On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications

Arslan Munir; Hüseyin Budak; Artion Kashuri; Saad Ihsan Butt Dr.

doi:10.32323/ujma.1918223

On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications

Abstract

Fractional integral inequalities play a fundamental role in modern analysisand have significant applications in the qualitative study of fractionaldifferential and integral equations. Motivated by the need for sharperestimates in fractional settings, this paper establishes several newMilne-Mercer type inequalities involving the Riemann-Liouville fractionalintegral operators. By employing the theory of $s$-convex functions, wederive a unified framework that extends and generalizes a variety ofclassical Mercer-type inequalities available in the literature. Furthermore,refined estimates are obtained for bounded functions and functionssatisfying Lipschitz-type conditions. The proposed results not only recovernumerous known inequalities as particular cases but also provide strongerbounds within the fractional calculus setting. To illustrate theapplicability and effectiveness of the theoretical findings, graphicalexamples are presented. In addition, applications to quadrature formulas, $q$-digamma functions, and modified Bessel functions are discussed,demonstrating the broad utility of the developed inequalities. The resultscontribute to the ongoing development of fractional integral inequalitiesand offer new tools for the analysis of special functions and relatedproblems in applied mathematics.

Keywords

Milne-mercer type, $s$-convex function, Riemann-liouville fractional integrals, Hölder's inequality, Power-mean inequality, Bounded function, Lipschitz function, $q$-digamma function, Modified bessel function

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Details

Primary Language

English

Subjects

Numerical and Computational Mathematics (Other)

Journal Section

Research Article

Authors

Arslan Munir
0009-0008-8148-5607
China

Hüseyin Budak
0000-0001-8843-955X
Türkiye

Artion Kashuri ^*
0000-0003-0115-3079
Albania

Saad Ihsan Butt Dr.
0000-0001-7192-8269
Pakistan

Early Pub Date

August 2, 2026

Publication Date

-

Submission Date

March 28, 2026

Acceptance Date

July 31, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.32323/ujma.1918223

IZ

https://izlik.org/JA98ER54UT

APA

Munir, A., Budak, H., Kashuri, A., & Dr., S. I. B. (2026). On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications. Universal Journal of Mathematics and Applications, Advanced Online Publication. https://doi.org/10.32323/ujma.1918223

AMA

1.Munir A, Budak H, Kashuri A, Dr. SIB. On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications. Univ. J. Math. Appl. 2026;(Advanced Online Publication). doi:10.32323/ujma.1918223

Chicago

Munir, Arslan, Hüseyin Budak, Artion Kashuri, and Saad Ihsan Butt Dr. 2026. “On Fractional Milne-Mercer Bounds for $s$-Convex Functions With Applications”. Universal Journal of Mathematics and Applications, no. Advanced Online Publication. https://doi.org/10.32323/ujma.1918223.

EndNote

Munir A, Budak H, Kashuri A, Dr. SIB (August 1, 2026) On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications. Universal Journal of Mathematics and Applications Advanced Online Publication

IEEE

[1]A. Munir, H. Budak, A. Kashuri, and S. I. B. Dr., “On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications”, Univ. J. Math. Appl., no. Advanced Online Publication, Aug. 2026, doi: 10.32323/ujma.1918223.

ISNAD

Munir, Arslan - Budak, Hüseyin - Kashuri, Artion - Dr., Saad Ihsan Butt. “On Fractional Milne-Mercer Bounds for $s$-Convex Functions With Applications”. Universal Journal of Mathematics and Applications. Advanced Online Publication (August 1, 2026). https://doi.org/10.32323/ujma.1918223.

JAMA

1.Munir A, Budak H, Kashuri A, Dr. SIB. On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications. Univ. J. Math. Appl. 2026. doi:10.32323/ujma.1918223.

MLA

Munir, Arslan, et al. “On Fractional Milne-Mercer Bounds for $s$-Convex Functions With Applications”. Universal Journal of Mathematics and Applications, no. Advanced Online Publication, Aug. 2026, doi:10.32323/ujma.1918223.

Vancouver

1.Arslan Munir, Hüseyin Budak, Artion Kashuri, Saad Ihsan Butt Dr. On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications. Univ. J. Math. Appl. 2026 Aug. 1;(Advanced Online Publication). doi:10.32323/ujma.1918223