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New insights on fractal–fractional integral inequalities: Hermite–Hadamard and Milne estimates

Author

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  • Lakhdari, Abdelghani
  • Budak, Hüseyin
  • Mlaiki, Nabil
  • Meftah, Badreddine
  • Abdeljawad, Thabet

Abstract

This paper investigates fractal–fractional integral inequalities for generalized s-convex functions. We begin by establishing a fractal–fractional Hermite–Hadamard inequality for such functions. In addition, a novel identity is introduced, which serves as the basis for deriving some fractal–fractional Milne-type inequalities for functions whose first-order local fractional derivatives exhibit generalized s-convexity. Subsequently, we provide additional results using the improved generalized Hölder and power mean inequalities, followed by a numerical example with graphical representations that confirm the accuracy of the obtained results. The study concludes with several applications to demonstrate the practicality and relevance of the proposed inequalities in various settings.

Suggested Citation

  • Lakhdari, Abdelghani & Budak, Hüseyin & Mlaiki, Nabil & Meftah, Badreddine & Abdeljawad, Thabet, 2025. "New insights on fractal–fractional integral inequalities: Hermite–Hadamard and Milne estimates," Chaos, Solitons & Fractals, Elsevier, vol. 193(C).
    • Handle: RePEc:eee:chsofr:v:193:y:2025:i:c:s0960077925001006
    DOI: 10.1016/j.chaos.2025.116087
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    References listed on IDEAS

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    1. Xiaoman Yuan & Hãœseyin Budak & Tingsong Du, 2024. "THE MULTI-PARAMETER FRACTAL–FRACTIONAL INEQUALITIES FOR FRACTAL (P,m)-CONVEX FUNCTIONS," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 32(01), pages 1-42.
    2. Yu, Shuhong & Zhou, Yunxiu & Du, Tingsong, 2022. "Certain midpoint-type integral inequalities involving twice differentiable generalized convex mappings and applications in fractal domain," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Huixia Mo & Xin Sui, 2014. "Generalized s‐Convex Functions on Fractal Sets," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    4. Huixia Mo & Xin Sui, 2014. "Generalized -Convex Functions on Fractal Sets," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-8, July.
    5. Huixia Mo & Xin Sui & Dongyan Yu, 2014. "Generalized Convex Functions on Fractal Sets and Two Related Inequalities," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    6. Yu, Yuping & Liu, Jun & Du, Tingsong, 2022. "Certain error bounds on the parameterized integral inequalities in the sense of fractal sets," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    7. Huixia Mo & Xin Sui & Dongyan Yu, 2014. "Generalized Convex Functions on Fractal Sets and Two Related Inequalities," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-7, June.
    8. Sarikaya, Mehmet Zeki & Tunc, Tuba & Budak, Hüseyin, 2016. "On generalized some integral inequalities for local fractional integrals," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 316-323.
    9. Wenbing Sun, 2021. "LOCAL FRACTIONAL OSTROWSKI-TYPE INEQUALITIES INVOLVING GENERALIZED h-CONVEX FUNCTIONS AND SOME APPLICATIONS FOR GENERALIZED MOMENTS," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(01), pages 1-12, February.
    10. Butt, Saad Ihsan & Khan, Ahmad, 2023. "New fractal–fractional parametric inequalities with applications," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    11. Chunyan Luo & Yuping Yu & Tingsong Du, 2021. "An Improvement Of Hã–Lder Integral Inequality On Fractal Sets And Some Related Simpson-Like Inequalities," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(05), pages 1-20, August.
    12. Shuhong Yu & Pshtiwan Othman Mohammed & Lei Xu & Tingsong Du, 2022. "An Improvement Of The Power-Mean Integral Inequality In The Frame Of Fractal Space And Certain Related Midpoint-Type Integral Inequalities," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(04), pages 1-23, June.
    13. Peng, Yu & Özcan, Serap & Du, Tingsong, 2024. "Symmetrical Hermite–Hadamard type inequalities stemming from multiplicative fractional integrals," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    14. Ai-Min Yang & Zeng-Shun Chen & H. M. Srivastava & Xiao-Jun Yang, 2013. "Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    15. Ai-Min Yang & Zeng-Shun Chen & H. M. Srivastava & Xiao-Jun Yang, 2013. "Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-6, November.
    16. Hongyan Xu & Abdelghani Lakhdari & Wedad Saleh & Badreddine Meftah, 2024. "Some New Parametrized Inequalities On Fractal Set," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 32(03), pages 1-14.
    17. Yong Zhang & Wenbing Sun, 2024. "On General Local Fractional Integral Inequalities For Generalized H-Preinvex Functions On Yang’S Fractal Sets," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-13.
    18. Wenbing Sun, 2021. "Hermite–Hadamard Type Local Fractional Integral Inequalities With Mittag-Leffler Kernel For Generalized Preinvex Functions," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-13, December.
    19. Wenbing Sun & Haiyang Wan, 2023. "Hermite–Hadamard-Type Inequalities Involving Several Kinds Of Fractional Calculus For Harmonically Convex Functions," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(09), pages 1-16.
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