Hermite-Hadamard, Fejér and Jensen-type inequalities via $ p $-harmonic convex functions with numerical validation and graphical insights

Faiza Azhar; Muhammad Imran Asjad; Imran Abbas Baloch; Manuel De la Sen; Faiza Azhar; Muhammad Imran Asjad; Imran Abbas Baloch; Manuel De la Sen

doi:10.3934/math.20251323

AIMS Mathematics

2025, Volume 10, Issue 12: 30109-30133. doi: 10.3934/math.20251323

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Hermite-Hadamard, Fejér and Jensen-type inequalities via $ p $-harmonic convex functions with numerical validation and graphical insights

1.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
2.
Jadara Research Center, Jadara University, Irbid 21110, Jordan
3.
Department of Mathematics, Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan
4.
Higher Education Department, Government Graduate College for Boys Gulberg, Lahore 54600, Pakistan
5.
Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, 48940-Leioa (Bizkaia), Spain

Received: 24 October 2025 Revised: 05 December 2025 Accepted: 11 December 2025 Published: 23 December 2025
MSC : 26A15, 26A51, 26D10, 26D15

This paper deals with the generalized version of Hermite-Hadamard and Fejér-type inequalities. Some classical results are extended to a newly introduced category of $ p $-harmonic convex functions for which explicit bounds are also established. Additionally, novel discrete versions of existing inequalities for univariate $ p $-harmonic convex functions over linear spaces are also provided. For validation, the classical results are retrieved by letting $ p\rightarrow1 $, which clearly indicates the accuracy of the achieved results. Finally, tabular and graphical analysis is presented to show the improved results with $ p $-harmonic convex functions over existing literature.
- convex functions,
- harmonic convex functions,
- $ p $-harmonic convex functions,
- Jensen's inequality,
- Fejér type inequality,
- Hermite-Hadamard inequality
Citation: Faiza Azhar, Muhammad Imran Asjad, Imran Abbas Baloch, Manuel De la Sen. Hermite-Hadamard, Fejér and Jensen-type inequalities via $ p $-harmonic convex functions with numerical validation and graphical insights[J]. AIMS Mathematics, 2025, 10(12): 30109-30133. doi: 10.3934/math.20251323

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Abstract

This paper deals with the generalized version of Hermite-Hadamard and Fejér-type inequalities. Some classical results are extended to a newly introduced category of $ p $-harmonic convex functions for which explicit bounds are also established. Additionally, novel discrete versions of existing inequalities for univariate $ p $-harmonic convex functions over linear spaces are also provided. For validation, the classical results are retrieved by letting $ p\rightarrow1 $, which clearly indicates the accuracy of the achieved results. Finally, tabular and graphical analysis is presented to show the improved results with $ p $-harmonic convex functions over existing literature.

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