What question did this study set out to answer?

The research aims to improve the approximation of Milne-type inequalities using convex functions and fractional integrals.

April 8, 2026Open Access

Better Approximation of Milne‐Type Inequalities via Convex Functions and ABK Fractional Integral Operators

Key Points

The research aims to improve the approximation of Milne-type inequalities using convex functions and fractional integrals.
Identified a new identity for twice differentiable functions.
Applied ABK fractional integral operators to derive inequalities.
Modified the order of the fractional integral to explore different scenarios.
Several new fractional Milne-type inequalities are established.
A comprehensive table summarizing results is provided.
Illustrative examples and graphical depictions clarify the findings.

Abstract

In this paper, we give an identity for the function which is twice differentiable. Through the applications of this identity and Atangana–Baleanu–Katugampola (ABK) fractional integrals, several fractional Milne‐type inequalities are derived for functions whose second derivatives inside the absolute value are convex. Furthermore, the table has also been created by changing the order of the fractional integral. Additionally, an example and a graphical depiction are supplied to elucidate the main conclusions of our study.

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Authors

Muhammad Bilal Ahmed

Khuram Ali Khan

Ammara Nosheen

Journals

Journal of Applied Mathematics

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Better Approximation of Milne‐Type Inequalities via Convex Functions and ABK Fractional Integral Operators

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Abstract

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Authors

Journals

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References and Citations

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