Fractal-fractional Maclaurin-type inequalities via generalized harmonic convexity and ANN-based computational illustrations

Bandar Bin-Mohsin; Bandar Bin-Mohsin

doi:10.3934/math.20261254

AIMS Mathematics

2026, Volume 11, Issue 9: 31861-31895. doi: 10.3934/math.20261254

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Research article

Fractal-fractional Maclaurin-type inequalities via generalized harmonic convexity and ANN-based computational illustrations

Bandar Bin-Mohsin ^,

Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 02 June 2026 Revised: 16 July 2026 Accepted: 15 September 2026 Published: 28 September 2026
MSC : 26A33, 26A51, 26D10, 26D15

This work developed a class of new integral inequalities within the framework of fractal-fractional calculus. The core contribution was the derivation of a new fractal-fractional auxiliary identity, which provides a unified basis for establishing several Maclaurin-type inequalities. The obtained results were formulated for functions whose local fractional derivatives satisfy generalized harmonic convexity. Furthermore, complementary upper bounds were derived by means of the generalized Hölder inequality and the power mean inequality, thereby extending the scope and flexibility of the proposed estimates. The validity of the theoretical results was illustrated through a representative example and graphical visualizations, which confirm the consistency of the developed inequalities. A feed-forward artificial neural network was also implemented to provide a complementary computational validation of one of the derived estimates by learning the dependence of the upper bound on the Hölder conjugate parameters. An application to selected special fractal means was also included to demonstrate the usefulness and practical significance of the proposed approach.
- fractal-fractional calculus,
- generalized harmonic convexity,
- Maclaurin's inequalities,
- generalized Hölder inequality,
- generalized power mean inequality
Citation: Bandar Bin-Mohsin. Fractal-fractional Maclaurin-type inequalities via generalized harmonic convexity and ANN-based computational illustrations[J]. AIMS Mathematics, 2026, 11(9): 31861-31895. doi: 10.3934/math.20261254

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Abstract

This work developed a class of new integral inequalities within the framework of fractal-fractional calculus. The core contribution was the derivation of a new fractal-fractional auxiliary identity, which provides a unified basis for establishing several Maclaurin-type inequalities. The obtained results were formulated for functions whose local fractional derivatives satisfy generalized harmonic convexity. Furthermore, complementary upper bounds were derived by means of the generalized Hölder inequality and the power mean inequality, thereby extending the scope and flexibility of the proposed estimates. The validity of the theoretical results was illustrated through a representative example and graphical visualizations, which confirm the consistency of the developed inequalities. A feed-forward artificial neural network was also implemented to provide a complementary computational validation of one of the derived estimates by learning the dependence of the upper bound on the Hölder conjugate parameters. An application to selected special fractal means was also included to demonstrate the usefulness and practical significance of the proposed approach.

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