Malaysian Journal of Mathematical Sciences, September 2025, Vol. 19, No. 3
A Study On Perturbed Newton-type Inequalities For Fractional Operator and Application
Munir, A., Qayyum, A., Baskonus, H. M., Alsulami, I. K., and Ghareeb, S.
Corresponding Email: dratherqayyum@um.edu.my
Received date: 10 January 2025
Accepted date: 24 March 2025
Abstract:
Fractional integral inequality is very important in the scientific and engineering disciplines which creates and expands various mathematical methods. In this article, the main aim is to develop the Perturbed Newton-type inequalities by using fractional operator. We derive new identity using the Caputo Fabrizio fractional operator. We use a specific Peano kernel to obtain the Perturbed Newton-type inequalities for the $s$-$\varphi $-convex and $\varphi $-quasi-convex functions. Based on this identity, we established several new error bounds and estimations which are appraised by employing some well-known inequalities Hölder and Power-mean. Furthermore, we introduce the refinement of Simpson's second formula, which is not developed by any researcher yet. We also present an application to some special means.
Keywords: perturbed Newton-type inequalities; $s-\varphi$-convex function; fractional integrals; $\varphi$-quasi-convex function; power-mean inequality
