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.