Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals

Nailə Namazova

doi:10.32323/ujma.1971125

Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals

Abstract

In this study, new Simpson-type integral inequalities are obtained by combining the flexible structure provided by variable-order fractional computation with the geometric properties of $(m_1,m_2)$-exponentially convex functions. First, a new integral identity involving variable-order Riemann–Liouville fractional integral operators is established for differentiable functions. This identity is used as a fundamental tool in the development of Simpson-type error estimators and integral inequalities. Then, various Simpson-type upper bounds are derived under the assumption that the absolute value of derivative of the function is $(m_1,m_2)$-exponentially convex. For this purpose, different types of error estimators are obtained using the Hölder inequality, the power-averaging approach, and appropriate integral techniques. The presented results include many classical and fractional Simpson inequalities in the literature as special cases and provide important generalizations within the framework of variable-order fractional integral operators. In addition, appropriate special cases and results are given to demonstrate the applicability of the obtained findings. The obtained results enrich the theory of Simpson-type inequalities within the framework of variable-order fractional calculus and provide a unified approach for deriving new error estimates associated with exponentially convex structures.

Keywords

Simpson-type inequalities, Fractional integrals of variable order, Riemann–Liouville fractional integral operators, (m1, m2)-Exponentially convex functions

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Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Nailə Namazova ^*
0009-0001-8960-2889
Azerbaijan

Early Pub Date

August 10, 2026

Publication Date

September 13, 2026

Submission Date

June 15, 2026

Acceptance Date

August 6, 2026

Published in Issue

Year 2026 Volume: 9 Number: 3

DOI

https://doi.org/10.32323/ujma.1971125

IZ

https://izlik.org/JA57KG88JC

APA

Namazova, N. (2026). Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals. Universal Journal of Mathematics and Applications, 9(3), 167-176. https://doi.org/10.32323/ujma.1971125

AMA

1.Namazova N. Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals. Univ. J. Math. Appl. 2026;9(3):167-176. doi:10.32323/ujma.1971125

Chicago

Namazova, Nailə. 2026. “Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals”. Universal Journal of Mathematics and Applications 9 (3): 167-76. https://doi.org/10.32323/ujma.1971125.

EndNote

Namazova N (September 1, 2026) Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals. Universal Journal of Mathematics and Applications 9 3 167–176.

IEEE

[1]N. Namazova, “Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals”, Univ. J. Math. Appl., vol. 9, no. 3, pp. 167–176, Sept. 2026, doi: 10.32323/ujma.1971125.

ISNAD

Namazova, Nailə. “Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals”. Universal Journal of Mathematics and Applications 9/3 (September 1, 2026): 167-176. https://doi.org/10.32323/ujma.1971125.

JAMA

1.Namazova N. Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals. Univ. J. Math. Appl. 2026;9:167–176.

MLA

Namazova, Nailə. “Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals”. Universal Journal of Mathematics and Applications, vol. 9, no. 3, Sept. 2026, pp. 167-76, doi:10.32323/ujma.1971125.

Vancouver

1.Nailə Namazova. Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals. Univ. J. Math. Appl. 2026 Sep. 1;9(3):167-76. doi:10.32323/ujma.1971125