Document Type : Research Paper
Authors
Department of Mathematics, Faculty of Mathematics, Computer Science and Sciences Matter, 08 May 1945 University, Guelma, Algeria.
Abstract
In this study, we first establish two new identities for multiplicative differentiable functions. Based on these identities, we derive the midpoint and trapezoid type inequalities. The acquired outcomes improve and refine upon Khan and Budak's findings. At the conclusion, some applications to special means are provided.
Keywords
Main Subjects
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 journal of mathematics, 12 (3) (2019), pp. 1-11.
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), pp. 111-122.
3. M.A. Ali, H. Budak, M.Z. Sarikaya and Z. Zhang, Ostrowski and Simpson type inequalities for multiplicative integrals, Proyecciones, 40 (3) (2021), pp.743-763.
4. A.E. Bashirov, E.M. Kurpinar and A. Özyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (1) (2008), pp. 36-48.
5. S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (5) (1998), pp. 91-95.
6. M. Grossman and R. Katz, Non-Newtonian calculus, Lee Press, Pigeon Cove, Mass., 1972.
7. A. Kashuri, B. Meftah and P.O. Mohammed, Some weighted Simpson type inequalities for differentiable s-convex functions and their applications, J. Frac. Calc. & Nonlinear Sys., 1 (1) (2021), pp. 75-94.
8. S. Khan and H. Budak, On midpoint and trapezoid type inequalities for multiplicative integrals, Mathematica, 64 (87) (2022), pp. 95-108.
9. B. Meftah, Ostrowski’s inequality for functions whose first derivatives are s-preinvex in the second sense, Khayyam J. Math., 3 (1) (2017), pp. 61-80.
10. B. Meftah, M. Merad, N. Ouanas and A. Souahi, Some new Hermite-Hadamard type inequalities for functions whose nth derivatives are convex, Acta Comment. Univ. Tartu. Math., 23 (2) (2019), pp. 163-178.
11. B. Meftah and K. Mekalfa, Some weighted trapezoidal type inequalities via h-preinvexity, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., 24 (2020), 81-97.
12. B. Meftah and K. Mekalfa, Some weighted trapezoidal inequalities for differentiable log-convex functions, J. Interdiscip. Math., 24 (3) (2021), pp. 505-517.
13. B. Meftah, Maclaurin type inequalities for multiplicatively convex functions, Proc. Amer. Math. Soc., 151 (5) (2023), pp. 2115-2125.
14. B. Meftah and A. Lakhdari, Dual Simpson type inequalities for multiplicatively convex functions, Filomat, 37 (2023), bo. 22, 7673-7683.
15. S. Özcan, Hermite-hadamard type ınequalities for multiplicatively s-convex functions, Cumhuriyet Science Journal, 41 (1) (2020), pp. 245-259.
16. S. Özcan, Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions, AIMS Math., 5 (2) (2020), pp. 1505-1518.
17. S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-convex functions, Konuralp J. Math., 8 (1) (2020),pp. 158-164.
18. S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-preinvex functions, Turk. J. Anal. Number Theory., 9 (3) (2021), pp. 65-70.
19. C.E.M. Pearce and J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formulæ, Appl. Math. Lett., 13 (2) (2000), pp. 51-55.
20. J.E. Pečarić, F. Proschan and Y.L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
21. V. Volterra and B. Hostinsky, Operations Infinitesimales Lineaires, Gauthier-Villars, Paris, 1938.
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