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

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.

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  • 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).
  • 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.
  • 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.
  • 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.
  • 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.
  • Breckner, W. W., Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numér. Théor. Approx., 22 (1) (1993), 39–51.
  • 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.
  • 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).
  • 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).
  • 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.
  • Chen, F., A note on Hermite-Hadamard inequalities for products of convex functions., Journal of Applied Mathematics (2013).
  • Costa, T., Jensen’s inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets and Systems, 327 (2017), 31–47.
  • Costa, T., Román-Flores, H., Some integral inequalities for fuzzy-interval-valued functions, Information Sciences, 420 (2017), 110–125.
  • 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).
  • Dragomir, S., Pecaric, J., Persson, L.-E., Some inequalities of Hadamard type, Soochow J. Math, 21 (3) (1995), 335–341.
  • Dragomir, S. S., Two mappings in connection to Hadamard’s inequalities, Journal of Mathematical Analysis and Applications, 167 (1) (1992), 49–56.
  • Dragomir, S. S., Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Proyecciones (Antofagasta), 34 (4) (2015), 323–341.
  • 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.
  • 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.
  • Guo, Y., Ye, G., Zhao, D., Liu, W., Some integral inequalities for log-h-convex interval-valued functions, IEEE Access, 7 (2019), 86739–86745.
  • 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).
  • Lupulescu, V., Fractional calculus for interval-valued functions, Fuzzy Sets and Systems, 265 (2015), 63–85.
  • Markov, S., Calculus for interval functions of a real variable, Computing, 22 (4) (1979), 325–337.
  • Markov, S., On the algebraic properties of convex bodies and some applications, Journal of convex analysis, 7 (1) (2000), 129–166.
  • Mitroi, F.-C., Nikodem, K., Wasowicz, S., Hermite–Hadamard inequalities for convex setvalued functions, Demonstratio Mathematica, 46 (4) (2013), 655–662.
  • 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.
  • Moore, R. E., Interval analysis, Prentice-Hall, Englewood Clifs, 1966.
  • Moore, R. E., Kearfott, R. B., Cloud, M. J., Introduction to interval analysis, Siam, Philadelphia, P. A., 2009.
  • 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.
  • Noor, M. A., Qi, F., Awan, M. U., Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis, 33 (4) (2013), 367–375.
  • Pachpatte, B., On some inequalities for convex functions, RGMIA Res. Rep. Coll, 6 (1) (2003), 1–9.
  • 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.
  • Román-Flores, H., Chalco-Cano, Y., Lodwick, W., Some integral inequalities for interval valued functions, Computational and Applied Mathematics, 37 (2) (2018), 1306–1318.
  • 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.
  • Sadowska, E., Hadamard inequality and a refinement of Jensen inequality for set valued functions, Results in Mathematics, 32 (3-4) (1997), 332–337.
  • Sarikaya, M. Z., Yildirim, H., On generalization of the riesz potential, Indian Jour. of Math. and Mathematical Sci, 3 (2) (2007), 231–235.
  • Tseng, K.-L., Hwang, S.-R., New Hermite-Hadamard-type inequalities and their applications, Filomat, 30 (14) (2016), 3667–3680.
  • 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.
  • 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.
  • 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.
  • 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).
  • 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.