Milne's Estimates via Conformable Fractional Multiplicative Integrals

Abdelghani Lakhdari; Juan Eduardo Napoles Valdes; Badreddine Meftah; Hüseyin Budak

doi:10.32323/ujma.1767392

Milne's Estimates via Conformable Fractional Multiplicative Integrals

Abstract

This paper explores Milne’s inequality in the context of multiplicative conformable fractional integrals, a recent extension of classical fractional calculus rooted in non-Newtonian analysis. Leveraging recent developments in multiplicative calculus, we establish a new fundamental identity that underpins the derivation of Milne-type inequalities for two classes of functions: (i) those whose $^*$-derivatives in $^*$-absolute value are multiplicative convex, and (ii) those with bounded $^*$-derivatives. A numerical example, accompanied by graphical illustrations, is included to demonstrate the validity and effectiveness of the theoretical findings. Further, we extend our analysis to functions satisfying specific derivative constraints and present some applications to special means. The paper concludes with a summary of the main contributions and a discussion of promising avenues for future research in multiplicative fractional analysis.

Keywords

Conformable fractional multiplicative integrals, Functions with bounded $^*$-derivatives, Milne's rule, Multiplicative calculus, Multiplicative convexity

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Details

Primary Language

English

Subjects

Numerical and Computational Mathematics (Other)

Journal Section

Research Article

Authors

Abdelghani Lakhdari ^*
0000-0003-2943-2678
Türkiye

Juan Eduardo Napoles Valdes
0000-0003-2470-1090
Argentina

Badreddine Meftah
0000-0002-0156-7864
Algeria

Hüseyin Budak
0000-0001-8843-955X
Türkiye

Early Pub Date

January 11, 2026

Publication Date

January 11, 2026

Submission Date

August 17, 2025

Acceptance Date

November 25, 2025

Published in Issue

Year 2026 Volume: 9 Number: 1

DOI

https://doi.org/10.32323/ujma.1767392

IZ

https://izlik.org/JA85DE79HZ

APA

Lakhdari, A., Napoles Valdes, J. E., Meftah, B., & Budak, H. (2026). Milne’s Estimates via Conformable Fractional Multiplicative Integrals. Universal Journal of Mathematics and Applications, 9(1), 1-18. https://doi.org/10.32323/ujma.1767392

AMA

1.Lakhdari A, Napoles Valdes JE, Meftah B, Budak H. Milne’s Estimates via Conformable Fractional Multiplicative Integrals. Univ. J. Math. Appl. 2026;9(1):1-18. doi:10.32323/ujma.1767392

Chicago

Lakhdari, Abdelghani, Juan Eduardo Napoles Valdes, Badreddine Meftah, and Hüseyin Budak. 2026. “Milne’s Estimates via Conformable Fractional Multiplicative Integrals”. Universal Journal of Mathematics and Applications 9 (1): 1-18. https://doi.org/10.32323/ujma.1767392.

EndNote

Lakhdari A, Napoles Valdes JE, Meftah B, Budak H (March 1, 2026) Milne’s Estimates via Conformable Fractional Multiplicative Integrals. Universal Journal of Mathematics and Applications 9 1 1–18.

IEEE

[1]A. Lakhdari, J. E. Napoles Valdes, B. Meftah, and H. Budak, “Milne’s Estimates via Conformable Fractional Multiplicative Integrals”, Univ. J. Math. Appl., vol. 9, no. 1, pp. 1–18, Mar. 2026, doi: 10.32323/ujma.1767392.

ISNAD

Lakhdari, Abdelghani - Napoles Valdes, Juan Eduardo - Meftah, Badreddine - Budak, Hüseyin. “Milne’s Estimates via Conformable Fractional Multiplicative Integrals”. Universal Journal of Mathematics and Applications 9/1 (March 1, 2026): 1-18. https://doi.org/10.32323/ujma.1767392.

JAMA

1.Lakhdari A, Napoles Valdes JE, Meftah B, Budak H. Milne’s Estimates via Conformable Fractional Multiplicative Integrals. Univ. J. Math. Appl. 2026;9:1–18.

MLA

Lakhdari, Abdelghani, et al. “Milne’s Estimates via Conformable Fractional Multiplicative Integrals”. Universal Journal of Mathematics and Applications, vol. 9, no. 1, Mar. 2026, pp. 1-18, doi:10.32323/ujma.1767392.

Vancouver

1.Abdelghani Lakhdari, Juan Eduardo Napoles Valdes, Badreddine Meftah, Hüseyin Budak. Milne’s Estimates via Conformable Fractional Multiplicative Integrals. Univ. J. Math. Appl. 2026 Mar. 1;9(1):1-18. doi:10.32323/ujma.1767392