Generalized trapezoid inequality for multiplicatively convex functions
DOI:
https://doi.org/10.5269/bspm.66248Abstract
In this paper, we first prove a new identity for multiplicative differentiable functions. Based on this identity, we establish a generalized trapezoid inequality for multiplicatively convex functions. Applications to special means are also given.
References
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[16] 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 (2019), no. 2, 163-178.
[17] B. Meftah and K. Mekalfa, Some weighted trapezoidal type inequalities via h-preinvexity. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 24 (2020), 81-97.
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[19] S. Özcan, Hermite-hadamard type inequalities for multiplicatively s-convex functions. Cumhuriyet Science Journal, 41 (2020), no. 1, 245-259.
[20] S. Özcan, Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions. AIMS Math. 5 (2020), no. 2, 1505–1518.
[21] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-convex functions. Konuralp J. Math. 8 (2020), no. 1, 158–164.
[22] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-preinvex functions. Turk. J. Anal. Number Theory. 9 (2021), no.3, 65-70.
[23] C. E. M. Pearce and J. Peˇcari´c, Inequalities for differentiable mappings with application to special means and quadrature formulæ. Appl. Math. Lett. 13 (2000), no. 2, 51–55.
[24] J. E. Peˇcari´c, 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.
[25] D. Stanley, A multiplicative calculus, Primus, Volume IX (4), pp 310326, 1999.
[26] V. Volterra and B. Hostinsky, Operations Infinitesimales Lineaires, Gauthier-Villars, Paris, 1938.
[27] Z. Zhou and T. Du, Analytical properties and related inequalities derived from multiplicative Hadamard k-fractional integrals, Chaos Solitons Fractals 189 (2024), part 2, Paper No. 115715.
[2] M. A. Ali, M. Abbas and A. A. Zafer, On some Hermite-Hadamard integral inequalities in multiplicative calculus. J. Ineq. Special Func. 10 (2019), no.1, 111-122.
[3] M. A. Ali, H. Budak, M. Z. Sarikaya and Z. Zhang, Ostrowski and Simpson type inequalities for multiplicative integrals. Proyecciones 40 (2021), no. 3, 743-763.
[4] M.B. Almatrafi, W. Saleh, A. Lakhdari, F. Jarad and B. Meftah, On the multiparameterized fractional multiplicative integral inequalities, J. Inequal. Appl. 2024, Paper No. 52, 27 pp.
[5] A. E. Bashirov, E.M. Kurpinar and A. Özyapici, Multiplicative calculus and its applications. J. Math. Anal. Appl. 337 (2008), no. 1, 36–48.
[6] A. Berkane, B. Meftah and A. Lakhdari, Right-Radau-Type Inequalities for Multiplicative Differentiable s-Convex Functions. Journal of applied mathematics & informatics, 42(4) (2024), 785-800.
[7] H. Budak and K. Özçelik, On Hermite-Hadamard type inequalities for multiplicative fractional integrals. Miskolc Math. Notes 21 (2020), no. 1, 91–99.
[8] 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 (1998), no. 5, 91–95.
[9] T. S. Du and Y. Long, The multi-parameterized integral inequalities for multiplicative Riemann-Liouville fractional integrals, J. Math. Anal. Appl. 541 (2025), no. 1, Paper No. 128692, 41 pp.
[10] A. Frioui, B. Meftah, A. Shokri, A. Lakhdari and H. Mukalazi, Parametrized multiplicative integral inequalities, Adv. Contin. Discrete Models 2024, Paper No. 12, 18 pp.
[11] H. Fu, Y. Peng and T. Du, Some inequalities for multiplicative tempered fractional integrals involving the λ-incomplete gamma functions. AIMS Math. 6 (2021), no. 7, 7456–7478.
[12] M. Grossman and R. Katz, Non-Newtonian calculus. Lee Press, Pigeon Cove, Mass., 1972.
[13] 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 (2021) no. 1, 75-94.
[14] H. Kavurmaci, M. Avci and M. E. Özdemir, New inequalities of Hermite-Hadamard type for convex functions with applications. J. Inequal. Appl. 2011, 2011:86, 11 pp.
[15] A. Lakhdari and B. Meftah, Some fractional weighted trapezoid type inequalities for preinvex functions. Int. J. Nonlinear Anal. 13 (2022) 1, 3567-3587.
[16] 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 (2019), no. 2, 163-178.
[17] B. Meftah and K. Mekalfa, Some weighted trapezoidal type inequalities via h-preinvexity. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 24 (2020), 81-97.
[18] B. Meftah, A. Lakhdari, W. Saleh and D.C. Benchettah, Companion of Ostrowski inequality for multiplicatively convex functions. Sahand Commun. Math. Anal. 21(2) (2024), 289-304.
[19] S. Özcan, Hermite-hadamard type inequalities for multiplicatively s-convex functions. Cumhuriyet Science Journal, 41 (2020), no. 1, 245-259.
[20] S. Özcan, Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions. AIMS Math. 5 (2020), no. 2, 1505–1518.
[21] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-convex functions. Konuralp J. Math. 8 (2020), no. 1, 158–164.
[22] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-preinvex functions. Turk. J. Anal. Number Theory. 9 (2021), no.3, 65-70.
[23] C. E. M. Pearce and J. Peˇcari´c, Inequalities for differentiable mappings with application to special means and quadrature formulæ. Appl. Math. Lett. 13 (2000), no. 2, 51–55.
[24] J. E. Peˇcari´c, 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.
[25] D. Stanley, A multiplicative calculus, Primus, Volume IX (4), pp 310326, 1999.
[26] V. Volterra and B. Hostinsky, Operations Infinitesimales Lineaires, Gauthier-Villars, Paris, 1938.
[27] Z. Zhou and T. Du, Analytical properties and related inequalities derived from multiplicative Hadamard k-fractional integrals, Chaos Solitons Fractals 189 (2024), part 2, Paper No. 115715.
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2025-09-22
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