Some integral inequalities for generalized preinvex functions with applications

Muhammad Tariq; Soubhagya Kumar Sahoo; Fahd Jarad; Bibhakar Kodamasingh; Muhammad Tariq; Soubhagya Kumar Sahoo; Fahd Jarad; Bibhakar Kodamasingh

doi:10.3934/math.2021805

AIMS Mathematics

2021, Volume 6, Issue 12: 13907-13930. doi: 10.3934/math.2021805

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Research article

Some integral inequalities for generalized preinvex functions with applications

1.
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan
2.
Department of Mathematics, Institute of Technical Education and Research, Siksha O Anusandhan University, Bhubaneswar 751030, Odisha, India
3.
Department of Mathematics, Çankaya University 06790, Ankara, Turkey
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 29 July 2021 Accepted: 13 September 2021 Published: 27 September 2021
MSC : 26A33, 26A51, 26D07, 26D10, 26D15

The main objective of this work is to explore and characterize the idea of $ s $-type preinvex function and related inequalities. Some interesting algebraic properties and logical examples are given to support the newly introduced idea. In addition, we attain the novel version of Hermite-Hadamard type inequality utilizing the introduced preinvexity. Furthermore, we establish two new identities, and employing these, we present some refinements of Hermite-Hadamard-type inequality. Some special cases of the presented results for different preinvex functions are deduced as well. Finally, as applications, some new inequalities for the arithmetic, geometric and harmonic means are established. Results obtained in this paper can be viewed as a significant improvement of previously known results. The awe-inspiring concepts and formidable tools of this paper may invigorate and revitalize for additional research in this worthy and absorbing field.
- preinvex function,
- Hölder's inequality,
- Hölder-İşcan inequality,
- improved power-mean integral inequality,
- $ s $-type preinvexity
Citation: Muhammad Tariq, Soubhagya Kumar Sahoo, Fahd Jarad, Bibhakar Kodamasingh. Some integral inequalities for generalized preinvex functions with applications[J]. AIMS Mathematics, 2021, 6(12): 13907-13930. doi: 10.3934/math.2021805

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Abstract

The main objective of this work is to explore and characterize the idea of $ s $-type preinvex function and related inequalities. Some interesting algebraic properties and logical examples are given to support the newly introduced idea. In addition, we attain the novel version of Hermite-Hadamard type inequality utilizing the introduced preinvexity. Furthermore, we establish two new identities, and employing these, we present some refinements of Hermite-Hadamard-type inequality. Some special cases of the presented results for different preinvex functions are deduced as well. Finally, as applications, some new inequalities for the arithmetic, geometric and harmonic means are established. Results obtained in this paper can be viewed as a significant improvement of previously known results. The awe-inspiring concepts and formidable tools of this paper may invigorate and revitalize for additional research in this worthy and absorbing field.

References

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