article · Mathematical Problems in Engineering
This research investigates fractional variants of several classical mathematical inequalities by integrating the Mercer concept. Specifically, the study establishes new Hermite, Hadamard, and Mercer type inequalities tailored for harmonically convex functions using fractional integral operators with an exponential kernel. In addition, weighted Hadamard, Fejér, and Mercer type inequalities featuring exponential functions in their kernels are rigorously proved. The investigation concludes by constructing Pachpatte and Mercer type inequalities that apply to the products of harmonically convex functions, also through the lens of fractional integral operators with exponential kernels. The resulting mathematical frameworks extend the theoretical properties of harmonically convex functions within fractional calculus.
Mathematical inequalities are foundational tools in mathematical analysis, optimisation, and engineering modelling. By expanding these inequalities to harmonically convex functions using fractional operators and exponential kernels, this theoretical work broadens the mathematical toolkit available for analysing complex, non-linear functional relationships and systems governed by fractional calculus.
The abstract does not indicate an application pathway or any direct commercial use, representing early-stage theoretical mathematical research.
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In the present study, fractional variants of Hermite–Hadamard, Hermite–Hadamard–Fejér, and Pachpatte inequalities are studied by employing Mercer concept. Firstly, new Hermite–Hadamard–Mercer-type inequalities are presented for harmonically convex functions involving fractional integral operators with exponential kernel. Then, weighted Hadamard–Fejér–Mercer-type inequalities involving exponential function as kernel are proved. Finally, Pachpatte–Mercer-type inequalities for products of harmonically convex functions via fractional integral operators with exponential kernel are constructed.
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DOI: 10.1155/2022/7269033
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