Midpoint Type Inequality for Multiplicatively Strong Convex Function via Multiplicative Calculus

Authors

  • Dawood Khan School of Mathematical Sciences, Universiti Sains Malaysia
  • Ilyas Khan Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu, India. . . . . . . . . . . . . . Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan. . . . . . . . . . . . . . . Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia
  • Juan Eduardo Nápoles Valdes National University of the Northeast Corrientes, Argentina
  • Waleed Mohammed Abdelfattah Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt
  • Muhammad Azeem Faculty of Engineering, department of Mechanical, aerospace and civil engineering, the University of Manchester, United Kingdom.

Abstract

In this paper, we first establish a new identity for multiplicatively differentiable functions within the framework of multiplicative calculus. By exploiting this identity, we derive a novel midpoint-type inequality for multiplicatively strong convex functions. The obtained result extends the existing theory of multiplicative integral inequalities and provides a unified approach for investigating midpoint estimates in multiplicative calculus

References

[1] S. Karamardian. The nonlinear complementarity problem with applications, Part II, J. Optim. Theory Appl., 4(3), (1969), 167–181.

[2] B.T. Polyak. Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Sov. Math. Dokl., 7, (1966), 72–75.

[3] G.H. Lin.; M. Fukushima. Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 118(1), (2003), 67–80.

[4] H.M. Srivastava.; Z.H. Zhang.; Y.D. Wu. Some further refinements and extensions of the Hermite–Hadamard and Jensen inequalities in several variables, Math. Comput. Model., 54(11–12), (2011), 2709–2717.

[5] S.K. Mishra.; N. Sharma. On strongly generalized convex functions of higher order, Math. Inequal. Appl., 22(1), (2019), 111–121.

[6] M.A. Noor.; K.I. Noor. Strongly exponentially convex functions, UPB Sci. Bull., Ser. A, Appl. Math. Phys., 81(4), (2019), 75–84.

[7] S.I. Butt. Generalized Jensen–Hermite–Hadamard–Mercer type inequalities for generalized strongly convex functions on fractal sets, Turkish Journal of Science, 8(2), (2024), 51–63.

[8] N. Merentes.; K. Nikodem. Remarks on strongly convex functions, Aequ. Math., 80(1–2), (2010), 193–199.

[9] M.A. Ali.; M. Abbas.; Z. Zhang.; I. B. Sial.; R. Arif. On integral inequalities for product and quotient of two multiplicatively convex functions, Asian Res. J. Math., 12(3), (2019), 1–11.

[10] S. Özcan.; S.I. Butt. Hermite–Hadamard type inequalities for multiplicatively harmonic convex functions, J. Inequal. Appl., 2023(1), (2023), 120.

[11] H. Kadakal.; M. Kadakal. Multiplicatively preinvex P-functions, J. Sci. Arts, 23(1), (2023), 21–32.

[12] S. Khan.; H. Budak. On midpoint and trapezoid type inequalities for multiplicative integrals, Mathematica, 64(87), (2022), 95–108.

[13] J.Q. Xie.; M.A. Ali.; T. Sitthiwirattham. Some new midpoint and trapezoidal type inequalities in multiplicative calculus with applications, Filomat, 37(20), (2023), 6665–6675.

[14] B. Meftah. Maclaurin type inequalities for multiplicatively convex functions, Proc. Amer. Math. Soc., 151(5), (2023), 2115–2125.

[15] M.A. Ali.; H. Budak.; M.Z. Sarikaya.; Z. Y. Zhang. Ostrowski and Simpson type inequalities for multiplicative integrals, Proyecciones, 40(3), (2021), 743–763.

[16] T. Abdeljawad.; M. Grossman, On geometric fractional calculus, J. Semigroup Theory Appl., 2016(2016) ID 2, 14 p.

[17] A.E. Bashirov.; E.M. Kurpinar, A. Özyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337(1)(2008), 36-48.

[18] S.I. Butt.; D. Khan.; V.D. Breaz. Fractional integral inequalities for strongly convex functions via multiplicative calculus with applications, Bound. Value Probl., 2025, (2025), 102.

[19] S.I. Butt.; D. Khan. Superquadraticity via multiplicative calculus, J. Comput. Appl. Math., 485, (2026), 117528.

[20] D. Khan.; S.I. Butt. Multiplicatively h-superquadratic functions and their applications in probability, special functions, and means, Math. Methods Appl. Sci., (2026), 1–36.

[21] M. Tariq.; W. Afzal.; M. Nadeem.; A.E. Munoz-Zavala.; J.E. Mac. Novel fractional Hermite–Hadamard and product-type inequalities via Raina function and preinvex mappings with entropy applications, Eur. J. Pure Appl. Math., 18(3) (2025), 6556–6556.

[22] H. Ahmad.; M. Tariq.; A. Asghar.; W. Afzal.; M. Aphane.; R. Efendiev. Some new notions of mathematical integral inequalities: theory and applications, Int. J. Anal. Appl., 24 (2026), 175.

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Published

30-06-2026

Issue

Vol. 1 No. 01 (2026): June 2026

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Articles

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Copyright (c) 2026 Dawood Khan, Ilyas Khan, Juan Eduardo Nápoles Valdes, Waleed Mohammed Abdelfattah, Muhammad Azeem (Author)

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This work is licensed under a Creative Commons Attribution 4.0 International License.

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How to Cite

Midpoint Type Inequality for Multiplicatively Strong Convex Function via Multiplicative Calculus. (2026). International Journal of Applied Mathematics , Optimization and Artificial Intelligence, 1(01), 31-39. https://ijamoai.org/ijamoai/index.php/ojs/article/view/15