Hermite-Hadamard Type Inequalities for Multiplicatively $h$-Convex Functions

Abstract

In this paper, some Hermite-Hadamard type inequalities for multiplicatively $h$-convex functions are established. Also, new integral inequalities involving multiplicative integrals are obtained for product and quotient of multiplicatively $h$-convex and convex positive functions.

Keywords

• Convex functions,,multiplicatively $h$-convex functions,multiplicative calculus,Hermite-Hadamard type inequalities

References

1. [1] M.A. Ali, M. Abbas, Z. Zhang, I.B. Sial and R. Arif, On Integral Inequalities for Product and Quotient of Two Multiplicatively Convex Functions, Asian Research J. Math., 12(3) (2019), 1-11.
2. [2] M.A. Ali, M. Abbas and A.A. Zafer, On Some Hermite-Hadamard Integral Inequalities in Multiplicative Calculus, J. Ineq. Special Func., 10(1) (2019), 111-122.
3. [3] A.E. Bashirov and M. Rıza, On Complex Multiplicative Differentiation, TWMS J. Appl. Eng. Math., 1(1) (2011), 75-85.
4. [4] A.E. Bashirov, E. Kurpınar, Y. Tando and A. O¨ zyapıcı, On Modeling with Multiplicative Differential Equations, Appl. Math., 26(4) (2011), 425-438.
5. [5] A.E. Bashirov, E.M. Kurpınar and A. Ozyapıcı, Multiplicative Calculus and Applications, J. Math. Anal. Appl., 337(1) (2008), 36-48.
6. [6] W.W. Breckner, Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Raumen, Publ. Inst. Math. (Beograd) (N.S.), 23(37) (1978), 13-20. (German)
7. [7] H. Budak and M.Z. Sarıkaya, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Mathematical Notes, 17(2) (2016), 1049-1059.
8. [8] Y.L. Daletskii and N.I. Teterina, Multiplicative Stochastic Integrals, Uspekhi Matematicheskikh Nauk, 27(2:164) (1972), 167-168.

9. [9] S.S. Dragomir, J. Pecaric and L.E. Persson, Some Inequalities of Hadamard Type, Soochow J. Math., 21(3) (1995), 335-341.
10. [10] S.S. Dragomir and C.E.M. Pearce, Quasi-Convex Functions and Hermite-Hadamard’s Inequality, Bull. Austral. Math. Soc., 57 (1998), 377-385.
11. [11] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
12. [12] M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
13. [13] J. Hadamard, Etude sur les proprietes des fonctions entieres en particulier d’une fonction consideree par Riemann, J. Math. Pures Appl., 58 (1893), 171-215.
14. [14] ˙I.˙Is¸can, S. Turhan and S. Maden, Hermite-Hadamard and Simpson-like Type Inequalities for Differentiable p-Quasi-Convex Functions, Filomat, 31(19) (2017), 5945-5953.
15. [15] H. Kadakal, Multiplicatively P-Functions and Some New Inequalities, New Trends in Math. Sci., 6(4) (2018), 111-118.
16. [16] M. Kadakal, Hermite-Hadamard and Simpson Type Inequalities for Multiplicatively Harmonically P-Functions, Sigma, 37(4) (2019), 1311-1320.
17. [17] R.L. Karandikar, Multiplicative Decomposition of Non-Singular Matrix Valued Continuous Semimartingales, The Annals of Probability, 10(4) (1982), 1088-1091.
18. [18] M. Kunt and İ. İşcancan, Hermite-Hadamard-Fejer Type Inequalities for p-Convex Functions, Arab J. Math. Sci., 23 (2017), 215-230.
19. [19] M.A. Noor, Advanced Convex Analysis, Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2010.
20. [20] M.A. Noor, F. Qi and M.U. Awan, Some Hermite-Hadamard Type Inequalities for log􀀀h-Convex Functions, Analysis, 33 (2013), 1-9.
21. [21] S. Ozcan, Some Integral Inequalities for Harmonically (a; s)-Convex Functions, J. Func. Spaces, Vol. 2019 (2019) Article ID 2394021, 8 pages.
22. [22] S. Ozcan and I.Iscan, Some New Hermite-Hadamard Type Inequalities for s-Convex Functions and Their Applications, J. Ineq. Appl., Article number: 2019:201 (2019).
23. [23] M.E. Ozdemir, H.K. O¨ nalan and M.A. Ardıc¸, Hermite-Hadamard Type Inequalities for (h(a;m))-Convex Functions, J. Conc. Appl. Math., 13(1) (2015), 96-107.
24. [24] A. Ozyapıcı and E. Mısırlı, Exponential Approximation on Multiplicative Calculus, 6th ISAAC Congress, p. 471, 2007.
25. [25] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992.
26. [26] M. Rıza, A. O¨ zyapıcı and E. Kurpınar, Multiplicative Finite Difference Methods, Quart. Appl. Math., 67(4) (2009), 745-754.
27. [27] E. Set, ˙I.˙Is¸can, M.Z. Sarıkaya and M.E. Ozdemir, On New Inequalities of Hermite-Hadamard-Fejer Type for Convex Functions via Fractional Integrals, Appl. Math. Comp., 259 (2015), 875-881.
28. [28] S. Varosanec, On h-Convexity, J. Math. Anal. Appl., 326(1) (2007), 303-311.
29. [29] B.-Y. Xi and F. Qi, Some Inequalities of Hermite-Hadamard Type for h-Convex Functions, Adv. Inequal. Appl., 2(1) (2013), 1-15.
30. [30] B.-Y. Xi and F. Qi, Some Integral Inequalities of Hermite-Hadamard Type for s-Logarithmically Convex Functions, Acta Mathematica Scientis, English Series, 35A(3) (2015), 515-526.