Document Type : Research Paper

Authors

1 Department of Mathematics, 8 may 1945 University, 24000, Geulma, Algeria.

2 Department CPST, National Higher School of Technology and Engineering, 23005, Annaba, Algeria.

3 Department of Mathematics, Taibah University, 42353, Al-Medina, Saudi Arabia.

4 Higher School of Management Sciences, 23000, Annaba, Algeria.

Abstract

The objective of this paper is to examine integral inequalities related to multiplicatively differentiable functions. Initially, we establish a novel identity using the two-point Newton-Cotes formula for multiplicatively differentiable functions. Using this identity, we derive Companion of Ostrowski's inequalities for multiplicatively differentiable convex mappings. The work also provides the results' applications.

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 Res. J. Math., 12 (3) (2019), pp. 1-11.
2. 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.
3. M.W. Alomari, M.E. Özdemir and H. Kavurmac, On companion of Ostrowski inequality for mappings whose first derivatives absolute value are convex with applications, Miskolc Math. Notes, 13 (2) (2012), pp. 233–248.
4. D. Aniszewska, Multiplicative runge–kutta methods, Nonlinear Dyn., 50 (2007), pp. 265-272.
5. A.E. Bashirov, E.M. Kurpinar and A. Özyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (1) (2008), pp. 36–48.
6. A.E. Bashirov, On line and double multiplicative integrals, TWMS J. Appl. Eng. Math., 3 (1) (2013), pp. 103–107.
7. A.E. Bashirov and S. Norozpour, On complex multiplicative integration, TWMS J Appl Eng Math., 7 (1) (2017), pp. 82–93.
8. A. Berhail and B. Meftah, Midpoint and trapezoid type inequalities for multiplicatively convex functions, arXiv preprint (2022).
9. A.H. Bhat, J. Majid, T.R. Shah, I.A. Wani and R. Jain, Multiplicative Fourier transform and its applications to multiplicative differential equations, J. Comput. Math. Sci., 10 (2) (2019), pp 375-383.
10. A.H. Bhat, J. Majid and I.A. Wani, Multiplicative Sumudu transform and its Applications, JETIR, 6 (1) (2019), pp. 579-589.
11. H. Budak and K. Özçelik, On Hermite-Hadamard type inequalities for multiplicative fractional integrals, Miskolc Math. Notes, 21 (1) (2020), pp. 91–99.
12. H. Boulares, B. Meftah, A. Moumen, R. Shafqat, H. Saber, T. Alraqad and E.E.A. Ahmad, Fractional Multiplicative Bullen-Type Inequalities for Multiplicative Differentiable Functions, Symmetry, 15 (2023).
13. 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.
14. H. Fu, Y. Peng and T. Du, Some inequalities for multiplicative tempered fractional integrals involving the λ-incomplete gamma functions, AIMS Math., 6 (7) (2021), pp. 7456–7478.
15. M. Grossman and R. Katz, Non-Newtonian calculus, Lee Press, Pigeon Cove, Mass., 1972.
16. A. Hassan, A.R. Khan, N. Irshad and S. Khatoon, Fractional Ostrowski-type Inequalities via (α, βγ, δ)-convex Function, Sahand Commun. Math. Anal., 20 (4) (2023), pp. 1-20.
17. 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.
18. H. Kavurmaci, M. Avci and M.E. Özdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, J. Inequal. Appl., 2011 (2011), pp. 11.
19. A. Lakhdari and B. Meftah, Some fractional weighted trapezoid type inequalities for preinvex functions, Int. J. Nonlinear Anal., 13 (1) (2022), pp. 3567-3587.
20. J.A. Machado, A. Babaei and B.P. Moghaddam, Highly accurate scheme for the Cauchy problem of the generalized Burgers-Huxley equation, Acta Polytech., 13 (6) (2016).
21. B. Meftah, A. Lakhdari and D.C. Benchettah, Some New Hermite- Hadamard Type Integral Inequalities for Twice Differentiable s- Convex Functions, Comput. Math. Model., (2023).
22. B. Meftah and A. Lakhdari, Dual Simpson type inequalities for multiplicatively convex functions, Filomat., 37 (22) (2023), pp. 7673–7683.
23. B. Meftah and C. Marrouche, Ostrowski Type Inequalities for n- Times Strongly m-MT-Convex Functions, Sahand Commun. Math. Anal., 20 (3) (2023), pp. 81-96.
24. E. Misirli and Y. Gurefe, Multiplicative Adams Bashforth-Moulton methods, Numer. Algorithms, 57 (4) (2011), pp. 425–439.
25. P. Mokhtary, B.P. Moghaddam, A.M. Lopes and J.A. T. Machado, A computational approach for the non-smooth solution of non-linear weakly singular Volterra integral equation with proportional delay, Numer. Algorithms, 83 (2020), pp. 987-1006.
26. A. Moumen, H. Boulares, B. Meftah, R. Shafqat, T. Alraqad, E.E. Ali and Z. Khaled, Multiplicatively Simpson Type Inequalities via Fractional Integral, Symmetry, (2023), 15,460.
27. Z.S. Mostaghim, B.P. Moghaddam and H.S. Haghgozar, Computational technique for simulating variable-order fractional Heston model with application in US stock market, Math. Sci., (2018).
28. Z.S. Mostaghim, B.P. Moghaddam and H.S. Haghgozar, Numerical simulation of fractional-order dynamical systems in noisy environments, Comp. Appl. Math., 37 (2018), pp. 6433-6447.
29. S. Özcan, Hermite-hadamard type ınequalities for multiplicatively s-convex functions, Cumhuriyet Sci. J., 41 (1) (2020), pp. 245-259.
30. S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-convex functions, Konuralp J. Math., 8 (1) (2020), pp. 158–164.
31. 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.
32. 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.
33. M. Riza, A. Özyapici and E. Misirli, Multiplicative finite difference methods, Quart. Appl. Math., 67 (4) (2009), pp. 745–754.
34. V. Volterra and B. Hostinsky, Operations Infinitesimales Lineaires, Gauthier-Villars, Paris, 1938.