give some examples to illustrate our main results. We also discuss special cases of our main results which are the extension of already established results.

Keywords

• Convex functions
• logarithmically convex function
• interval-valued functions
• multiplicative integral operator
• Hermite-Hadamard inequalities

References

1. Ali, M. A., Abbas, M., Budak, H., Kashuri, A., Some new Hermite–Hadamard integral inequalities in multiplicative calculus, TWMS Journal of Applied and Engineering Mathematics (2020, In Press).
2. Ali, M. A., Abbas, M., Zafar, A. A., On some Hermite-Hadamard integral inequalities in multiplicative calculus, Journal of Inequalities and Special Functions, 10 (1) (2019), 111– 122.
3. Ali, M. A., Abbas, M., Zhang, Z., Sial, I. B., Arif, R., On integral inequalities for product and quotient of two multiplicatively convex functions, Asian Research Journal of Mathematics (2019), 1–11.
4. Aubin, J.-P., Cellina, A., Di¤erential inclusions: set-valued maps and viability theory, Springer Science & Business Media, Springer-Verlag Berlin Heidelberg New York Tokyo, 2012.
5. Bashirov, A. E., Kurp¬nar, E. M., Özyapıcı, A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337 (1) (2008), 36–48.
6. Breckner, W. W., Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numér. Théor. Approx., 22 (1) (1993), 39–51.
7. Budak, H., Ali, M. A., Tarhanaci, M., Some new quantum Hermite–Hadamard-like inequalities for coordinated convex functions, Journal of Optimization Theory and Applications, 186(3) (2020), 899–910.
8. Budak, H., Erden, S., Ali, M. A., Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Mathematical Methods in the Applied Sciences (2020).

9. Chalco-Cano, Y., Flores-Franulic, A., Román-Flores, H., Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Computational & Applied Mathematics, 31 (3) (2012).
10. Chalco-Cano, Y., Lodwick, W. A., Condori-Equice, W., Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Computing, 19 (11) (2015), 3293–3300.
11. Chen, F., A note on Hermite-Hadamard inequalities for products of convex functions., Journal of Applied Mathematics (2013).
12. Costa, T., Jensen’s inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets and Systems, 327 (2017), 31–47.
13. Costa, T., Román-Flores, H., Some integral inequalities for fuzzy-interval-valued functions, Information Sciences, 420 (2017), 110–125.
14. Dragomir, S., Pearce, C., Selected topics on Hermite-Hadamard inequalities and applications, rgmia monographs, victoria university, 2000, ONLINE: http://rgmia. vu. edu. au/monographs (2004).
15. Dragomir, S., Pecaric, J., Persson, L.-E., Some inequalities of Hadamard type, Soochow J. Math, 21 (3) (1995), 335–341.
16. Dragomir, S. S., Two mappings in connection to Hadamard’s inequalities, Journal of Mathematical Analysis and Applications, 167 (1) (1992), 49–56.
17. Dragomir, S. S., Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Proyecciones (Antofagasta), 34 (4) (2015), 323–341.
18. Ertugral, F., Sarikaya, M. Z., Simpson type integral inequalities for generalized fractional integral, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 113 (4) (2019), 3115–3124.
19. Flores-Franuliµc, A., Chalco-Cano, Y., Román-Flores, H., An ostrowski type inequality for interval-valued functions, In 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) (2013), IEEE, pp. 1459–1462.
20. Guo, Y., Ye, G., Zhao, D., Liu, W., Some integral inequalities for log-h-convex interval-valued functions, IEEE Access, 7 (2019), 86739–86745.
21. Kara, H., Ali, M. A., Budak, H., Hermite–Hadamard-type inequalities for interval-valued coordinated convex functions involving generalized fractional integrals, Mathematical Methods in the Applied Sciences (2020).
22. Lupulescu, V., Fractional calculus for interval-valued functions, Fuzzy Sets and Systems, 265 (2015), 63–85.
23. Markov, S., Calculus for interval functions of a real variable, Computing, 22 (4) (1979), 325–337.
24. Markov, S., On the algebraic properties of convex bodies and some applications, Journal of convex analysis, 7 (1) (2000), 129–166.
25. Mitroi, F.-C., Nikodem, K., Wasowicz, S., Hermite–Hadamard inequalities for convex setvalued functions, Demonstratio Mathematica, 46 (4) (2013), 655–662.
26. Mohammed, P., Some new Hermite-Hadamard type inequalities for mt-convex functions on differentiable coordinates, Journal of King Saud University-Science, 30 (2) (2018), 258–262.
27. Moore, R. E., Interval analysis, Prentice-Hall, Englewood Clifs, 1966.
28. Moore, R. E., Kearfott, R. B., Cloud, M. J., Introduction to interval analysis, Siam, Philadelphia, P. A., 2009.
29. Nikodem, K., Sanchez, J. L., Sanchez, L., Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Mathematica Aeterna, 4 (8) (2014), 979–987.
30. Noor, M. A., Qi, F., Awan, M. U., Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis, 33 (4) (2013), 367–375.
31. Pachpatte, B., On some inequalities for convex functions, RGMIA Res. Rep. Coll, 6 (1) (2003), 1–9.
32. Peajcariaac, J. E., Tong, Y. L., Convex functions, partial orderings, and statistical applications, Academic Press, Boston San Diego New York London Sydney Tokyo Toronto, 1992.
33. Román-Flores, H., Chalco-Cano, Y., Lodwick, W., Some integral inequalities for interval valued functions, Computational and Applied Mathematics, 37 (2) (2018), 1306–1318.
34. Román-Flores, H., Chalco-Cano, Y., Silva, G. N., A note on Gronwall type inequality for interval-valued functions, In 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) (2013), IEEE, pp. 1455–1458.
35. Sadowska, E., Hadamard inequality and a refinement of Jensen inequality for set valued functions, Results in Mathematics, 32 (3-4) (1997), 332–337.
36. Sarikaya, M. Z., Yildirim, H., On generalization of the riesz potential, Indian Jour. of Math. and Mathematical Sci, 3 (2) (2007), 231–235.
37. Tseng, K.-L., Hwang, S.-R., New Hermite-Hadamard-type inequalities and their applications, Filomat, 30 (14) (2016), 3667–3680.
38. Vivas-Cortez, M., Aamir Ali, M., Kashuri, A., Bashir Sial, I., Zhang, Z., Some new Newton’s type integral inequalities for co-ordinated convex functions in quantum calculus, Symmetry, 12 (9) (2020), 1476.
39. Wang, J., Li, X., Zhu, C., et al., Refinements of Hermite-Hadamard type inequalities involving fractional integrals, Bulletin of the Belgian Mathematical Society-Simon Stevin, 20 (4) (2013), 655–666.
40. Zhao, D., Ali, M. A., Kashuri, A., Budak, H., Generalized fractional integral inequalities of Hermite–Hadamard type for harmonically convex functions, Advances in Difference Equations, 2020 (1) (2020), 1–14.
41. Zhao, D., Ali, M. A., Murtaza, G., Zhang, Z., On the Hermite–Hadamard inequalities for interval-valued coordinated convex functions, Advances in Difference Equations, 2020, 570 (2020).
42. Zhao, D., An, T., Ye, G., Liu, W., New Jensen and Hermite–Hadamard type inequalities for h-convex interval-valued functions, Journal of Inequalities and Applications, 2018 (1) (2018), 302.