New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator

Fractal and Fractional

article

Muhammad Tariq¹  Sotiris K. Ntouyas²  Asif Ali Shaikh¹ 

[1] Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology;Department of Mathematics, University of Ioannina

关键词: preinvex functions;    fractional operator;    Hadamard inequality;    Fejér inequality;    Pachpatte inequality26A51;    26A33;    26D07;    26D10;    26D15;   

DOI  :  10.3390/fractalfract7050405

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the ( m , t g s )-type s-preinvex function, Godunova–Levin ( s , m )-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings.

【 授权许可】

CC BY