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Peer-reviewedCC BY 4.0Axioms

On a New Generalized Convexity and Its Related Midpoint-, Trapezoid-, and Simpson-Type Inequalities via Multiplicative Calculus

  • Mohammad Akram1
  • Inna Kalchuk2
  • Muhammad Tariq3,4
  • Waqar Afzal5,6
6 affiliations
  1. 1Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
  2. 2Faculty of Information Technologies and Mathematics, Lesya Ukrainka Volyn National University, 43025 Lutsk, Ukraine
  3. 3Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
  4. 4Mathematics Research Center, Near East University, Near East Boulevard, Mersin 10, 99138 Nicosia, Turkey
  5. 5Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54000, Pakistan
  6. 6Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan
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Abstract

In this paper, we introduce and investigate a refined generalization of convexity, termed the multiplicatively strongly s-convex function, and establish several of its fundamental algebraic properties. Within the framework of multiplicative calculus, this generalized convexity structure is employed to derive new midpoint-, trapezoid-, and Simpson-type inequalities, thereby providing unified refinements of several classical integral inequalities. Particular choices of the underlying parameters yield a variety of meaningful special cases and corresponding novel inequalities. Furthermore, we demonstrate the applicability of the established results through inequalities involving several important special means, including the arithmetic, geometric, logarithmic, and m-logarithmic means. To complement the theoretical developments, selected consequences are illustrated numerically and graphically, providing comchosenational evidence for the validity and effectiveness of the proposed inequalities. The results contribute to the development of multiplicative convexity theory and offer a flexible framework for deriving further inequalities within multiplicative calculus.

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Cite

BibTeXRIS
BibTeX
@article{akram2026generalized,
  title = {{On a New Generalized Convexity and Its Related Midpoint-, Trapezoid-, and Simpson-Type Inequalities via Multiplicative Calculus}},
  author = {Mohammad Akram and Inna Kalchuk and Muhammad Tariq and Waqar Afzal},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100769},
  url = {https://www.mdpi.com/2075-1680/15/10/769},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On a New Generalized Convexity and Its Related Midpoint-, Trapezoid-, and Simpson-Type Inequalities via Multiplicative Calculus
AU  - Mohammad Akram
AU  - Inna Kalchuk
AU  - Muhammad Tariq
AU  - Waqar Afzal
PY  - 2026
DA  - 2026/10/14
JO  - Axioms
DO  - 10.3390/axioms15100769
UR  - https://www.mdpi.com/2075-1680/15/10/769
AB  - In this paper, we introduce and investigate a refined generalization of convexity, termed the multiplicatively strongly s-convex function, and establish several of its fundamental algebraic properties. Within the framework of multiplicative calculus, this generalized convexity structure is employed to derive new midpoint-, trapezoid-, and Simpson-type inequalities, thereby providing unified refinements of several classical integral inequalities. Particular choices of the underlying parameters yield a variety of meaningful special cases and corresponding novel inequalities. Furthermore, we demonstrate the applicability of the established results through inequalities involving several important special means, including the arithmetic, geometric, logarithmic, and m-logarithmic means. To complement the theoretical developments, selected consequences are illustrated numerically and graphically, providing comchosenational evidence for the validity and effectiveness of the proposed inequalities. The results contribute to the development of multiplicative convexity theory and offer a flexible framework for deriving further inequalities within multiplicative calculus.
SN  - 2075-1680
ER  -