Fractional Newton-type integral inequalities for the Caputo fractional operator

Abstract

In this paper, we present a set of Newton-type inequalities for n-times differentiable convex functions using the Caputo fractional operator, extending classical results into the fractional calculus domain. Our exploration also includes the derivation of Newton-type inequalities for various classes of functions by employing the Caputo fractional operator, thereby broadening the scope of these inequalities beyond convexity. In addition, we establish several fractional Newton-type inequalities by using bounded functions in conjunction with fractional integrals. Furthermore, we develop specific fractional Newton-type inequalities tailored to Lipschitzian functions. Moreover, the paper emphasizes the significance of fractional calculus in refining classical inequalities and demonstrates how the Caputo fractional operator provides a more generalized framework for addressing problems involving non-integer order differentiation. The inclusion of bounded and Lipschitzian functions introduces additional layers of complexity, allowing for a more comprehensive analysis of function behaviors under fractional operations.