Bullen–Simpson-Type Inequalities for Fractal-Fractional Integrals Involving Generalized Harmonic Convex Functions

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Abstract
This article is devoted to the derivation of new inequalities within the theory of fractal–fractional calculus. Its principal contribution is the construction of a new auxiliary identity of fractal–fractional type. By exploiting this identity, we establish a variety of Bullen–Simpson-type inequalities associated with functions whose local fractional derivatives possess generalized harmonic convexity. Moreover, by employing the generalized Hölder and power mean inequalities, we derive supplementary bounds that further enhance the scope of the analysis. The accuracy of the theoretical developments is illustrated through a representative example, together with graphical visualizations that substantiate the consistency of the proposed results. Finally, an application to certain special fractal means is presented in order to demonstrate the effectiveness and practical relevance of the obtained inequalities.
Funding Information
  • King Saud University (ORF-2026-158)

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