Fractional Maclaurin Type Inequalities for Functions whose First Derivatives are s-Convex Functions

Abstract

Classical and fractional integral inequalities have become a popular
method and a powerful tool for estimating errors of quadrature formulas. Several
studies on various types of inequality have been conducted and the literature in this area is vast and diverse. The current study intends to investigate one of the open three-point Newton-Cotes formulae, known as Maclaurin’s formula, using Riemann-Liouville fractional operators. To accomplish so, we first created a new identity. 
From this identity and through the s-convexity, we have established some new Maclaurin-type inequalities, we also discussed the cases that can be derived of our finding. Furthermore, various applications for error estimates are offered to demonstrate the efficacy of our primary results.

Key words and phrases. Maclaurin’s formulae, Newton-Cotes quadrature, s-convex functions.

2010 Mathematics Subject Classification. 26D10, 26D15, 26A51